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PROPERTY OP
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P*' J *~
A R T fc S i C I E M r 1 A VEMTAi
• • •
SHORE PROCESSES
AND
SHORELINE DEVELOPMENT
BY
DOUGLAS WILSON JOHNSON
Associate Professor of Physiography, Columuia University
FIRST EDITION
NEW YORK
JOHN WILEY & SONS, Inc.
London: CHAPMAN & HALL, Limited
1910
Copyright, 1919, by DOUGLAS W. JOHNSON
Copyrighted in Great Britain
Stanbope $rcss
H.G1LSON COMPANY BOSTON, U.S.A.
PKEFACE
The present work was born of a need experienced by the author in connection with his shoreline studies. In the course of a critical examination of the arguments supposed by many to demonstrate a progressive subsidence of the Atlantic coast of North America within historic time, it developed that in respect to certain of these arguments agreement between students of the problem could not be reached because there was not sufficient agreement as to what features are normally characteristic of a stable coast, and what features are peculiar to coasts which are rising or subsiding. No work existed which combined with an extended analysis of the forces operating along the shore, a full and systematic discussion of the cycle of shoreline development and such further discussion of the modifying effects of changes of level as would enable one to differentiate stable, rising, and subsiding coasts.
It seemed necessary, therefore, to enquire somewhat fully into the fundamental principles of shore processes and shoreline de- velopment; for it would not be profitable to add to the already overburdened literature on changes of level another essay which should merely add quantitatively to the volume of evidence pre- viously discussed by many earlier writers, and again assert as the correct interpretation of that evidence conclusions which some geologists and geographers accept and others reject. Profit could come from the study only in case the discussion of principles was such as to bring geologists and geographers into substantial agree- ment as to what shore features are, and what are not indicative of changes of level. Once this measure of agreement was reached, I could not doubt that a critical analysis of the arguments supposed to prove the progressive subsidence within historic time of the coast of southeastern Canada, the Atlantic coast of the United States, and certain other marginal areas of the continents, would demonstrate to the impartial and critical student the inadequacy of those arguments. I therefore set myself the task of bringing together the results of shoreline studies published in different languages, of analyzing and criticising the conclusions reached
• • •
111
iv PREFACE
where this might appear to be profitable, and of presenting a digest of those fundamental principles which should prove to be best established by the independent work of different students and best supported by my own field observations. At the srme time I purposed to develop somewhat fully certain important aspects of the physiography of shorelines which have hitherto received little consideration.
The magnitude of the task proved to be greater than antici- pated, partly because of the wide divergence of expert opinion regarding the manner in which shore processes operate, and partly because of the great volume and scattered distribution of the writings dealing with the subject. It was, indeed, the desire to relieve others who might have occasion to study shore processes and shoreline forms, of the burden of duplicating the work involved in my undertaking which first suggested to me the desirability of placing on record, in compact form for their use, the results of my enquiry, even where these results did not relate to the original problem of coastal subsidence. The present volume is the con- crete product of this desire to render a service to my fellow students.
The engineer will find in the chapters on waves and currents a summary of the widely conflicting opinions and observations re- lating to those most puzzling forces with which he has to deal. In the later chapters he will also find, I hope, not a few discussions of shore forms and of the method of their development which will prove useful to him in his work on marine engineering structures. The dynamic geologist will find in the first chapters an extended account of two of the forces of nature with which he is much con- cerned, and in the remaining chapters abundant illustrations of the manner in which those forces operate near the margins of the lands. The geographer will be mainly concerned with the last seven chapters where the forms of the shoreline receive a syste- matic treatment which, if not adequate, is at least somewhat more detailed and complete than any hitherto attempted. Throughout the volume the reader will note that repeatedly conclusions reached and principles established are briefly applied to the prob- lem of changes of level; and he will understand that this is the thread, appearing now and then, which is to connect parts of the present study with a later volume devoted exclusively to the much mooted question of coastal subsidence.
PREFACE V
As a rule an advance summary precedes, and a brief risumi con- eludes the text of each chapter. This will enable engineer, geolo- gist and geographer to determine in some measure the extent to which matters pertinent to their respective fields are discussed. A bibliography, arranged alphabetically according to authors and placed at the end of the volume, supplements the list of references given at the close of each chapter. Finally, an index of authors and an index of subjects are provided in the form which it is hoped will prove most serviceable to the reader.
In any attempt to give proper credit for the aid rendered by others during the preparation of this volume the writer is much embarrassed. The work of preparation has extended over several years, during which time a number of students, colleagues and friends have been most generous in rendering valuable assistance. It would be impossible to make specific acknowledgments to all of them, so great is the measure of my indebtedness. Special thanks are due to my cousin, Miss Laura Dale Johnson, for assuming the labor of reading the proofs and seeing the book through the press during my absence; to Miss Florrie Holzwasser of the De- partment of Geology of Barnard College, and others among my graduate studerfts, for assistance in reviewing and abstracting the literature relating to the subject in hand; and to Dr. A. K. Lobeck for preparing the five block diagrams showing successive stages in the development of a shoreline of submergence. Acknowledg- ments should be made to "The Geographical Review," "Science/' the "Bulletin of the Geological Society of America," and the "Journal of Geology" for the use of certain material originally published in their pages. Many of the observations recorded in this volume were made in the course of a special Shaler Memorial Investigation of the problem of coastal subsidence undertaken with the support of the Shaler Memorial Fund of Harvard Uni- versity; and observations on the New Jersey coast were obtained in connection with a study in progress for the Geological Survey of New Jersey under the direction of Dr. H. B. Kiimmel. It is a pleasure to express special obligations to Professor W. M. Davis for helpful criticism of the manuscript, and to acknowledge the debt which all physiographers owe to his studies of shoreline topography, which were the first to demonstrate the value of applying the idea of the cycle to the history of shore forms. To Professor Joseph Barrell, whose studies have to some extent paral-
vi PREFACE
leled certain of my own, I am indebted for many valuable sugges- tions and for his generous courtesy in giving to the manuscript a careful and critical reading, from the results of which I have greatly profited.
In conclusion it is but fair to acknowledge the author's keen appreciation of certain defects which the reader may discover in his perusal of the text. The volume goes to press under circum- stances which absolutely prevent that careful attention to details which every work of this kind should receive. On entering the service of his country the writer was forced to choose between publishing his studies without the final supervision which he had hoped to give the proofs, and postponing publication indefinitely. In view of the uncertainties attending service in the zones of mili- tary operations, it has seemed wiser to allow the work to go to press, in the hope that the indulgent reader will not find the value of the volume materially affected by such errors in execution as the presence of the author alone could have prevented.
DOUGLAS WILSON JOHNSON.
On Board Troop Ship, April 3, 1918.
CONTENTS
Pact
List of Plates ix
List of Illustrations xiii
CHAPTER I
Water Waves 1
Advance summary, 1; Scope of subject, 2; Literature, 4; Waves of oscillation, 7; Origin, 7; Wave motion, 8; Wave form, 12; Wave height, 21; Wave length, 27; Wave velocity, 29; Waves of transla- tion, 33; Earthquake and explosion waves, 38; Tidal waves, 41; Standing waves; seiches, 42; Boundary waves, 44; Resume^ 45; References, 46.
CHAPTER II
The Work of Waves 5*5
Advance summary, 55; Wave energy, 55; Nature of wave attack, 57; Wave dynamometer, 62; Measurements of wave energy, 63; Damage by storm waves, 65; Conditions affecting wave energy, 72; Wave refraction, 74; Depth of wave action, 76; Resum6, 83; References, 83.
CHAPTER m
Current Action 87
Advance summary, 87; Types of currents, 88; Wave currents, 90; Tidal currents, 106; Seiche currents, 122; Wind currents, 123; Planetary currents, 128; Pressure currents, 130; Convection cur- rents, 131; Salinity currents, 131; River currents, 136; Reaction currents, 138: Eddy currents, 139; Deflection of currents, 141; Resume^ 148; References, 149.
CHAPTER IV
Terminology and Classification of Shores 159
Advance summary, 159; Terminology of shores, 159; Plains, planes, and peneplanes, 164; Classification of shorelines, 169; I. Shorelines of submergence, 173; II. Shorelines of emergence, 186; III. Neutral shorelines, 187; IV. Compound shorelines, 190; Stages of shoreline development, 192; Resume*, 192; References, 192.
vu
VU1 CONTENTS
CHAPTER V
Page
Development of the Shore Profile 199
Shorelines of submergence, 199; Advance summary, 199; Initial stage, 201; Young stage, 203; Mature stage, 210; Old stage, 224; Validity of the theory of a marine cycle, 228; Correlation of the marine and fluvial cycles, 242; Independence of marine and fluvial cycles, 245; Comparative rapidity of marine and fluvial planation, 249; Probability of marine planation, 253; Interruptions and acci- dents during the marine cycle, 257; Shorelines of emergence, 258; Initial stage, 258; Young stage, 259; Mature and old stages, 262; Neutral shorelines, 262; Compound shorelines, 265; Resume^ 267; References, 268.
CHAPTER VI
Development of the Shoreline 272
A. Shorelines of submergence, 272; Advance summary, 272; Initial stage, 272; Young stage, 275; Stages of development of shore details, 328; Relative importance of different marine forces in the formation of bars, forelands, etc., 333; Mature stage, 339: Old stage, 344; Resume^ 345; References, 345.
«
CHAPTER VH
Development of the Shoreline (Continued) 348
B. Shorelines of emergence, 348; Advance summary, 348; Initial stage, 348; Young stage, 350; Effect of progressive subsidence on lagoon history, 383; Effect of progressive elevation on lagoon his- tory, 386; Offshore bars not an evidence of subsidence, 386; Mature stage, 389; Old stage, 390; Resume*, 392; References, 392.
CHAPTER Vm
Development of the Shoreline (Continued) 395
C. Neutral and compound shorelines, 395; Neutral shorelines, 395; Compound shorelines, 400; Contraposed shorelines, 401; References, 403.
CHAPTER IX
Shore Ridges and Their Significance 404
Advance summary, 404; Origin of beach ridges, 404; Rate of beach ridge formation, 414; Beach ridges as records of changes of level, 439; ReaumS, 453; References, 454.
CHAPTER X
Minor Shore Forms 458
Advance summary, 458; Beach cusps, 458; Low and ball, 486; Ripple marks, 489; Rill marks, 512; Swash marks, 513; Backwash marks, 517; Sand domes, 518; Shore dunes, 519; Resume\ 525; References, 525.
LIST OF PLATES
Pagb I. Storm waves breaking against seawall at Hastings, England. 14 II. Combing wave, showing water completing orbital move- ment although insufficient in quantity to fill the wave form 17
III. Waves breaking against seawall at Scarborough, England. . 19
IV. Storm wave striking face of seawall at Scarborough, England m 59 V. Water forced vertically upward by wave breaking against
seawall at Scarborough, England 61
VI. Marine Cliff at Highland Light, Cape Cod, rapidly retreat- ing under wave attack 64
VII. Shakespeare's Cliff near Dover, England 70
VIII. Wire fence undermined by wave attack and left hanging in
mid-air 73
IX. Cobblestones cast into a high ridge well above sealevel by *
storm waves 78
X. Cobblestones cast upon the beach from deep water through the combined effect of wave action and the buoyancy of
attached seaweeds 92
XI. Surf breaking on the shore of Cape Canaveral, Florida 95
XH. Winthrop Great Head, a wave-cliffed drumlin near Boston,
Massachusetts 98
XIII. Lowestoft Ness on the east coast of England 102
XTV. Giant sand ripples produced by strong ebb tide current in
the Avon River estuary near Windsor, Nova Scotia 110
XV. Salt marsh near Green Harbor, Massachusetts 112
XVI. Hornviken, a small fjord nefcr North Cape, Norway, showing
typical oversteepened walls of a glacial trough 175
XVll. Idde Fjord near Fredriksten, Norway, showing rectangular
pattern characteristic of many fjord coasts 178
XVIII. The Naerd Fjord, Norway, a partially submerged glacial
trough 180
XIX. Lake Loen, Norway, occupying a glacial trough and practi- cally continuous with the upper part of Nord Fjord .... 183 XX. Stromstad Harbor, Sweden, showing characteristics of a
fiard coast 185
XXI. Northwestern coast of France near Fe'camp, showing youth- ful cliff profiles on a mature shoreline of submergence . . . 200 XXII. Same view as Plate XXI, but at low tide, showing marine
cliff, wave-cut bench, and narrow beach at base of cliff. . 202 XXIII. Marine cliff cut in glacial drift near Plymouth on the Mas- sachusetts coast 205
ix
X LIST OF PLATES
Page
XXIV. Marine cliff cut in sand and clays of the coastal plain near
Beaufort, North Carolina 207
XXV. Marine cliff cut in sand dunes on the shore of Cape Cana- veral, Florida 1 209
XXVI. View near Rhuvaal, Islay, off the west coast of Scotland, showing former marine cliff and wave-cut bench, recently
elevated above sealevel 227
XXVII. Elevated marine cliff near Oban, Scotland 233
XXVIII. Marine cliff and wave-cut rock bench on the Pacific coast
south of Cape Flattery, Washington 236
XXIX. Marine peneplane and monadnock near Madura, east coast
of India 239
XXX. Base of monadnock in Plate XXIX, showing effects of
marine erosion 241
XXXI. Monadnock on marine peneplane of the east coast of India . 251 XXXII. Imposing marine cliff at North Cape, Norway 252
XXXIII. Rocky headland and drowned valley of a young shoreline of
submergence, near Clifton, Massachusetts 255
XXXIV. Stack or chimney in front of a young cliff on the coast of
France 277
XXXV. The Dogstone, near Oban, Scotland, a stack in front of a
marine cliff, now elevated 25 feet above sealevel 279
XXXVI. Fingal's Cave on the Island of Staff a, western Scotland 280
XXXVII. Ancient sea cave on an elevated shoreline of western Scot- land, transformed into a stable 282
XXXVIII. Wave-cut arch on the northwest coast of France 284
XXXIX. Third Cliff near Scituatc, Massachusetts, showing small
landslide due to wave erosion of cliff base 286
XL. Bay-head or pocket beach near Rye, New Hampshire 288
XLI. Deltas of cobblestone formed by overwash of storm waves
near Marblehead, Massachusetts 305
XLI I. Tombolo connecting the former Island of Marblehead with
the Massachusetts mainland 314
XLIII. Hanging valleys on the chalk coast of southeastern England, where waves cut back the shore faster than the streams
can deepen their valleys 342
XLIV. Former offshore bar near Wrightsville, North Carolina, rising above the marsh surface back of the present off- shore bar 353
XLV. Surface of a salt marsh near Boston, Massachusetts, which overlies a peat deposit 20 feet deep composed, of remains
of high-tide vegetation 384
XLVI. Contraposed shoreline south of Rye Beach, New Hampshire. 402 XL VII. Ancient beach ridge (center of view) connecting former
island in distance with one in foreground 406
XLVIII. Ancient series of dune ridges on Cape Canaveral, truncated at right angles by a later series in the foreground, on one member of which the man is sitting 415
LIST OF PLATES xi
Paqb
XLIX. Shingle beach ridges of the Dungeness cuspate foreland,
England 421
L. Shingle beach ridges on the island of Rugen, Germany .... 425 LI. Forested dune ridges on the Darss cuspate foreland, Ger- many, showing marked inequality in altitude of crestlines 427 LII. Road-cut through an old dune ridge at the back of the
present shore at Daytona, Florida 432
LIII. Successive curving beach ridges and swales forming lines of
growth on Cape Canaveral 441
LIV. Level surface of Skanor peninsula, southwestern Sweden. 448 LV. Beach cusps of gravel on the shore of Carmel Bay, California 452 LVI. Giant sand cusps on Melbourne Beach, Florida, truncated
J by wave erosion 464
LVII. Cusps on Nantaskct Beach, Massachusetts 468
LVTII. Beach cusps on the west coast of Porto Rico, near Melones
Point 472
LIX. Sandstone slab showing fossil oscillation ripples 490
LX. Current ripples formed by an ebbing tide 492
LXI. Current ripples on the shore of Nantaskct Qcach, Massa- chusetts 493
LXII. Giant current ripples near Annisquam, Massachusetts, showing irregular pattern due to interference of wave and
tidal currents 499
LXI11. Current ripples near Windsor, Nova Scotia 503
LXIV. Current ripples modified by later oblique wave or current
action 506
LXV. Plaster cast of interference ripple mark formed by two
systems of waves crossing nearly at right angles 507
LXVI. Rill marks on a sandy beach 511
LXVII. Plaster cast of swash marks left by four successive waves
on the sandy shore of Lake Erie 514
LXV11I. Cast of backwash marks 515
LXIX. Plaster cast of backwash marks 516
LXX. Shore dunes of the Holland coast near Katwijk 520
LXXI. Shore dunes near Scheveningen, Holland 521
LXXII. Dune of barchane form overwhelming trees on the Province- lands of Cape Cod 522
LXXIII. Shore dunes near Cape Henry, Virginia, migrating inland
over the forest 524
LIST OF ILLUSTRATIONS
Frontispiece. Wave-cut bench and marine cliff bordering Desecheo
Island, Porto Rico, recently elevated above sealevel.
Fio. Paob
1. Diagram showing the motion of water particles in an oscillatory
wave 9
2. Diagram showing the elliptical orbits of water particles in shallow-
water waves, and the decrease in size of orbits with increasing depths 10
3. Diagram showing theoretical form of a cycloidal wave, and the
rapid decrease in size of the orbits (through which the water par- ticles move) with increasing depth 12
4. Theoretical profiles of three trochoidal waves having different sized
orbits (solid-line profile and broken-line profile), or different spacing of orbits (broken-line profile and dotted-line profile). 13
5. Diagram showing how two regular series of waves of different
heights and lengths combine to form an irregular series 26
6. Diagram showing movement of water particles in a wave of trans-
lation 34
7. Diagram showing how oscillatory waves breaking on a subaqueous
terrace produce waves of translation 37
8. Diagram to illustrate the movement of water particles in standing
waves, such as the seiche , 43
9. Boundary wave formed by local air current over liquids of different
densities 44
10. Diagram showing movement of water particles in overlying fresh
water and underlying salt water during the passage of bound- ary waves from left to right 44
11. Stevenson's wave dynamometer 62
12. Diagram to illustrate the process of wave refraction, whereby wave
attack is concentrated on headlands . . 75
13. Section of beach slope showing by dotted lines the so-called zig-zag
path of debris particles during beach drifting and by solid lines
the parabolic paths actually followed 94
14. Parabolic paths of large and small particles of d6bris subject to
beach drifting , 96
15. Parabolic paths followed by d6bris particles impelled by tne com-
bined action of on-shore swells and oblique wind waves 97
16. Diagram to illustrate relation of beach drifting to wind directions .100
17. Parabolic paths of d6bris particles subject to beach drifting 101
18. Elliptical orbit of water particle during passage of the tide wave
over a sloping seabottom 106
19. Course followed by a nearly submerged float under the influence
of tidal currents in New York harbor 116
• • •
xm
XIV LIST OF ILLUSTRATIONS
Fia. Pagb
20. Theoretical course calculated by Parsons for the float whose actual
course is shown in Figure 19 118
21. Eddy currents in the Gulf of Honduras and Mosquito Gulf 140
22. Elements of the shore zones during an early stage of development. 160
23. Elements of the shore zones in an advanced stage of development. 162
24. Embayed coastal plain of Chesapeake Bay region, showing example
of ria shoreline 174
25. A typical section of the fjord coast of Norway, showing angular
pattern attributed to fault-control 177
26. Mississippi Delta 187
27. Delta of the Tiber 188
28. Niger Delta 189
29. Initial stage of a fault shoreline 190
30. Coast of North Carolina, showing one type of compound shoreline . 191
31. Compound shoreline due to faulting and partial submergence of up-
throw block 191
32. Stages in the development of the shore profile of a shoreline of sub-
mergence 201
33. Successive profiles of equilibrium on a retrograding shore 211
34. Profiles of equilibrium off the Madagascar coast as plotted from
charts by Barrell 212
35. Stages in the development of the shore profile of a shoreline of sub-
mergence 213
36. Variations in the beach profile of equilibrium due to variations in
the different shore forces 218
37. Mature and old shore profiles of a shoreline of submergence 224
38. San Clemente Island off the coast of southern California, showing
series of uplifted wave-cut terraces 229
39. Cross-section of the western coast and continental shelf of Norway. 232
40. Stages in the reduction of a land mass 256
41. Overlapping of marine deposits upon the abrasion platform of a
slowly subsiding land mass 258
42. Elements of the profile of a shoreline of emergence 258
43. Stages in the development of the shore profile of a fault coast .... 264
44. Profile of a shoreline of emergence 266
45. Shoreline of submergence, initial stage 273
46. Shoreline of submergence in Chesapeake Bay region 274
47. Early youth of a shoreline of submergence, showing crenulate shore-
line 275
48. Crenulate shoreline of the southwest coast of Ireland 276
49. Young shoreline of submergence near Idzuhara, Japan, showing
crenulate stage 278
50. Young shoreline of submergence, showing types of beaches, bars,
spite, and forelands 283
II. Initial unorganized condition of currents along a young shoreline ol submergence compared with organized condition which ob- tains when the stages of submaturity or maturity are reached. . . 285 52. Sand spits on the shore of Port Orchard, Washington 287
LIST OF ILLUSTRATIONS XV
Fig. Pagb
53. Simple spit and compound recurved spit at entrance to Port
Moller, Alaska . 289
54. Successive stages in the development of one type of compound re-
curved spit 291
55. Compound recurved spit enclosing Toronto Harbor 293
56. Development of the Cape Cod shoreline 294
57. Development of Sandy Hook spit 296
58. Lagoons and ridges of the Presque Isle compound recurved spit.. . 299
59. Spits built by converging currents 300
60. Spits converging to form a bay bar on the Alaskan coast 300
61. Bay-mouth bars on the Marthas Vineyard coast 301
62. Bay-mouth bar on the Marthas Vineyard coast 302
63. Bay-head bar near Duluth 303
64. Mid-bay bar in Hempstead Harbor, Long Island 304
65. Winged headland near Sag Harbor, Long Island 306
66. Offsets, overlaps, and stream deflection 308
67. Looped bar on shore of Shapka Island, Alaska 310
68. Renard Island near Seward, Alaska, showing embankment growing
from island toward mainland 311
69. Inner Iliasik Island, Alaska, showing embankment which may be
upbuilding toward the surface simultaneously along its entire
length 311
70. Single tombolo connecting former island of Marblehead with the
mainland 312
71. Former island of Big Nahant tied to Little Nahant, and the latter
to the mainland by single tombolo ' 313
72. Duxbury and Saquish Neck tombolos uniting former islands with
the mainland of Massachusetts near Plymouth 316
73. Monte Argentario, Italy, tied to the mainland by a double tombolo 317
74. Morro del Puerto Santo, Venezuela, a Y-tombolo 318
75. Former islands, many of which were wholly or completely eroded by
wave action, and the resulting d6bris used by the waves to build
a complex tombolo tying the remaining islands to the mainland. 319
76. Nantasket Beach, Massachusetts, the complex tombolo formed by
wave erosion of the islands shown in Fig. 75, with deposition of the d6bris to give connecting beaches uniting the remaining
islands with each other and with the mainland at the south. 320
77. A strongly recurved spit on the Washington coast, about to become
a cuspate bar 321
78. Cuspate bars on the Narragansett Bay shore 321
79. Cuspate bar, showing enclosed marsh near Providence, Rhode
Island 321
80. Cuspate bar originally built as a tombolo tying to the mainland an
island since removed by wave erosion 321
81. Cuspate bars on the shores of Port Discovery, Washington 323
82. Cuspate foreland near Port Townsend, Washington 324
83. Types of cuspate foreland bars 324
84. The former Cape Canaveral, now known as False Cape 326
XVI LIST OF ILLUSTRATIONS
Fia. Paob
85. Marsh bars on the Delaware Bay shore 327
Comparison of text figures to facilitate correlations of successive
stages in the development of a shoreline of submergence 331
86. Lake Balaton (Platten Lake), showing position of cuspate bar . . . 337
87. Shoreline of submergence, submature stage 340
88. Shoreline of submergence, mature stage 341
89. Valleuses on the northwest coast of France 343
90. Theoretical profile through offshore bar 357
91. Theoretical profile through offshore bar 357
92. Theoretical profile through offshore bar 357
93. Theoretical profile through offshore bar 358
94. Theoretical profile through offshore bar 358
95. Theoretical profile through offshore bar 359
96. Theoretical profile through offshore bar 359
97. Profile through offshore bar 361
98. Profile through offshore bar 361
99. Profile through offshore bar 361
100. Profile through offshore bar 361
101. Profile through offshore bar 361
102. Profile through offshore bar 363
103. Profile through offshore bar 363
104. Profile through offshore bar 363
105. Profile through offshore bar 363
106. Profile through offshore bar 363
107. Profile through offshore bar 363
108. Profile through offshore bar 364
109. Profile through offshore bar 364
110. Profile through offshore bar 364
111. Profile through offshore bar 364
112. Profile through offshore bar 364
113. Profile through offshore bar 364
114. Profile through offshore bar 364
115. Offshore bar and lagoon of the Long Island coast, showing distribu-
tion of tidal inlets 371
16. Offshore bar and lagoon of the New Jersey coast 371
17. Tidal delta at Ocracoke Inlet, North Carolina coast 375
18. Stages in the development and retrogression of an offshore bar. 376
119. Stages in the normal history of an offshore bar, due account being
taken of the effect of migrating inlets 378
120. Diagram showing how wave erosion of a lobate delta may transform
it into an arcuate delta (broken line) 396
121. Fault shoreline bordering a scarp which dies out toward the right 397
122. Similar to Fig. 121, except that the fault traversed a little dissected
plain of faint relief 398
123. Successive stages in the retrogression of a fault shoreline bordering
rocks of varying resistance 399
124. Compound shoreline, combining essential features of a shoreline of
submergence and a fault shoreline 401
LIST OF ILLUSTRATIONS Xvii
Fio. Paob
125. Stages in the formation of a contraposed shoreline 403
126. Cuspatc delta of the Tagliamento River, Italy, showing parallel
beach ridges 410
127. Successive stages in the development of Rockaway sand spit, Long
Island 417
128. Diagram of cliffed headland and associated beach ridge plain, show-
ing that one series of ridges truncating another does not neces- sarily imply a longer lapse of time than an equal number of
parallel ridges 418
129. Ridges of the Cape Canaveral cuspate foreland 420
130. The Dungeness cuspate foreland, showing shingle beach ridges and
swales 423
131. Dune ridges of the Darss cuspate foreland, Germany 429
132. Dune ridges of the Swinemunde tombolo 434
133. Beach ridges indicating coastal emergence 446
134. Beach ridges indicating coastal submergence 447
135. Hypothetical case in which beach ridges on a rising coast may give
a false indication of stability 449
136. Hypothetical case in which beach ridges on a sinking coast give a
false indication of stability 460
137. Types of beach ridges formed on a stable coast 451
138. Beach ridges of equal height separated by swales of different depths
due to variations in spacing of ridges 453
139. Diagram illustrating Branner's theory of beach cusp formation . . . 461
140. Diagram illustrating Branner's theory of the formation of unequally
spaced beach cusps 461
141. Variations in the form of beach cusps 465
142. Partially eroded older cusps and respaced later series 466
143. Normal and inverted beach cusps 473
144. Artificial beach cusps 476
145. Beach cusps (after Jefferson) showing compound cusps at right. . . 484
146. Oscillation ripples 491
147. Current ripples 494
148. Vortices involved in the formation of current ripple mark 497
L49. Sand dome 518
SHORE PROCESSES AND SHORELINE DEVELOPMENT
CHAPTER I WATER WAVES
Advance Summary. — No adequate appreciation of the many problems presented by the shoreline can be gained until one is familiar with the work of waves and currents. The relative importance of these two forces in shaping the shore is a much disputed point; and the difficulties involved can best be set forth, and an attempt at their solution can best be made, if we review the essential characters of waves and currents with some care, and critically examine the manner in which each operates. We will first turn our attention to the phenomena of water waves of different types; then we will be in a position to discuss the work accomplished by such waves; after which currents and their work will be considered.
In this first chapter, after a note on the general scope of the present treatment of waves, there is presented to the reader a brief survey of the literature on the subject, which may be useful in showing the growth of our knowledge of waves since the time of Leonardo da Vinci. Attention is then directed to the two types of waves which are most effective in shore proc- esses: the wave of oscillation and the wave of translation. In each case the origin and nature of the water movement are ex- plained and the elements of wave form are described. The depths at which waves break on approaching a coast determine the position of certain shore forms, and therefore receive con- sideration. The factors affecting the height of waves are of vital interest to the engineer employed on harbor works or coast defenses, and to the student of shore forms produced under varied conditions of wave attack; hence these factors
1
2 WATER WAVES
are discussed with some fullness. A wave's capacity for de- struction depends upon both its height and its length, and the velocities of certain waves vary with wave length according to definite laws. The lengths of waves and their velocities are therefore matters of importance to engineer and geologist, and the natural laws which govern them possess a fascinating in- terest for the laymen interested in one of the most impressive of Nature's destructive forces.
Earthquakes and explosion waves are comparatively rare phe- nomena, but their spectacular character, the popular interest which attaches to them, and the disasters for which they are responsible, entitle them to consideration in any treatise on waves. The great wave known as the "tide " is of importance to the student of shore processes only in connection with the currents which it produces, and is accordingly given scant space in Chapter I. A treatment of tidal currents will be found in Chapter III. Standing waves including the seiche, and the so-called "boundary waves " produced at the contact of liquids having different densities, are of theoretical rather than prac- tical interest in the present connection, and are discussed very briefly.
Scope of Subject. — Water waves may be produced in a variety of ways. The bow of a vessel pushing through the water, or a strong wind blowing over the sea, or a rain-drop falling into it, will each produce waves; but in each case the waves are of essentially different character, and behave according to distinctly different laws. One of the waves generated by a submarine displacement of the earth's crust is similar to the wave pushed out from the bow of a moving ship, but is unlike those produced during a storm at sea. Waves which form when a fine wire is drawn through the water behave very differently from the ship's waves, but are like the ripples set in motion by a falling drop of water. The great wave known as the " tide " is a compound wave, combining some of the characteristics of the two groups of waves first mentioned. When wind-made waves break on a shallow shore they give rise to a new series of waves similar to those produced by the bow of a vessel. Such facts as these are sufficient to show that the subject of waves is an extremely complicated one. Op- portunity for making direct observations of wave motion below the surface of water bodies in nature is very limited, while the
SCOPE OF SUBJECT 3
theoretical treatment of wave motion carries one into the realms of higher mathematics.
It is not within the scope of the present report to enter into a discussion of all the interesting series of water waves known to science. The beautiful and complicated wave pattern developed by a moving ship has little to do with the modelling of shore forms, and the reader who would follow that phase of the sub- ject further is referred to Lord Kelvin's popular lecture " On Ship Waves"1, the second chapter of J. A. Fleming's little volume on " Waves and Ripples "2, and the twelfth chapter of Vaughan Cornish's book entitled " Waves of the Sea and other Water Waves "3, in which latter place will be found exquisite photo- graphic illustrations of ship waves. The phenomena of ripples are treated at some length by Fleming4, and a more technical account is given by J. Scott Russell5. Other interesting forms of water waves are described at length by Russell6 and Vaughan Cornish7. We must confine our discussion to those types of wave motion which have a significant effect upon the shore.
But even if we limit ourselves to a consideration of those waves of practical importance to the engineer and physiographer, our task is by no means an easy one. The literature of the subject is extensive and much of it highly technical in character. Different authorities employ different formulae in deriving some of the elements of wave motion, and the results they obtain agree neither with each other nor with the results obtained by experi- mentation. Airy commends the experimental work of J. Scott Russell as being the best ever done, but warns the reader against accepting that author's theoretical expressions, claiming that his own formulae express the true relations and are verified by Russell's results8. Russell, in turn, demonstrates the inaccu- racy of Airy's formulae, and deplores the fact that the methods of investigation employed by that able authority should not have led him to better conclusions9. Hagen likewise opposes with some vigor certain of the suppositions made by Airy, while the experimental observations of Caligny and Russell disagree on important points. Krummel has well expressed the present con- dition of the subject in the words: " In short analysis, observa- tion and experiment are not yet in the desired agreement "10. Fortunately, a number of the disputed points are not of special importance to the student of shore forms, however much he
4 WATER WAVES
may be interested in the complex but beautiful laws which govern the motions of waves.
Literature. — Some of the principal sources of information upon which I have relied, smdto which the student of waves is referred for elaborate discussions, may briefly be mentioned. Of historical interest are the work of Leonardo da Vinci, who in the latter part of the fifteenth century recognized many of the fundamental principles of wave motion, and advanced theories which are similar to those of modern investigators; and of Newton, who a century later gave us the first exact mathematical treatment of waves. Among more recent works the publications of Franz Gerstner, which appeared in the early years of the nine- teenth century, are especially important. I have not seen the original papers of these authors, but their work is reviewed by the Weber brothers, Oialdi, Wheeler and others, in reports mentioned below.
In 1809 Bremontier's able essay entitled " Recherches sur le Mouvement des Ondes "n was published. This early report of experimental work on the laws of wave action and of observations on wave action in nature, contains the first effective demon- stration of the power of waves to affect the bottom at considerable depths. The important volume of the two Weber brothers on " Wellenlehre auf Experimente gegrundet ' 12, based on elaborate experimental studies and published in 1825, contains a review of practically everything written on waves from the time of Newton up to 1820, and adds much to the sum of previous knowledge on the subject. Six years later Emy's treatise " Du Mouvement des Ondes et des Travaux Hydrauliques Maritimes "l3 refuted Bremontier's conclusion that during wave movement the water particles rose and fell in vertical paths, substituted the more nearly correct opinion that the particles moved in vertical ellipses, and developed at great length the theory that a special type of " bottom wave " (flot de fond) was the principal cause of changes in the forms of the coast and of the destruction of maritime engineering structures. Emy does not appear to have been familiar with the work of the Weber brothers. J. Scott Russell's two reports on " Waves "14, made to the British Asso- ciation in 1837 and 1842-1843, present the results of admirable experimental work made under conditions more favorable than those attending the experiments of the Weber brothers, although
LITERATURE 5
Russell directed his attention principally to the waves of trans- lation. In reading Russell's reports the student must guard against misapprehension arising from the fact that the text references to plate numbers and to the lettering of illustrations are full of errors. The same author's great monograph on " Naval Architecture "15 contains several valuable chapters on waves. In 1865 there were published the results of experiments made during the preceding decade by Bazin and Darcy16 on a much more extensive scale than those performed by Russell.
Airy's elaborate treatise " On Tides and Waves "17 appeared in the Encyclopedia Metropolitana in 1845, and has since been recognized as the standard mathematical discussion of the theory of waves, although the validity of some of his assumptions has been assailed. In spite of its technical character the non- mathematical student will find in it much of value. Two papers by Stokes18 which appeared a few years later and which have since been included in the first volume of his " Mathematical and Physical Papers," are important because of their con- tributions to the theory of oscillating waves. Rankine gave a mathematical analysis of the " Exact Form of Waves near the Surface of Deep Water "19 in 1863. Fourteen years later Bous- sinesq produced his exhaustive treatise entitled " Essai sur la Th^orie des Eaux Courantes ,,2° which includes an extended mathematical discussion of waves. Bertin's long " Etude sur la Houle et le Roulis "21 and still more elaborate " Donn^es Th6oriques et Exp6rimentales sur les Vagues et le Roulis "M appeared in sections during the decade 1869-1879 in the M6moires de la Soci6t£ Nationale des Sciences Naturelles de Cherbourg, a publication which in the same period carried articles on the same or related subjects by de Saint- Venant23, Mottez24 and others. All of these papers except the last mentioned are mathematical in character, but contain matter of importance for the non- mathematical student of wave action, the later sections of Bertin's second memoir including the results of experiments made by himself and Caligny upon the effects of waves breaking on sloping beaches, either with or without the disturbing effects of seawalls.
In 1866 Cialdi published his important book " Sul Moto Ondoso del Mare e su le Correnti di esso"25, in which he reviews the works of many previous writers, particularly those of Italian
6 WATER WAVES
authors, and discusses wave action from the standpoint of the engineer. Caligny's important work on " Oscillations de l'Eau,"26 published in 1883, includes the results of valuable experimental work on waves, particular interest attaching to his contributions to our knowledge of waves of translation. Stevenson's treatise on " The Design and Construction of Harbours ',27 contains a large number of facts which have materially increased our familiarity with the mechanical work of waves, and from the engineering point of view is one of the best published treatises on wave action. A little book on " Waves and Ripples in Water, Air and Aether ,,2S by Fleming, although representing a course of lectures given before a juvenile audience, presents in simple form many laws of wave motion which will interest the older reader. Wheeler's " Practical Manual of Tides and Waves "29 reviews a few of the important works on waves, and discusses the principles of wave action at some length. A large number of interesting facts concerning the behavior of waves will also be found in the same author's volume on "The Sea Coast"30. Vaughan Cornish's beautifully illustrated book entitled " Waves of the Sea and other Water Waves "31 does not consider the principles of wave motion very fully, but presents a wealth of facts concerning the height, length, and other elements of waves, and discusses the action of waves on shore detritus.
The best general review of the principles of wave action which has come to my notice is to be found in the second volume of Krummel's " Handbuch der Ozeanographie "32. Gaillard's trea- tise on " Wave Action in Relation to Engineering Structures ,,3S contains a fairly extended review of the most important work of previous writers and discusses the results of the author's own excellent researches. The book loses part of its value as a ref- erence work because many of the quotations from the works of previous writers are unaccompanied by such citations of the original sources as would enable the reader to find them. White's " Manual of Naval Architecture ',34 has a valuable chapter on deep sea waves. The numerous papers by Vaughan Cornish, published in the London Geographical Journal and elsewhere, contain many interesting facts not stated in his book above mentioned; and the volumes of the "Proceedings of the In- stitution of Civil Engineers " (London) include a number of extended articles on the action of waves and currents upon
ORIGIN 7
shore debris, which together with the voluminous discussions appended, present various facts and theories of interest to the student of wave action. Many other sources on which I have drawn are mentioned in the pages which follow.
WAVES OF OSCILLATION
Origin. — The waves produced by the action of the wind are the most important type of sea waves. When wind acts upon a water surface it subjects it to irregular, unequal pressure because winds never blow with constant velocity, but always in irregular gusts. Unequal pressures deform the water surface, giving it an undulatory form. The wind can then act directly upon the undulations, pressing strongly against the sides of the elevations, but acting less effectively against the partially protected de- pressions. The water in the elevations is moved forward, both by direct pressure and by friction with the passing air. This action causes the undulations to advance and to increase in size until the limit of wave height for the given wind velocity is reached, providing the breadth and depth of the water body are sufficiently great.
If one watches the surface of a pond when a faint breeze first springs up, he will note that the once glassy surface suddenly becomes covered with tiny ripples, which disappear almost as suddenly if the breeze dies down. But if the breeze continues, it will be seen that these miniature waves increase in size pro- gressively toward the leeward side of the pond, those on the windward side remaining the original size. If the breeze now ceases suddenly, the tiny ripples on the windward side quickly vanish, but the larger waves developed where the wind blew across a greater expanse of water continue to agitate the surface of the pond for some time. It can be shown that the wind has produced two distinct types of waves. The tiny ripples belong to the class known as capillary waves, are like the ripples pro- duced by a falling raindrop or a fine wire moved through the water, are due to surface tension rather than to gravity, and move the more rapidly the smaller the wave length. On the other hand, the larger waves on the leeward side of the pond belong to the class usually denominated by the term "waves of os- cillation," are due entirely to gravity, move the more rapidly
8 WATER WAVES
the greater the wave length, and very large examples in the ocean may travel for hours or days without any sensible loss of energy due to viscosity. There is a certain length of wave, therefore, on the border line between large ripples and small waves of oscillation, which has the slowest rate of motion. Progressively shorter waves travel with increasing velocities and belong to the class of ripples. Those of progressively greater length also travel with increasing velocities, but belong to the class of true waves of oscillation36. A good brief summary of the principal points in the theory of oscillatory waves will be found in a paper published by Lyman in 186838.
Wave Motion. — In all types of waves, the wave form moves far over the surface of the water while the individual water particles move but a comparatively short distance; just as " waves " may be seen to sweep across a wheat-field with every gust of wind, although the individual stalks of wheat merely bend slightly and then return to the'r original positions. The contrast between wave movement and water movement is strikingly exhibited when waves advance up an estuary during the ebbing of the tide. In typical waves of oscillation in deep water each water particle moves through a circular orbit, the particle •moving forward on the crest of the wave, downward on the back, backward in the trough, and upward on the wave front. The relation of the orbital paths of the water particles to the direction of wave propagation is shown in Figure 1. It is important to note with care both the direction of orbital motion, ahd the part of the orbit in which a water particle has a given direction, as these points frequently are incorrectly represented. For ex- ample, one of our best known college texts on " Physiography " contains a figure illustrating wave motion which erroneously shows the direction of orbital movement at the crest of the wave as opposite to the direction of wave propagation, while the black dots representing the water particles are "in the wrong positions in all of the orbits except those showing the particle at the top of wave crest and bottom of wave trough.
A cork or piece of seaweed floating on the water, and moving with the water particles, may be seen to describe a circular orbit when a wave form passes under it. The cork is at the top of its orbit as the crest of a wave passes, reaches the bottom as the trough passes, and attains the top. when the next crest
WAVE MOTtON
arrives. Thus the time re- quired for the cork to move through its orbit is precisely that required for the crest of the wave to advance a dis- tance equal to one complete wave length, i.e., the distance from the crest of one wave to the crest of the next. Now in a wave 20 feet high, having a length of 1000 feet or more, it is evident that the water particle travels through its circular orbit a distance of but little more than 60 feet while the wave form travels a fifth of a mile. As we shall see in a later paragraph, the velocity of waves is often so great that the ocean would be unnavigablc were it not for the fortunate fact that the water does not travel with the wave form.
Although emphasis is prop- erly laid upon the fact that the particles of water move in a limited orbit while the wave form progresses, the common statement that in the open sea the water parti- cles have no progressive mo- tion is not quite accurate. In 1847 Stokes demonstrated from the mathematical stand- point that " the particles, in addition to their motion of oscillation, will have a pro- gressive motion in the direc- tion of propagation of the waves "", the forward motion of the
10
WATER WAVES
particles being not altogether compensated by their backward motion. According to Stokes this progressive motion, in deep water at least, decreases rapidly as the depth of the particle considered increases. Cialdi later discussed this progressive mo- tion of the water particles at much length, and sought to ex- plain it as in part a consequence of the increase in density of the particles brought about by the cooling due to evaporation and radiation at the crests of the waves38. It is certain that the wind by pressing more upon the posterior parts of the waves than upon the anterior parts, gives a distinct progressive motion to the water involved in oscillatory waves, and that this motion is greatest at the surface, decreasing with depth. Stokes has developed a formula for calculating the extent to which a ship may be drifted from her course by the progressive motion of the water particles in waves of this class, although he does not regard the formula as of practical importance39.
In water of limited depth the water particles move round and round in ellipses whose major axes are horizontal, (Fig. 2),
Fig. 2. — Diagram showing the elliptical orbits of water particles in shallow- water waves, and the decrease in size of orbits with increasing depths. (After Krummel.)
and at the bottom the ellipses are reduced to straight lines, the water particles simply moving forward and backward40. In somewhat deeper water the particles near the surface will move in circles, those farther down in ellipses, and those on the bottom in straight lines. It is this back-and-forth movement on the bottom which Emy41 was considering when he proposed his theory of " ground waves " or " bottom waves " (flots de fond), although he apparently included in addition certain phenomena of waves of translation. This theory was assigned an undue importance, and was greatly elaborated by Cialdi42 and Cor- naglia43, and by others of the Italian school whose works discuss the " flutto di fondo " at much length. The latter author lays
WAVE MOTION 11
much stress on the existence of a u neutral line " where the land- ward and seaward components of the groundwave are supposed to be exactly balanced; and considered that inside this line the motion of debris is landward, while outside it is seaward. Thou- let44 applies the term " lames de fond " to waves of an entirely different type, — waves originating from seismic disturbances, the discussion of which will be taken up on a later page. On a level sea-bottom covered by a limited depth of water, it is evident that oscillatory waves would cause sand to shift back and forth, but would give to it no progressive motion, were there no progressive motion of the water particles themselves. If we admit the existence of the progressive motion discussed in the preceding paragraph as characteristic of normal waves of oscillation, it would seem to follow that this motion will still obtain when the orbits are reduced to straight lines, and that we should therefore expect, in the absence of opposing forces, a slow but progressive transfer of sand in the direction of wave advance. Caligny investigated a series of waves formed by raising and lowering a cylinder in the end of a wooden trough, and found that the water particles moved in elliptical orbits which had their greatest diameters vertical instead of horizontal. It is possible that the orbital motion of this type of wave is responsible for those illustrations of sea waves appearing in certain text-books of physical geography, in which the orbital paths are shown as ellipses, with major axes vertical. But according to Caligny44 these waves are peculiar in several respects: they belong to the class of waves of translation, although they have an oscillatory movement; and experiments showed that grains of sand and other material were slowly transported along the bottom of the trough in a direction opposite to that of the wave propagation. It would seem inadmissible to compare these waves with those formed by the wind in the open ocean. Bremontier46 supposed that in normal wave motion the water particles rose and fell in vertical paths, while Emy47 presents arguments to show that the paths must be ellipses with the major axes vertical. In both cases the arguments are evidently unsound, and the con- clusions opposed by the results of more modern studies of deep- sea waves. In short, I have not found a satisfactory basis for those illustrations of deep-sea waves showing elliptical orbits with major axes vertical.
12
WATER WAVES
As will readily appear from Figures 2 and 3 the size of the orbits through which the water particles move decreases rapidly with increase in depth. At the depth of one wave length below the surface, the water particles of an oscillatory wave are moving in orbits whose diameters are only ^^ as great as the diameter of the orbits at the surface48. We may express this relation in
~~o
21
+
Fig. 3. — Diagram showing theoretical form of a cycloidal wave, and the rapid decrease in size of the orbits (through which the water particles move) with increasing depth.
the following rule49: For each additional I of the wave length below the mid-height of the surface wave, the diameter of the orbit is decreased by £. Thuis:
Depth below mid-height of surface wave in frac- tions of wave length 0, J, j, I, J, etc.
Proportionate diameter of orbit 1, J, J, J, A> etc.
For the diameter of an orbit situated one wave length below the surface, the rule would give a value of yyj oi the surface orbit, which is approximately correct and is the figure quoted by Cornish60 and others. If the sea is disturbed by waves having a height of 20 feet and a length of 400 feet, the water parti- cles at the surface move in circles having a diameter of 20 feet, while the particles at a depth of 400 feet move in circles only *fo of an inch in diameter. The importance of this principle will appear when we come to consider the depths at which waves may erode the sea-bottom and transport material.
Wave Form. — The theoretical form of oscillatory waves in the open se$ is indicated by Figure 4 which represents the profiles
WAVE FORM 13
of three such waves. The profiles are trochoidal curves61, or the curves which would be described by points within a circle which is rolled along the under side of a straight line. In the figures this curve is produced by drawing a series of circular orbital
Fig. 4. — Theoretical profiles of three trochoidal waves having different sized orbits (solid-line profile and broken-line profile), or different spacing of orbits (broken-line profile and dotted-line profile). Modified after Grabau.
paths, indicating the proper position of the water particle in each orbit, and connecting these positions by a curved line. As will appear from the figures, the sharpness of the wave crests varies according as the series of orbits having water particles in the same given positions, is closely or widely spaced. From the mathematical standpoint, the curve will be sharp crested or not according as the point within the rolling circle is at or near the circumference, or near its center. If at the circumference, the curve developed will be the very sharp crested form called the cycloid (Fig. 3). This is the shortest and steepest form which a true wave theoretically can have62. As a matter of fact no wave approximating the form of the common cycloid can be produced in nature, as Gaillard has shown63. In the steepest deep-sea waves observed the ratio of height to length is only about one-half that demanded by the cycloidal wave form64. It is doubtful whether the precise form of the flatter trochoid is ever achieved, for it can be shown that the trochoidal theory of waves does not adequately satisfy all the conditions of wave formation66. Nevertheless, the deviation of deep-water waves from the true trochoidal form is so slight, and the trochoidal theory, especially as modified by Stokes66, is so superior to all other theories of wave formation, that we shall not go far wrong if we consider such waves as having the form of the trochoid and call them trochoidal waves.
In any trochoidal wave the crest is steeper and narrower than the trough and contains an insufficient amount of water to fill the trough. The level of the water during calm is there- fore lower than the level of the centers of the orbits which the
WATER WAVES
WAVE FORM 15
surface water particles describe during wave action. In other words, half the height of the waves does not give the true sea level, that level being somewhat lower. Stevenson gives a for- mula prepared by Rankine, for calculating the position of mean sea level when height and length of wave are known; he also observes that large waves in Wick Bay had about two-thirds of their height above still-water level, and one-third below57. On the basis of extensive observations Gaillard has devised more satisfactory formulae for determining the still-water level, taking due account of the fact that a larger percentage of wave height is above still-water level in shallow water than would be indicated by a formula which, like Stevenson's, is applicable to deep-water waves. Gaillard found that in shallow water about three-quarters of the wave height is above still-water level just before the wave breaks58. The importance of this fact will be apparent when it is remembered that the effective salt-water level of the sea may thus be raised a number of feet above high tide level, and also that floating logs or blocks of ice may accom- plish considerable work to any height reached by the crest of the waves.
When a strong wind is blowing, the trochoidal profile of the waves is seen to be materially altered. If the wind is in the direction of wave propagation, as is more commonly the case in the open sea, the forward motion of the water particles on the wave crest is accelerated, while the backward motion in the trough is retarded. Since the troughs are somewhat protected from the wind, the retardation is less effective than the accelera- tion of the wave crests. The net result is a steepening of the front of the wave, so that the profile becomes noticeably asym- metrical. Winds of sufficient velocity may even force some of the water on the wave crest out of its orbital path, blowing it forward into the adjacent trough in the form of foam and spray. When asymmetrical waves pass out of the region of the storm winds which generated them, they decrease in height, become more rounded and symmetrical, and closely approach the tro- choidal form, although the steeper front has been observed on deep-water waves in calm weather59. These waves may be propagated hundreds or thousands of miles from the storm center where they originated, and ultimately become the gentle undu- lations known as the " swell," or " ground swell."
16 WATER WAVES
Surf. — A very important alteration of form occurs when the oscillatory wave passes into shallow water. The wave becomes higher and shorter, the front steepens, the crest arches forward and, finding itself unsupported by sufficient water on the front of the wave, dashes downward with a roar, producing the phenom- enon known as the " surf." An individual breaking wave is known as a " breaker," or less frequently asa" combing wave "; the latter term is also applied to a deep-water wave whose crest is pushed over forward by a strong wind. The commonly accepted explanation of surf is that the wave is retarded by friction when it enters shallow water, the lower part " dragging " on the bottom while the upper part advances unimpeded, until the wave becomes so steep in front that it falls forward. There seem to be fatal objections to this theory of surf action. In the first place the amount of friction necessary to produce the observed result does not seem to exist. Experimental studies of waves in shallow water of uniform depth under conditions favorable for the development of frictional retardation fail to show it60. On the other hand, it will later be shown that wave velocity decreases with decreasing depth. It is equally certain that the size of the orbital paths increases as waves enter shallower water, while at the same time the volume of water is decreasing. With constantly enlarging orbits and diminishing water supply, there must come a time when the volume of water is insufficient to build up the entire wave form, the deficiency manifesting itself asa" hollowing " of the front of the wave. The water available endeavors to curve around through the entire orbit, but on reach- ing the top of the circle finds itself unsupported and collapses.
The form of a breaking wave is not that which should exist if friction were the principal cause of the surf. If the observer can secure a position where the wave profile is discernible, he will find that there is a steepening of the wave front, to be sure, but the form does not suggest a steepening due to forward in- clination of the whole wave mass resulting from " bottom drag," so much as it does a steepening due to the absence of water on, and consequent hollowing of the front side of the wave. When the wave finally breaks, masses of foam floating on the water surface appear to describe an orbit that is more symmetrical than one should expect in a wave deformed by great bottom friction, while the forward arching crest tries to complete a
WAVE FORM
18 WATER WAVES
wave form which, if achieved, would not show excessive steepen- ing on the front. (Plate II.) The credit for first stating the above explanation of surf action belongs to Hagen61.
Depth at Which Waves Break. — The depth of water in which the oscillatory wave assumes the form of a breaker is a mat- ter of some interest. As in the case of the wave of translation, described below, Russell62 found that breaking occurred when the depth of the water equalled the height of the wave, a rule not wholly confirmed by the experiments of Bazin63, who found that breaking occurred more frequently when the height of the wave exceeded two-thirds of the total depth. Russell states that his rule also holds good for oscillatory waves, but unfortunately he is neither clear nor consistent in his method of calculating wave height and water depth in the case of these waves. In one place we read that " every wave broke exactly when its height above the antecedent hollow was equal to the depth of the water," the method of calculating water depth not being stated; on another page both wave height and water depth are apparently mea- sured from mean water level; according to a third statement the author never saw a wave as much as 10 feet high in 10 feet of water, nor 20 feet high in 20 feet of water, although he has seen waves approach very nearly to those limits64. Cornish expresses the rule as follows: waves break when the depth of water reck- oned from the undisturbed sealevel is equal to the height of the crest above the trough65. In other words, a wave entering shallowing water increases in height as the water decreases in depth until the height of the wave above the trough, and the mean water depth reach approximate equality, when the wave breaks. According to this rule the navigator who sees waves 8 or 9 feet high (or about G feet above still-water level) breaking over a certain submarine bar, may know that he can count on but 8 or 9 feet of mean water depth, or G feet of depth below the trough, at the place in question. Some other factor or factors, however, combine with water depth to determine the breaking of a wave, with the result that the above rule does not always hold. De- partures from the rule are noted by Stevenson66. Cialdi67 cites a great number of cases in which waves have been known to break in water many times deeper than the wave height, and both Thoulet68 and Krummel69 have placed some of these in tabular form. The latter author suggests that the frequent
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20 WATER WAVES
breaking of waves in deep water just above the outer edge of a submarine terrace may be due to an upward push imparted to the lower water when it comes against the terrace face, this push being transmitted to the surface and causing the waves to break70. Gaillard found that while oscillatory waves some- times break quite uniformly when the true height of the wave equals the depth of the water measured from still-water level, in other cases they break when the ratio of water depth to wave height is from 1.16 to 2.71. He observed that the depth at which breaking occurs varies with variations in wind velocity, slope of bottom, smoothness of bottom, and wave length; and suggests that the strength of the undertow is probably another important factor in determining the depth at which waves break. In addition to his own observation Gaillard quotes those of many other observers71. The depth of breaking is of importance in determining the position of barrier beaches and other related shore forms.
Intersecting Waves. — Thus far we have considered the form of waves from the standpoint of changes in profile. If now we turn to their variations in form along the crest line, we have first to note that the typical oscillatory wave can not be traced far in the direction indicated. The'crest soon descends at either end and is lost in the maze of other waves. In the open sea one experiences the greatest difficulty in determining the end limits of a given crest, and also in following the progressing crest for any length of time. The reason for this is found in the fact that more than one set of waves are always disturbing the ocean surface, and the several sets intersect each other at various angles. Even with two intersecting series it is evident that the water will rise very high where crest coincides with crest, will fall very low where trough coincides with trough, and will have all in- termediate elevations where different parts of the front and back of one wave intersect different parts of another wave. Imagine several series of waves crossing each other at distinctly different angles, and we have an adequate explanation for all the great irregularity in wave form observed in the open ocean. Only when the observer is stationed high above the tossing waters, and then only under favorable conditions, can he distinguish the several orderly systems of waves which are responsible for the apparent chaos.
WAVE HEIGHT 21
But even in a single wave system the crests are not of in- definite extent. This is because the wind which causes the waves is never of uniform strength, and because the large waves result in part from unequal combinations of smaller waves, as shown on a later page. The wind comes in gusts of varying strength and somewhat varying direction, and so irregular a force could not produce a regular wave crest stretching far over the ocean. Instead we have a large number of short, nearly parallel, over- lapping crests which in course of time combine into a smaller number of larger but decidedly irregular waves. Even in the region of the trade winds, where the winds blow with an un- usual degree of regularity, " the open sea does not present a series of parallel ridges, each one of uniform height, with a lat- eral extension many times greater than the distance from crest to crest "72. On the contrary, there is no evidence of any contin- uous approximation toward regularity.
Wave Height. — In discussing the sizes of waves we have to do with two principal elements of wave form : the height measured from the bottom of the trough to the top of the crest; and the length measured from crest to crest, or from trough to trough. The initial height of the oscillatory waves depends on: (1) the strength of the wind, (2) its duration, and (3) the extent of open water over which it blows. A faint breeze sets in motion very small waves which increase in size to a certain limit, but which would never become great billows. In the trade wind belt the maximum height of wave for a certain strength of wind is soon reached, and although the wind may continue steadily for days at the given strength, there is no increase in the size of the waves. In a general way, the velocity of the wind in statute miles per hour divided by 2.05 will give the height of the waves in feet73. Thus the average height of waves in a gale blowing 44 statute miles per hour is
44 -h2.05 = 21.5 feet.
It should be noted, however, that in very severe storms the highest waves may not occur when the wind velocity is at a maximum, but are seen to develop as the wind begins to subside. The explanation of this phenomenon is probably to be found in the fact that the excessive force of a violent wind blows off the tops of the waves and casts them into the preceding troughs,
22 WATER WAVES
thereby materially diminishing the wave height. It is possible also that as the storm subsides^the waves, which were com- pelled to remain independent and irregular under the gusty force of the storm wind, gradually combine into a smaller number of larger waves which are little affected by the failing strength of the dying wind74.
Effect of Wind Duration. — Wind duration is another factor in increasing wave height up to the limiting height for a given wind strength. When a breeze springs up, small ripples first appear over the water surface, but gradually develop to larger size with- out any increase in the strength of the breeze. If a large swell is already running in the direction of the wind, a sudden increase in wind velocity results in increased height of waves; but in this case the wind does not have to endure very long to bring about a very remarkable increase in height. Cornish has recorded an increase of 7 feet in the height of waves during a squall lasting 4 minutes, and an increase of 2 feet per minute in the height of waves during another squall75. The precise method by which small wind waves grow to large ones is not wholly understood, but the Weber brothers give the following four causes for wave enlargement: (1) the continuous horizontal pressure of the wind upon the wave crest, thus tending to enlarge the orbital movement of the water particles; (2) the combining of several smaller waves moving in the same direction; (3) the pressure exerted by a large wave upon the next following smaller wave, by which the latter is enlarged; and (4) the crossing of waves proceeding in different directions76. Cornish thus states Another theory of wave en- largement: " The horizontal velocity of the air being greatest at the crest, the downward pressure of the atmosphere is least there. Conversely at the trough, where horizontal velocity, is least, downward pressure is greatest. Hence the trough is pushed farther down and the crest is sucked up "77.
Effect of Length of Fetch. — Of corresponding importance is the effect of " length of fetch " of the wind across open water upon wave height. We have already seen that when a breeze blows across a pond there first appear small ripples over all its surface but that 'these soon increase in size progressively toward the leeward side of the pond. The ripples on the windward side, where the wind has blown across a small expanse of water only, remain small no matter how long or how strong the breeze may
WAVE HEIGHT
23
blow. But those on the lecyvard side, where the fetch of the wind across open water is greater, soon develop into waves of some size because here the waves due to the direct effect of the wind are combined with the waves originating on the opposite side of the pond and propagated by gravity in the direction of the wind. This illustrates on a small scale a matter of much impor- tance in the case of sea waves. Stevenson has shown that for ordinary gales and distances the height of the waves in feet is 1.5 times the square root of the distance in nautical miles which the wind has blown over open water78; or
height = 1.5 Vdistance.
Gaillard observed waves 23 feet high near Duluth with a length of fetch of 259 nautical miles79. This agrees fairly well with the calculated height of 24.1 feet based on the formula. Des Bois prepared a table to show the heights of waves corre- sponding to different wind velocities, based on his observation that a wave 2 meters high corresponded to a wind velocity of 5 meters per second, and the provisional theory that " the square of the velocity of the wind will be proportional to the cube of the height of the wave"; and he found that this table corre- sponded roughly with the results he obtained from a large number of direct measurements80.
For short distances a modification of Stevenson's formula is necessary. The following table is condensed from one given by that author, and shows the appproximate heights of waves as determined by length of fetch, assuming great depth of water and a strong gale of wind.
TABLE SHOWING APPROXIMATE HEIGHTS OF WAVES DUE TO DIFFERENT LENGTHS OF FETCH
|
Nautical |
Height* in feet |
i Nautical |
Heights |
Nautical |
Heights in feet |
|
miles |
miles |
in feet |
miles |
||
|
1 |
3 |
5 |
4.3 |
50 |
10.6 |
|
2 |
3.4 |
10 |
5.6 |
100 |
15 |
|
3 ' |
3.8 |
20 |
7.1 |
200 |
21.2 |
|
4 |
4.1 |
30 40 |
8.4 9.5 |
300 |
26 |
For expanses of open water exceeding 500 or 600 miles in length the height of storm waves does not appear to increase
24 WATER WAVES
according to Stevenson's empirical formula. With a fetch of 3600 miles the waves should reach a height of 90 feet, but so great a height is probably never attained. The reason for this discrepancy is doubtless to be found in the fact that we have no storm winds blowing steadily for a long period in the same di- rection over so great a stretch of water81. The facts that the wind direction may be approximately the same over a long stretch of water, or that it may have a constant direction for several days at a given place, as noted by Redfield and by Stevenson82, are not alone sufficient. The winds must blow with the strength of a strong gale in a constant direction over the entire distance for several days, if the full effect of a 2000 or 3000 mile " fetch " is to be realized, since the waves formed to windward must have time to travel the long distance to leeward and produce the cumulative effect which results in maximum wave height. In our cyclonic storms the greatest distance traversed by heavy winds in a reasonably constant direction and for a period of time sufficient for large waves to develop, probably does not exceed 600 or 700 miles. The " effective fetch," therefore, is much more limited than the absolute distance across open water; and Vaughan Cornish has estimated, from a study of charts illus- trating weather conditions in the North Atlantic Ocean for nine weeks of exceptionally stormy weather, that the greatest effective length of fetch during that period was about 600 nautical miles83. But while waves formed on greater expanses of open water do not reach the heights calculated from the formula given above, they do exceed the altitude of about 37 feet cal- culated for the greatest effective fetch, because they may com- bine with an already existing swell and thereby increase their height.
Recorded Wave Heights. — Observations of the heights of waves are often unreliable, but the approximate height under different conditions has been pretty well established by a number of compe- tent observers. On Lake Superior, waves reach a height of from 20 to 25 feet84; in the Mediterranean Sea, 25 to 30 feet86. Sjjoresby's oft-quoted observations on the North Atlantic give a height of 43 feet for the largest waves86, and Cornish reports waves 43 feet high from the same ocean87. When two great waves intersect, peaks of water may rise momentarily to a height of 50 or even 60 feet88. Although the North Pacific Ocean has a breadth of open deep sea
WAVE HEIGHT 25
much greater than that of the North Atlantic, the waves do not appear to reach any greater height89; but in the Southern Ocean waves attain heights of from 45 to 50 feet90. White refers to trustworthy observations of waves of a single series having heights of 44 to 48 feet, and mentions waves formed by the com- bination of two or more series said to attain from 58 to 65 feet91. Gaillard gives an interesting tabulation of the height, length, and period of ocean waves recorded by a number of different observers, the highest figure for wave height in the table being " greater than 50 feet," in the case of a wave photographed by Capt. Z. L. Tanner of the U. S. Navy92. By means of a barom- eter Abercromby measured waves 46 feet high in the Southern Ocean, and concluded that some waves certainly attain a height of 60 feet93. Airy was of the opinion that under no circum- stances does the height of an unbroken wave exceed 30 or 40 feet94; but against this theoretical opinion we may safely accept the figures of competent observers, and conclude that waves 40 feet high are of fairly frequent occurrence in the open ocean, while heights of 50 feet or more are rare, but not unknown.
When these high storm waves run out of the storm area, they gradually decrease in altitude, and in the form of swells usually do not exceed a height of 15 or 20 feet. By the time they are neanng a distant coast they may have been reduced to heights of a few feet on!y, and so have become almost imperceptible. Entering shallowing water they seem to awaken to new life, crowding closer together and increasing in height until they break. At the time of breaking the wave height may be anywhere from a few feet up to 25 feet or more. If a wave comes in contact with a vertical wall or cliff the base of which reaches down to deep water, the wave is reflected back without breaking. The water next the wall moves up and down through a vertical distance equal to twice the original height of the wave, as does also the water half a wave length from the wall. Similarly, a wave run- ning in a direction parallel to a vertical or steep wall has that portion of the wave next the wall notably increased in height95.
Combined Waves. — Waves which appear to belong to the same series vary greatly in height. The larger figures given above are for individual waves, and in each case the average height for the series to which the waves belonged was much less. Thus a wave 40 feet high may occur in a series of waves having an average height of
26
WATER WAVES
but 20 or 25 feet96. This inequality in wave height is probably due in considerable part to the fact that what appears to be a single series of waves of irregular height is really the combined effect of two or more series of waves moving in the same direction, each series having different but fairly constant height and length. Figure 5 from Cornish's work on "Waves" shows, in the third line, the pro- file of an apparently irregular series of waves (c) resulting from the combination of the two regular series (a and b) shown in the first and second lines. By holding the page with the figure nearly on a level with the eye, but slightly inclined toward the observer, the marked irregularity of the combined series may easily be detected.
Fig. 5. — Diagram showing how two regular series of waves (a and 6) of different heights and lengths combine to form an irregular series (c).
The successive wave heights, in feet, measuring from each crest to the next trough, toward the right, are as follows: 22.50, 37.50, 18.75, 40.00, 27.50. The average wave height for an indefinite length of this irregular series will be 30 feet or pre- cisely the height of the dominant regular series. It is evident, therefore, that an observer might conclude that the sea was disturbed by a single series of waves 600 feet long and 30 feet in average height, and that the real presence of a swell 20 feet high would be undetected. This may explain the fact that during a storm at sea the long swell remains invisible, yet be- comes noticeable as soon as the shorter storm waves die down a little97. Gaillard suggests, however, that the waves first made by a strong wind are of unstable form and cannot travel far without being destroyed and contributing their energy to the more stable waves of nearly perfect trochoidal form, the " swell "; while Taylor is of the opinion that direct wind action causes the
WAVE LENGTH 27
water particles to move in orbits of varying amplitude and velocity, producing a confused sea; but that as soon as the wind ceases the viscosity of the water tends to make the orbits identical, and thus to produce a more uniform system of tro- choidal waves98.
The combination of two or more series of waves moving in the same direction explains the fact that when waves break upon the shore, there is a recurrence, at intervals, of waves of excep- tional height. It should be noted, however, that while the popular idea that every seventh wave is a big one rests upon a basis of fact, the ratio of wave lengths in the combining series is just as likely to make every third, or ninth, or some other wave the largest; or if three sets of waves combine, the large waves may arrive at irregular intervals.
Wave Length. — The total energy of a wave has been shown to vary nearly as the square of the height and as the first power of the length, so that these dimensions may be said to measure the capacity of a wave for destruction". Of these two important elements of wave form we have just considered the height, and may now turn our attention to the length.
The ratio of wave height to wave length is a matter of con- siderable interest. Inasmuch as storm waves usually appear higher and steeper than those in a moderate sea, we should expect this ratio to increase with increasing roughness of the sea. Lieutenant Paris found that in a light sea the ratio of height to length is only 1 to 39, in a rough sea 1 to 21, while in a heavy sea it rises to 1 to 19100. Schott compared the ratios directly with the strength of the wind, and found that with a moderate wind the ratio of height to length was 1 to 33, with a strong wind 1 to 18, and with a storm wind 1 to 17 or even as high as 1 to 13101. On the other hand, White compares the ratios with the lengths of the waves, and shows that as the lengths increase the ratios diminish. Thus he finds from an analysis of 179 published French observations that with a wave length of less than 100 feet, the average ratio of height to length is 1 to 17; with a length of 100-200 feet, the ratio is 1 to 20; with a length of 200-300 feet, 1 to 25; with a length of 300-400 feet, 1 to 27. For greater wave lengths the figures are not wholly in accord with the theory, while in waves from 100 to 400 feet long the very small ratio of 1 to 50 has been observed102. Cornish has com-
28 WATER WAVES
pared the lengths of waves with the expanse of open water over which the wind blows and finds that " the length of the storm- waves is increased when the length of the sheet of water is increased, but more slowly "l03.
The lengths of deep-water waves are quite definitely related to their velocities and to their periods, as will be shown more fully on a later page; but we may note here that the wave length (in feet) is roughly equal to 5 J times the square of the period (in seconds). Thus, if waves pass a given point at the rate of one in every 4 seconds, the wave length must be approximately 82 feet; for
Length = 5| (period)2
= 5* (4)*
= 82 feet.
Recorded Wave Lengths. — The greatest trustworthy measurement of wave length is that recorded by Capt. Mottez of the French Navy, for a wave in the North Atlantic, measuring 2750 feet from crest to crest. In the English Channel Cornish observed waves whose period indicated a length of 2594 feet104. Ross observed a wave in the South Atlantic 1920 feet long105. The greatest length reported by Des Bois is 1640 feet106, while Major Leonard Darwin found the waves of an exceptionally severe storm in the Southern Ocean to be 1200 feet in length107. Some of these high figures are probably due to the combination of two sets of waves in such a manner as to give an abnormally long stretch of low water between two crests, for storm waves in the open sea are not usually more than 600, and very rarely more than 700 feet long. Scoresby found the extreme length of the great storm waves measured by him to be 790 feet108. Officers on the North Atlantic liners regard 600 feet as an enormous wave length, although they agree that larger lengths are to be found in the Southern Ocean, where in one exceptional storm Lieutenant Paris found the greatest aver- age length was 771 feet, with not a few waves over 900 feet, and several surpassing 1312 feet in length109.
There seems to be little doubt, however, that the swell has a length often more than double that of storm waves, and at least one of the figures given above, that of 2594 feet for the length of waves observed by Cornish, refers to the swell. When the swell enters shallow water the velocity and wave length are
WAVE VELOCITY 29
diminished, but the period remains the same. Since the period bears a definite relation to the length of the waves in deep water, it is possible, by counting the number of breakers arriving at the shore in a given time, to determine the lengths of the waves in the open sea. In this manner it has been established that the swell in the open sea must not infrequently have lengths of from 1000 to 2000 feet, and occasionally more110. Now in deep-water waves a great wave length means a great velocity, and some authorities doubt whether short storm waves will lengthen to form the longer swells, since this would mean that the speed of the waves was accelerated after the wind ceased to act upon them. Antoine111, however, believes that just such an acceleration does occur. Others suppose that the waves are propagated by gravity at the same rate of speed given them by the wind, or even that their velocity suffers a slight diminution. Cornish concludes that the longer swells are present during storms, but are obscured by the shorter waves which are then more prominent112. We shall find later that the longer waves, while they agitate the surface less than storm waves, agitate the deeper waters much more, and have an im- portant effect upon the shoreline.
Wave Velocity. — The velocity of oscillatory waves is a mat- ter *of considerable interest in various connections. We have already observed that the wave form travels at a speed very much greater than that of the water particles themselves. Thus, a wave 400 feet long and 15 feet high will have a velocity of about 45 feet per second, while the surface water particles will move round in their orbits at a speed of but 5£ feet per second. For ocean waves of large size the wave velocity is apt to be six or seven times as great as the orbital velocity; but it is im- possible to give any definite rule for the relations of these two elements of wave motion113.
We can correlate the velocity of wave motion with wave length more precisely, however, for in deep water the velocity of the wave depends on its length, and is proportional to the square root of its length114. The velocity of any wave whose length is known may be calculated approximately by very simple for- mulae. Thus, the velocity in miles per hour is equal to the square root of 2\ times the wave length measured in feet115. If it is desired to have the result expressed in feet per second, then
30 WATER WAVES
the velocity in feet per second is equal to 2\ times the square root of the length in feet116. According to the first formula a wave 100 feet long will have a velocity of 15 miles per hour;
for
Velocity = V2\ X length
= \/2\ X 100 = V225 = 15 miles per hour.
According to the second formula the same wave will have a velocity of 22.5 feet per second; for
Velocity = 2\ Vlength
= 21 V 100 = 2\ X 10 = 22.5 feet per second.
If we reduce the 15 miles per hour, derived from the first formula, to feet per second, we get 22 feet per second, whifch agrees fairly well with the .result obtained by the second formula. We may also determine the approximate velocity of a wave in feet per second by the formula:
Velocity = V5| X length
which becomes, in the case of the wave described above,
Velocity = V5J X 100
= 22f feet per second.
As Gaillard has pointed out in commenting on the above formula, the velocity of a deep-water wave is practically the same as that which a body would acquire in falling through a distance equal to 8 per cent of the wave length117.
Because of the relations existing between wave velocity, wave length, and the period of the waves, we may determine the velocity of waves in other ways. Thus the velocity of the wave in knots per hour is roughly equal to three times the period (in seconds)118. Or if we transform the period of the wave into the number of waves per minute (wave-frequency), then the velocity in feet per minute is equal to the wave length multiplied by the frequency. The velocity in miles per hour may be found by dividing the frequency into 1981W. Thus if the wave 100 feet
WAVE VELOCITY 31
in length, considered above, have a period of about 4| seconds, then the velocity in knots per hour is roughly 13£, for
Velocity = 3 X period
= 3X4|
= 13^ knots per hour.
The velocity of this same wave in feet per minute will be 1333; for a period of 4| seconds means a frequency of 13 J (60 -4- 4£ = 13$), wrhence we have the following:
Velocity = wavelength X frequency = 100 X 13£ = 1333 feet per minute.
This agrees roughly with the velocities previously obtained,
since it is equivalent to a speed of 22.2 feet per second. The
velocity as determined from the frequency alone is 14.85 miles
per hour; for
Velocity = 198 -r- frequency
= 198 4- 13*
= 14.85 miles per hour.
In order to determine the velocity of a set of waves by this last method it is only necessary to count the number of times per minute some floating object bobs up and down as the waves pass under it, or to count the waves as they rise against some fixed object. The result is in sufficiently close agreement with the velocity of 15 miles per hour determined by a preceding formula.
The periods of waves are more easily determined than are length or velocity, for which reason it is convenient to have in tabular form the lengths and velocities of deep-water waves corresponding to given periods. The table on the following page, taken from White's " Naval Architecture "l2°, covers all waves of ordinary size.
Velocities of Shallow-water Waves. — In the preceding pages we have discussed the laws controlling the velocities of deep-water waves. Shallow-water waves, or waves whose lengths are great compared to the depth of the water, obey different laws. It is a well-known lact that such waves move less rapidly than deep- water waves, and Gaillard has expressed in tabular form the relative velocities of the two types, assuming equal wave lengths, but varying depths of water for the shallow-water wave, with a minimum depth equal to .05 of the wave length121. The velocities
32
WATER WAVES
LENGTH AND VELOCITY OF DEEP-WATER WAVES
(After While.)
|
«"* • 1 _1 |
T iL # *. |
Speed of advance |
|
|
Period, seconds |
Length, feet |
Feet per second Knots per hour |
|
|
1 |
5.12 |
5.12 |
3.03 |
|
2 |
20.49 |
10.24 |
6.07 |
|
3 |
46.11 |
15.37 |
9.10 |
|
4 |
81.97 |
20.49 |
12.14 |
|
5 |
128.08 |
25.62 |
15.17 |
|
6 |
184.44 |
30.74 |
18.21 |
|
7 |
251.04 |
35.86 |
21.24 |
|
8 |
327.89 |
40.99 |
24.28 |
|
0 |
414.99 |
46.11 |
27.31 |
|
10 |
512.33 |
51.23 |
30.35 |
|
11 |
619.92 |
56.36 |
33.38 |
|
12 |
737.76 |
61.48 |
36.42 |
|
13 |
865.84 |
66.60 |
39.45 |
|
14 |
1004.17 |
71.73 |
42.49 |
|
15 |
1152.74 |
76.85 |
45.52 |
|
16 |
1311.56 |
81.97 |
48.56 |
of shallow-water waves of this type must be calculated by means of a formula less simple than those given for deep-water waves, since the formula must be applicable to varying depths of water Such a formula, and numerous comparisons of the observed velocities of shallow-water waves with the velocities computed by the formula, are given in Gaillard's treatise on " Wave Action "122. When the wave length is more than 1000 times the depth of the water, the velocity depends wholly upon the depth according to Airy, and is proportional to the square root of the depth. The velocity of such a wave is the same as the velocity which a body would acquire by falling through a distance equal to half the depth of the water128. This is the law for the velocity of the wave of translation as determined by Russell124; and it should be noted that Airy is inclined to regard the wave of translation as merely a variety of the wave of oscillation128. It is also interesting to note that while this law is called Airy's law or formula by some, and is named for Russell by others, it was really applied by Lagrange to water waves at least as early as 1788126, and is therefore better known as the Lagrange Formula. The law does not hold good for very shallow depths, according to Caligny127; nor in moving water, according to Moller128.
WAVES OF TRANSLATION 33
The waves generated in the ocean by earthquakes and sub- marine volcanic explosions have lengths which are great in com- parison to the depth of the ocean, and must therefore obey the laws controlling the movements of shallow-water waves. If we determine the velocity of such a wave, therefore, we should be able to secure some idea of the depth of the ocean it traverses. This was first done by Bache, who estimated the mean depth of the North Pacific Ocean (4200 to 4500 meters) from the veloc- ity of a wave produced by the Simoda earthquake in 1854; and later others followed his example in the cases of the Iquique earthquake and the Krakatoa explosion129. The calculations are necessarily inaccurate for various reasons, but are nevertheless of considerable interest.
WAVES OF TRANSLATION
Thus far we have confined our attention to waves of oscil- lation, in which the water particles move forward on the crest and backward in the trough. There is another type of wave which is also of great interest to the student of shorelines, al- though its importance is not always appreciated. This is the " wave of translation," in which the water particles move for- ward as the wave passes, but do not exhibit a compensating backward motion. While not important on the open sea, this type of wave is extensively developed in the shallow waters along all coasts, the waves of oscillation generated in deep water frequently becoming more or less completely transformed into waves of translation as they approach the shore.
Form. — The wave of translation was discovered by Russell, and described at length by him in his reports to the British Asso- ciation130. He showed that when a volume of water was suddenly added to the still water in a canal, or when a portion of the canal water was displaced by suddenly plunging a solid body into it, or when the canal water was pushed into a mound by the shoving motion of a boat or of a plate held vertically, a single prominent wave rolled forward over the canal surface. The entire form of this wave rose above the still-water surface of the canal, and included no trough such as constitutes part of the wave of oscil- lation. A careful examination of the newly discovered wave showed that it differed widely from oscillatory waves in other
34 WATER WAVES
respects, and that the motion of its water particles made the name " wave of translation " appropriate. Let us consider briefly the essential characters of this wave, turning our attention first to the nature of the movements executed by the water particles.
Motion. — Immediately before and immediately after the pass- ing of a wave of translation, the surface of the water and the water particles in depth may be quite still. During the passage of the wave the surface water particles rise and move forward, descending again to the original level, but to an advanced po- sition horizontally, where they come to rest. Thus in Figure 6 the particle a rises, moves forward and descends to the position 6. Water particles below the surface move forwapd the same dis- tance, but their vertical rise diminishes with increase in depth.
Fig. C. — Diagram showing movement of water particles in a wave of trans- lation. (After Russell.)
The paths described by the water particles are semi-ellipses which have their major axes horizontal and equal, and their minor axes progressively shorter as the distance below the sur- face increases, until on the bottom the path becomes a straight line131. It will Ikj seen from the figure that water particles ver- tically above each other, as acey, come to rest in the same relative position farther on at bdfh. There is thus a real and permanent forward translation of the water itself through a short distance, in addition to the forward transmission of the wrave form through a very great distance. The space through which the water particles are moved forward is just large enough to contain the volume of water in the wave above still-water level. Manifestly there are several points connected with the motion of the water particles in waves of translation which will prove of importance when we come to discuss the effect of waves upon shores. The fact that the water particles advance, but do not have a compensating backward motion should result in effective transportation of sand and gravel on shallow sea-bottoms in the direction of wave propagation, unless other forces prevent. It
WAVES OF TRANSLATION 35
is likewise worthy of note that in waves of translation the bottom particles move forward just as far as do the surface particles, whereas we have already seen that in oscillatory waves the move- ment of the water particles dies out rapidly below the surface. Evidently waves of translation may profoundly affect the bottom to great depths, although this conclusion is subject to the quali- fication, subsequently to be discussed, that waves of translation traversing water bodies of great depth as compared to the size of the waves, tend to be transformed into waves of oscillation. We shall see later that only one-half of the energy of an oscil- latory wave is transmitted forward with the wave form, whereas the total energy of a wave of translation is thus transmitted.
Wave Length. — The length of the typical wave of translation is measured from the point where it begins to rise from the still- water level in front to the point where the back slope of the wave again merges with still-water level. These points arc not easily determined with accuracy, but according to Russell the wave length thus measured on artificial waves is " equal to about six times the depth of the fluid below the plane of repose. " The height of the wave above the still-water surface may be equal to the depth of the fluid in repose, but cannot exceed this measure, as the wave breaks whenever the height becomes equal to the depth132. Actual measurements of wave heights and lengths in nature are usually made upon the open sea or in other localities favorable to the formation of waves of oscillation; and while it is possible that some of the figures previously given are really those for waves of translation, no distinction is usually made by the observer, and we lack proper data for the range in size of natural waves of translation.
Velocity. — The velocity of the wave of translation depends upon the depth of the water measured from the crest of the wave, and varies as the square root of the depth. Otherwise expressed, the velocity of the wave is the same as the velocity which a heavy body will acquire by falling freely through a distance equal to half the depth of the fluid below the wave crest133. In the deep ocean such waves should have very high velocities, and doubtless many of the earthquake waves which traverse the ocean with velocities from a few hundred miles to nearly a thousand miles an hour134 are true waves of translation; while the tidal wave which has a velocity of from 480 to 660 miles an hour in depths of
36 WATER WAVES
between 12,000 and 20,000 feet186, is a compound wave having some of the characteristics of the wave of translation. In very shallow water the velocities of waves of translation must of necessity be very low.
Complexities of Waves of Translation. — According to Caligny the waves of translation are not always so simple in character as supposed by Russell. The French engineer made a series of ex- periments which led him to conclude that there are waves of translation in which the water particles describe closed orbits; that these orbits may approximate vertical ellipses, but that the backward movement of the water particles may slightly exceed the forward movement, causing material on the bottom to be transported in a direction opposite to that of wave propaga- tion; and that a solitary wave of translation may pass through a series of oscillatory waves, complicating their form and causing them to break. He also pointed out that it is possible to have a succession of solitary waves of translation which will resemble ordinary waves of oscillation, the spaces between the waves resembling troughs so closely as to mislead the observer136.
The investigation of waves of translation in nature is further complicated by the fact that notwithstanding Hunt's arguments to the contrary137, normal waves of oscillation appear to be grad- ually transformed into waves of translation when they enter water which slowly decreases in depth, and hence all intermediate phases between the two types of waves may be encountered. When the tops of breakers fall forward, the volume of water thus added to the water surface in front produces waves of translation which run on shore, mingling with waves of oscillation. If waves of translation encounter a cliff or steep shore, they may be reflected, the direction of the transport of water particles in the reflected wave being seaward. For these and other reasons which will presently appear, it may be practically impossible to determine the nature of the water movements which are affecting the distribution of sand and gravel along a shelving coast.
Such complications should not, however, make us lose sight of the importance of waves of translation as agents of shoreline changes. Under favorable conditions the operation of these waves may easily be observed.. Thus, when large swells en- counter the seaward margin of a submarine terrace (Fig. 7),
WAVES OF TRANSLATION
they break and form smaller waves of trans- lation which, on a calm day, may cross the shallow water to the shore without deforma- tion until they break as a secondary surf on the beach. The level water surface between any two waves of translation may be seen to differ distinctly from the true trough of the oscillatory wave in deeper water. Russell ob- served a striking example of waves of transla- tion, formed in the manner above described, on the shore of Dublin Bay, and thus de- scribes the phenomena:
" One of the common sea waves, being of the second order {waves of oscillation), approaches the shore, consisting as usual of a negative or hollow part, and of a positive part elevated above the level; .... At length the wave , breaks, and the positive part of the wave falls forward into the negative part, filling up the hollow .... After a wave has first been made to break on the shore, it does not cease to travel, but if the slope be gentle, and the beach shallow and very extended (as it sou times is for a mile inwards from the breaking point, if the waves be large), the whole inr portion of the beach is covered with positive ( waves of the first order (waves of translation), from among which all waves of the second order have disappeared. This accounts for the phenomenon of breakers transporting shingle and wreck, and other substances shorewards after a certain point." Then referring more particularly to the conditions at Dublin Bay, he says that the " waves coming in from the deep sea are first broken when they approach the shallow beach in the usual way; they give off residuary waves, which are positive (waves of translation); these are wide asunder from each other, are wholly positive (i.e., above still- water level), and the spaces between them,
II
38 WATER WAVES
several times greater than the amplitude of the wave, are per- fectly flat; and in this condition they extend over wide areas and travel to great distances "l38.
EARTHQUAKE AND EXPLOSION WAVES
In investigations of shoreline changes the student may have occasion to refer to another class of waves which are occasionally developed upon the ocean, and which arc improperly called " tidal waves." These are the waves of enormous size and destructive energy produced by submarine earthquakes and volcanic explo- sions, and for]which Hobbs139 has suggested adopting the Japanese name "tsunamis." They occur at such rare intervals, and oper- ate for such a brief period, that they are probably not of great importance in modeling the forms of the shore. But inasmuch as they temporarily raise the upper limit of salt water far above its normal position, and leave behind them records which may be mistaken as evidences of a former higher stand of the mean sealevel, it is important that we become familiar with the work of these waves.
Nature and Origin of Wave Motion. — A submarine earthquake may produce several types of waves. There arc first the short and quick oscillations which travel toward the surface with the velocity of sound in water, and which are felt by overlying vessels as a sharp and violent shock, often causing the sailors to believe that the ves- sel has struck a reef. Old charts contain many isolated shallows and reefs reported by vessels which had really experienced earth- quake shocks in deep water. Occasionally such shocks are severe enough to hurl the ship out of water, to break off its masts, or even to destroy the vessel entirely140. Those oscillations are not of the type which produce prominent surface waves, however. Other groups of waves are produced by the dislocation of the sea-bot- tom. While the mechanism of these dislocation waves is not well understood, it is probable that the uplifting of a portion of the sea-bottom raises a mound of water above the general sur- face of the sea, and that the settling back of this water generates a great wave of translation which traverses the ocean with high velocity. Sometimes several such waves are produced, possibly by the disintegration of a former single wave of translation after the manner described by Russell for some of his experi-
EARTHQUAKE AND EXPLOSION WAVES 39
ments141. The sudden settling of a submarine crust block may generate a negative wave of translation. On the other hand, the behavior of many earthquake waves upon reaching the coast suggests that they partake of the characters of oscillatory waves, the water particles moving backward in a sort of great trough toward the oncoming wave crest. According to Iieid the waves caused by the same earthquake first appear as a depression of the water at some ports, and as an elevation at others; a fact which he attempts to explain on the theory that the down- dropped block generates a negative wave and the upraised block a positive wrave142. It is possible that the phenomena in question may be explained as a result of the different velocities with which positive and negative waves are propagated, both having resulted from the return of a mass of water raised above the general level, in some such manner as that described by Russell for his " residuary negative waves "143. Our knowledge of earth- quake waves is still too meager, however, to enable us to speak with assurance on this and other questions concerning their behavior. An experimental study of their mode of piopa- gation will be found in the Weber brothers' " Wellenlehre "14S a full resume* of pur present knowledge of the subject in Kriinr- meFs " Ozeanographie "145, and a good brief statement in Thou- let's " Oceanographie Dynamique "146.
In submarine volcanic explosions there is also produced a sharp and powerful shock, corresponding exactly to the first mentioned effect of earthquakes. At this time small jets of water may be shot into the air; but there soon follows a doming or up-swelling of the ocean surface, and finally the whole mass of up-raised water may be hurled into the air by the escaping gases. The doming of the water, ihe push exerted by the gases, and the back-falling mass of water, all tend to produce waves, some of which arc waves of translation, and some probably oscil- latory or compound waves147. Explosion waves and dislocation waves cannot be distinguished, and the origin of many of these waves, often • designated collectively as " earthquake waves," remains in doubt. According to Krummel, Rudolph supposed that tho great wave which overwhelmed Lisbon following the earthquake of 1755 was due to a volcanic explosion near the Portuguese coast148. Most authorities agree that the waves which followed the eruption of Krakatoa in 1883 were due directly to
40 WATER WAVES
the force of the explosion itself, but some have argued that they resulted from the masses of rock falling back into the water149.
On the open sea the heights of earthquake and explosion waves quickly diminish, and since the lengths are very great, they soon become so low and flat as to be unnoticed by vessels. But when they enter shallow water they behave like other waves, the height increasing until the wave form breaks to produce a gigantic surf. The velocity of these waves is very great, as they may travel a distance of 9000 or 10,000 miles in 24 hours, and one instance is recorded in which a velocity of 900 miles an hour was attained150. Their periods range from 15 minutes to one or two hours, and by assuming them to be the periods of free waves in deep water it has been calculated that the lengths of earth- quake and dislocation waves vary from 100 miles to 600 miles or more151.
Recorded Heights. — As students of shoreline phenomena we are more interested in the height attained by this class of waves when they reach the coast. We can better appreciate the truly surpris- ing elevations at which they may leave evidences of their former presence if we review some of the actual cases of which we have authentic records. In the years 358 and 365 A.D., the eastern shore of the Mediterranean was visited by great waves which passed over islands and low shores, sweeping away buildings and thou- sands of people. Boats were left on the roofs of houses in Alexan- dria, and others were stranded nearly a mile inland near Modhoni where they were later found slowly decaying1'2. Following the Lisbon earthquake in 1755 a wave variously estimated as from 40 to 60 feet high broke on the coast at Cadiz. The great earth- quake at Lima in 1724 was followed by a wave said to have been 80 feet high and which carried four vessels far inland. In August, 1868, an earthquake on the coast of Peru resulted in large waves, one of which submerged the mainland 55 feet above high-water mark. A United States war vessel was carried a quarter of a mile inland at Arica, where it remained until an- other great wave carried it still farther inland in 1877. This last was the wave caused by the Iquique earthquake, and it is said to have varied in height from 20 to 80 feet. An earthquake on the island of Hondo, Japan, in 1854 was accompanied by a wave which rose 30 feet above the usual level of the water. In 1896 another disturbance on the same coast generated three
TIDAL WAVES 41
waves, the largest of which was 50 feet high on the shore. Ships were torn from their anchorage, and one two-masted schooner was washed nearly a third of a mile inland. The Messina earth- quake of December 28, 1908, produced waves which rose nearly 30 feet high on some of the adjacent coasts. Following the eruption of Krakatoa in 1883 waves of enormous height wrought destruction over great distances. On the southern end of Sumatra one wave was over 70 feet high, and carried a gunboat two miles inland where it was left 30 feet above sealevel. In Katimbong the wave rose 80 feet, and on the shallow shore of Merak, on the Java coast, reached the enormous height of 115 to 135 feet1"
It is evident that such great waves must leave many records of their presence far above the normal level of the sea. Not only large vessels and smaller boats, which readily attract the popular attention, but fish and other forms of marine life are left stranded far inland and high above the reach of the highest tides or greatest storm waves. The bones of whales, and well- preserved marine shells occasionally found high above the sea, must not too readily be accepted as proof of a very recent uplift of the land. Successive earthquake waves in a given ocean may deluge the coasts of all the surrounding continents; and we must therefore expect to find driftwood, shells, and bones of fish well above sealevel at occasional points on almost any shore.
TIDAL WAVES
The great periodic motion of the sea known as the tide com- bines some of the features of oscillatory waves with others be- longing to waves of translation. It has been described by Russell as a " compound wave of the first order " (wave of translation) having more of the characteristics of waves of this order than of oscillatory waves154. There is no necessity, however of our entering into a discussion of the origin and character of the tidal wave, since the onlv elements of its motion of vital in- terest to the student of shore forms are the currents it produces, and the height to which it rises; both of which points are con- sidered in another part of this volume.
Wheeler has expressed the belief that the rising and falling of the tide is accompanied by the production of " tidal wavelets "
42 WATER WAVES
which are not the result of wind action, but are in some way genetically related to the tide itself156. The explanation of their origin which he gives is not wholly satisfactory, and his theory seems to be based upon his observation that waves from 6 to 24 inches in height break upon the beach at the rate of ten to twenty a minute " when there is an entire absence of wind or other dis- turbing cause." In the absence of sufficient evidence to connect such wavelets with the tides, we may perhaps more safely regard them as due to the action of gentle breezes and occasional gusts of wind, possibly some distance away, which even on the calmest day never permit the ocean surface to become absolutely quies- cent. Haupt156 states that the flood tide produces waves which break obliquely on the beach, and speaks of " the angle at which the flood breaks upon the shore." But since he also speaks of these supposed tidal waves as u breakers racing along the shore," and quotes Mitchell's description of the manner in which the " larger class of swell or rollers " strike the shore as an example of tidal wave activity, it would appear that Haupt has mistaken the ground-swell of distant storms for tidal wraves. A similar misapprehension may have been responsible for Marsh's curious idea that " on most coasts the supply of sand for the formation of dunes is derived from tidal waves," since " the momentum acquired by the heavy particles in rolling in with the water tends to carry them even beyond the flow of the waves "157.
STANDING WAVES; SEICHES
Under certain conditions there may exist oscillations of the water known as standing waves, in which the water particles do not describe closed orbits, but return through the same paths by which they advance (Fig. 8). The surface water moves up- ward in all of the crest, and downward in all of the trough, and the vertical movement of the particles is at a maximum under the crest where the horizontal movement is nil158. Hori- zontal movement is at a maximum under the nodal lines (Fig. 8). An example of the standing wave is the seiche, typically developed in inland lakes, and extensively studied in Lake Geneva by Forel. This movement consists of a periodic rise and fall of the water surface which is initiated by winds piling up the water at one end of the lake, by sudden variations in atmospheric
' STANDING WAVES; SEICHES 43
pressure, by earthquakes, by landslides, or by some other dis- turbance; and which continues for some time with gradually diminishing intensity. Each body of water has its own period, appropriate to its dimensions, and the extent to which the water rises and falls depends on the dimensions of the water body and the nature of the disturbing force. The principal seiche on Lake Geneva has an amplitude of from 8 centimeters to 2 meters159.
// ; \ \
Fig. 8. — Diagram to illustrate the movement of water particles in standing
waves, such as the seiche.
Seiches also occur along the coasts of the ocean, especially in bays and straits. Examples of these and other types of seiches are described in Harris's " Manual of Tides "16°, and Thoulet's " Oc6anographie Dynamique "161. According to Dawson162 a seiche at Yarmouth, Nova Scotia, had an amplitude of from 128 to 143 centimeters, or a maximum change of level of nearly 5 feet. As a rule, however, most seiches have an amplitude of a few inches only. The period varies from a few minutes in small water bodies to many hours in large ones, and the velocity of the water particles participating in the oscillation is not great. Indeed, the direct effect of seiches upon shoreline processes is probably almost negligible. In the rare cases where the am- plitude is great the effect of seiches is temporarily to raise the zone of ordinary wave activity to an appreciable extent; and occasionally the rising and falling of the water will cause currents of some importance through narrow straits or inlets; but these are exceptional cases and do not justify us in devoting further space to the subject of seiches in this connection. A good ac- count of this type of wave motion, with a short bibliography, will be found in the work by Harris already referred to, while Darwin's volume on " Tides and Kindred Phenomena " gives a description of Forel's important researches and a list of his classic papers188.
WATER WAVES
BOUNDARY WAVES
Where a layer of lighter surface water overlies a heavier water stratum, any sudden wind which creates or accelerates movement of the surface water will cause a rise of the under- lying heavier water at the point affected, and a corresponding depression in the heavier water farther forward. The de- velopment of such " boundary waves " at the plane of con- tact of two liquids Fig. 9. — Boundary wave formed by local air having different den- current over ^^ ot different densities.
Bities can readily be (Aftcr Sandstrom>
demonstrated by repeating Sandstrom's experiment, in which one of the layers was colored in order to distinguish it from the other, and a local air current was artificially generated16*. {Fig. 9.)
When fresh water from some large river flows out over the heavier salt water of the sea, conditions favoring the formation of boundary waves exist. Such waves move very slowry, their velocities depending upon the difference in density of the two water layers and in- creasing with the square root of this difference LSS. If the generating wind cease, the boundary waves advance to the margins of the
Fia. 10. — Diagram showing movement of water containing basin, particles in overlying fresh water (white) and where they are par- underlying salt water (shaded) during the tia|[ destroyed and ™"» nf hmiiiil.™ o'tves from left to right ., a , ,, .
. W. Ekmnn.) part,y reflected back
beneath the surface. In their progress they give rise to surface waves of the same length, but much smaller height, the crests of the surface waves being directly above the troughs of the boundary waves"*. Since boundary waves have a low velocity, and the water particles involved move still more slowly, it may be doubted whether they
passage of boundary v Gong arrow). (After
are of importance in shore processes. A good brief summary of the character of these waves is published by Helland-Hansen and Nansen in their report on the Norwegian Sea167, while the mathematical theory applicable to them has been developed by Stokes188.
RfiSUMfi
•
In the foregoing pages we have gained some idea of the nature of that force which is the most important agent in the modeling of shore forms. We have considered the form and charac- teristics of waves on the deep sea in order that we might the better appreciate the changes which they undergo as they approach the coast and begin their geological work. The mo- tion of the water particles in different types of waves; the nature of wave motion in deep water and over shallow sea-bottoms; the origin of storm waves, swells, and surf; the magnitude of waves and the conditions which govern their size; and the velocity of wave advance in both deep and shallow water, have in turn received our attention. With these points in mind we are prepared to enquire into the energy expended by waves upon the shore, and the work thereby accomplished.
In spite of the apparently hopeless chaos presented by the surface of a stormy sea, we know that the waves are controlled by definite natural laws, and that the different elements of form and motion are in systematic relation to one another. So per- fect is this relationship that one may stand upon the beach and time the breakers as they dash themselves to pieces at his feet, and learn thereby the length and velocity which these same waves had, hours ago, far away upon the deep sea. On the other hand, we know that not all the laws which control the behavior of waves have been discovered; and we have seen that where different types of waves act simultaneously upon the same water body it may be difficult or even impossible to analyze the resultant movements of the water. We are there- fore prepared to find that through the work of waves upon a coast the shoreline is changed according to definite natural laws which are in part, at least, discoverable. But we shall not be surprised if, in the present state of our knowledge of waves, we find it impossible to explain all of the changes which take place upon a shore under their influence.
46 WATER WAVES
REFERENCES
1. Kelvin, Lord (Sir William Thompson). Popular Lectures and Ad-
dresses. Ill, Navigation, 511 pp., London, 1891.
2. Fleming, J. A. Waves and Ripples in Water, Air, and iEther. 299
pp., London, 1902.
3. Cornish, Vaughan. Waves of the Sea and Other Water Waves.
374 pp., Chicago, 1911.
4. Fleming, J. A. Waves and Ripples in Water, Air, and iEther. 299 pp,,
London, 1902.
5. Russell, J. Scott. Report on Waves, made to the Meetings in 1842
and 1843. Report of the British Association. XIV, 375-381, 1844 (1845).
6. Russell, J. Scott. Report of the Committee on Waves, appointed by
the British Association at Bristol in 1836, etc.
Report of the British Association. VII, 417-496, 1837 (1838). Russell, J. Scott. Report on Waves, made to the Meetings in 1842 and 1843. Report of the British Association. XIV, 311-390, 1844 (1845).
7. Cornish, Vaughan. Waves of the Sea and Other Water Waves.
374 pp., Chicago, 1911.
8. Airy, G. B. On Tides and Waves. Encyclopaedia Metropolitana. V,
345*-350,* 1848.
9. Russell, J. Scott. Report on Waves, made to the Meetings in 1842
and 1843. Report of the British Association. XIV, 337, 1844 (1845).
10. Krummel, Otto. Handbuch der Ozeanographie. II, Die Bewegungs-
formen des Meeres, p. 12, Stuttgart, 1911.
11. Bremontier, N. T. Recherches sur le Mouvement des Ondes. 122
pp., Paris, 1809.
12. Weber, Ernst Heinrich and Wilhelm. Wellenlehre auf Experi-
mente Gegriindet. 433 pp., Leipzig, 1825.
13. Emy, A. R. Du Mouvement des Ondes et des Travaux Hydrauliques
Maritimes. 188 pp., Paris, 1831.
14. Russell, J. Scott. Report of the Committee on Waves, appointed by
the British Association at Bristol in 1836, etc. Report of the British Association. VII, 417-496, 1837 (1838). Russell, J. Scott. Report on Waves, made to the Meetings in 1842 and 1843. Report of the British Association. XIV, 311-390, 1844 (1845).
15. Russell, J. Scott. The Modern System of Naval Architecture. 3
Vols., London, 1865.
16. Bazin, Henri. Recherches Expenmentales sur la Propagation des
Ondes. Mem. de l'Acad. des Sciences de l'lnst. de France XIX, 495-644, 1865.
17. Airy, G. B. On Tides and Waves. Encyclopaedia Metropolitana.
V, 241*-396,* 1848.
18. Stokes, George Gabriel. Report on Recent Researches in Hydro-
dynamics. Mathematical and Physical Papers. I, 157-187, 1880.
REFERENCES 47
Stokes, George Gabriel. On the Theory of Oscillatory Waves. Mathematical and Physical Papers. I, 197-229, 1880.
19. Rankin, W. J. M. On the Exact Form of Waves near the Surface of
Deep Water. Philosophical Transactions of the Royal Society of London. CLI1I, Pt. I, 127-138, 1863.
20. Boussinesq, J. Essai sur la Theorie des Eaux Courantes. Mem. de
l'Acad. dee Sciences de l'Inst. de France. XXIII, 1-680, 1877; XXIV, No. 2, 1-64, 1887.
21. Bertin, Emile. Etude sur la Houle et le Roulis. Memoires de la
Socio" t6 Impcriale des Sciences Naturelles de Cherbourg. XV, 5-44, 313-355, 1869.
22. Bertin, Emile. Donnces Th6oriques et Exp6rimentales sur les Vagues
et le Roulis. M6moires de la Soci&c' Nationalc des Sciences Natu- relles de Cherbourg. XVII, 209-352, 1873; XVIII, 1-128, 1874; XXII, 161-227, 1879.
23. Saint-Venant, Barre de. Du Roulis sur Mer Houleuse. Memoires/ie
la Soci6te* Nationale des Sciences Naturelles de Cherbourg. XVI, 5-66, 1871.
24. Mottez, A. Du Courant Alternatif dans la Houle. Memoires de la
Soci6te Nationalc des Sciences Naturelles de Cherbourg. XVI, 360-370, 1872.
25. Cialdi, Alessandro. Sul Moto Ondoso del Mare e su le Correnti di
esso. 695 pp., Rome, 1866.
26. Caligny, A. de. Oscillations de l'Eau. 964 pp., Paris, 1883.
27. Stevenson, Thomas. The Design and Construction of Harbours.
3rd Edition. 355 pp., Edinburgh, 1886.
28. Fleming, J. A. Waves and Ripples in Water, Air, and jEther. 299
pp., London, 1902.
29. Wheeler, W. H. A Practical Manual of Tides and Waves. 201 pp.,
London, 1906.
30. Wheeler, W. H. The Sea Coast: Destruction: Littoral Drift: Pro-
tection. 361 pp., London, 1902.
31. Cornish, Vaughan. Waves of the Sea and Other Water Waves.
374 pp., Chicago, 1911.
32. Krummel, Otto. Handbuch der Ozeanographie. II. Die Bewe-
gungsformen des Meeres. 766 pp., Stuttgart, 1911.
33. Gaillard, D. D., Wave Action in Relation to Engineering Struc-
tures. Corps of Engineers U. S. Army, Professional Paper No. 31. 232 pp., Washington, 1904.
34. White, W. H. Manual of Naval Architecture. 5th Edition, 731 pp.,
London, 1900. 35 Kelvin, Lord (Sir William Thompson). Popular Lectures and Ad- dresses. Ill, Navigation, p. 456, London, 1891. Fleming, J. A. Waves and Ripples in Water, Air, and iEther, p. [42,
London, 1902. Russell, J. Scott. Report on Waves, made to the Meetings in 1842 and 1843. Report of the British Association. XIV, 375, 1844 (1845).
48 WATER WAVES
36. Lyman, C. S. A New Form of Wave Apparatus. Jour, of the Frank-
lin Institute. LXXXVI, 187-194, 1868.
37. Stokes, George Gabriel. On the Theory of Oscillatory Waves.
Mathematical and Physical Papers. 1, 198, 208, 1880.
38. Cialdi, Alessandro. Sul Moto Ondoso del Mare e su le Correnti di
esso, p. 68, Rome, 1866.
39. Stokes, George Gabriel. On the Theory of Oscillatory Waves.
Mathematical and Physical Papers. I, 209, 1880.
40. Stokes, George Gabriel. Report on Recent Researches in Hydro-
dynamics. Mathematical and Physical Papers. I, 164-165, 1880.
41. Emy, A. R. Du Mouvement des Ondes et des Travaux Hydrauliques
Maritimes, p. 49, Paris, 1831.
42. Cialdi, Alessandro. Sul Moto Ondoso del Mare e su le Correnti di
esso. 695 pp., Rome, 1866.
43. Corn ag li a, P. Sul Regime delle Spiagge e sulla Regolazione dei Porti.
569 pp., Turin, 1891. Review, Nature, XLV, 362, 1892.
44. Thoulet, J. Oceanographie Dynamique, p. 54, Paris, 1896.
45. Caligny, A. de. Oscillations de l'Eau, pp. 195-197, Paris, 1883.
46. Bremontier, N. T. Recherches sur le Mouvement des Ondes. 122
pp., Paris, 1809.
47. Emy, A. R. Du Mouvement des Ondes et des Travaux Hydrauliques
Maritimes, p. 17, Paris, 1831.
48. Airy, G. B. On Tides and Waves. Encyclopaedia Metropolitaua.
V, 294, 1848. Fleming, J. A. Waves and Ripples in Water, Air, and jEther, p. 11, London, 1902.
49 White, W. H. Manual of Naval Architecture. 5th Edition, p. 199,
London, 1900.
50. Cornish, Vaughan. Waves of the Sea and Other Water Waves,
p. 142, Chicago, 1911.
51. Rankin, W. J. M. On the Exact Form of Waves near the Surface of
Deep Water. Philosophical Transactions of the Royal Society of London. CLIII, Pt. I, 127, 1863.
52. Fenneman, N. M. Development of the Profile of Equilibrium of the
Subaqueous Shore Terrace. Jour, of Geol. X, 4, 1902.
53. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, pp. 55, 123, Washington, 1904.
54. White, W. H. Manual of Naval Architecture. 5th Edition, p. 213,
London, 1900.
55. Gaillard, D. D. Wave Action in Relation to Engineering Struc-
tures. Corps of Engineers U. S. Army, Professional Paper No. 31, pp. 36, 55, Washington, 1904.
56. Stokes, George Gabriel. On the Theory of Oscillatory Waves.
Mathematical and Physical Papers. I, 197-229, 1880.
57. Stevenson, Thomas. The Design and Construction of Harbours.
3rd edition, pp. 78, 79, Edinburgh, 1886.
REFERENCES 49
58. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, pp. 110-114, Washington, 1904.
59. Cornish, Vauqhan. Waves of the Sea and Other Water Waves, p. 135,
Chicago, 1911. GO. Krummel, Otto. Handbuch der Ozeanographie. II, Die Bewegungs- formen des Meeres, p. 112, Stuttgart, 1911.
61. Hagen, G. Handbuch der Wasserbaukunst. 3. Teil. Das Meer. I,
pp. 19, 86, Berlin, 1863.
62. Russell, J. Scott. Report on Waves, made to the Meetings in 1842
and 1843.
Report of the British Association. XIV, 371, 1844 (1845).
63. Bazin, Henri. Recherches Expe'rimentales sur la Propagation des Ondes.
Mem. de PAcad. des Sciences de PInst. de France. XIX, 518, 1865.
64. Russell, J. Scott. Report of the Committee on Waves, appointed
by the British Association at Bristol in 1836, etc.
Report of the British Association. VII, 451, 1837 (1839). Russell, J. Scott. Report on Waves, made to the Meetings in 1842 and 1843.
Report of the British Association. XIV, 371, 1844 (1845).
65. Cornish, Vauqhan. Waves of the Sea and Other Water Waves, p.
170, Chicago, 1911.
66. Stevenson, Thomas. The Design and Construction of Harbours.
3rd Edition, pp. 77-78, Edinburgh, 1886.
67. Cialdi, Alessandro. Sul Moto Ondoso del Mare e su le Correnti di
esso, pp. 145-157, Rome, 1866.
68. Thoulet, J. Ocdanographie Dynamique, p. 51, Paris, 1896.
69. Krummel, Otto. Handbuch der Ozeanographie. II. Die Bewegungs-
forem des Meeres, p. Ill, Stuttgart, 1911.
70. Ibid., p. 112.
71. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, pp. 114-123, Washington, 1904.
72. Cornish, Vauqhan. Waves of the Sea and Other Water Waves, pp.
132, 133, Chicago, 1911.
73. Ibid., pp. Ill, 133.
Cornish, Vauohan. On the Dimensions of Deep Sea Waves, and their Relations to Meteorological and Geographical Conditions. Geographical Jour. XXIII, 643, London, 1904.
74. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 67, Washington, 1904.
75. Cornish, Vauohan. Waves of the Sea and Other Water Waves, pp.
128, 129, Chicago, 1911.
76. Weber, Ernst Heinrich and Wilhelm. Wellenlehre auf Experi-
mente Gegriindet, p. 25, Leipzig, 1825.
77. Cornish, Vaughan. Waves of the Sea and Other Water Waves,
p. 106, Chicago, 1911.
60 WATER WAVES
78. Stevenson, Thomas. The Design and Construction of Harbours.
3rd Edition, p. 29, Edinburgh, 1886.
79. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 69, Washington, 1904.
80. Bois, Coupvent des. Memoire sur la Hauteur des Vagues a la Sur-
face des Oceans. Comptes Rendus de l'Acad. des Sciences. LXII, pp. 86-87, 1866.
81. Cornish, Vaughan. On the Dimensions of Deep Sea Waves, and
their Relations to Meteorological and Geographical Conditions. Geographical Jour. XXIII, 636, London, 1904.
82. Stevenson, Thomas. The Design and Construction of Harbours.
3rd Edition, pp. 34, 35, Edinburgh, 1886.
83. Cornish, Vaughan. Waves of the Sea and Other Water Waves, p.
67, Chicago, 1911.
84. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 82, Washington, 1904.
85. Cornish, Vaughan. Waves of the Sea and Other Water Waves,
pp. 33, 40, Chicago, 1911.
86. Scorfsby, William. On Atlantic Waves, their Magnitude, Velocity,
and Phenomena.
Report of British Association for 1850, Pt. II, p. 28, 1851.
87. Cornish, Vaughan. Waves of the Sea and Other Water Waves, pp.
53, 60, Chicago, 1911.
88. Scoresby, William. On Atlantic Waves, their Magnitude, Velocity,
and Phenomena. Report of British Association for 1850, Pt. II,
p. 28, 1851. Cornish, Vaughan. Waves of the Sea and Other Water Waves,
p. 60, Chicago, 1911. Cornish, Vaughan. On the Dimensions of Deep Sea Waves, and
their Relations to Meteorological and Geographical Conditions.
Geographical Jour. XXIII, 627, London, 1904.
89. Cornish, Vaughan. Waves of the Sea and Other Water Waves, p.
62, Chicago, 1911.
90. Ibid., pp. 74-77.
91. White, W. H. Manual of Naval Architecture. 5th Edition, p. 212,
London, 1900.
92. Gaillard, D. D. Wave Action in Relation to Engineering Struc-
tures. Corps of Engineers U. S. Army, Professional Paper No. 31, pp. 76-79, Washington, 1904.
93. Abercromby, Ralph. Observations on the Height, Length, and Ve-
locity of Ocean Waves. Philosophical Magazine, XXV, 269, 1888.
94. Airy, G. B. On Tides and Waves. Encyclopaedia Metropolitana,
V, 351, 1848.
95. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 85, Washington, 1904.
REFERENCES 51
96. Cornish, Vaughan. On the Dimensions of Deep Sea Waves, and
their Relations to Meteorological and Geographical Conditions. Geo- graphical Jour. XXIII, 626, London, 1904.
97. Cornish Vaughan. Waves of the Sea and Other Water Waves, pp.
96-101, Chicago, 1911. Cornish, Vaughan. On the Dimensions of Deep Sea Waves, and their Relations to Meteorological and Geographical Conditions. Geographi- cal Jour. XXIII, 627-633, London, 1904.
98. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No, 31, p. 57, Washington, 1904.
99. Ibid., p. 70.
100. Paris, A. Observations sur Pfitat dc la Mer Recueillies a bord du
Dupleix et de la Minerve (1867-70). Revue Maritime et Coloniale. XXXI, 121, 1871.
101. Schott, Gerhard. Uber die Dimensionen der Meereswellen. Fest-
schrift Ferdinand Freiherrn von Richthofen zum Sechzigsten Geburts- tag, p. 250, Berlin, 1893.
102. White, W. H. Manual of Naval Architecture. 5th Edition, pp.
213-214, London, 1900.
103. Cornish, Vaughan. Waves of the Sea and Other Water Waves, pp.
30, 34, Chicago, 1911.
104. Ibid., p. 92.
105. White, W. H. Manual of Naval Architecture. 5th Edition, p. 211,
London, 1900.
106. Bois, Coupvent des. Memoire sur la Hauteur des Vagues a la Surface
des Oceans. Comptes Rendus de l'Acad. des Sciences. LXII, p. 83, 1866.
107. Cornish, Vaughan. Waves of the Sea and Other Water Waves, p.
73, Chicago, 1911.
108. Scoresby, William. On Atlantic Waves, their Magnitude, Velocity,
and Phenomena. Report of British Association for 1850. Pt. II, 29, 1851.
109. Cornish, Vaughan. Waves of the Sea and Other Water Waves,
pp. 70, 82, Chicago, 1911.
110. Ibid., pp. 88-94.
Cornish, Vaughan. On the Dimensions of Deep Sea Waves, and their Relations to Meteorological and Geographical Conditions. Geo- graphical Jour. XXIII, 627, London, 1904.
111. Antoine, Ch. Des Lames de Haute Mer, p. 3, Paris, 1879.
112. Cornish, Vaughan. Waves of the Sea and Other Water Waves, p. 87,
Chicago, 1911.
113. White, W. H. Manual of Naval Architecture. 5th Edition, p. 205,
London, 1900.
114. Airy, G. B. On Tides and Waves. Encyclopedia Metropolitana.
V, 292, 1848.
115. Fleming, J. A. Waves and Ripples in Water, Air, and jEther, p. 10,
London, 1902.
52 WATER WAVES
116. White, W. H. Manual of Naval Architecture. 5th Edition, p. 204,
London, 1900.
117. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 38, Washington, 1904.
118. White, W. H. Manual of Naval Architecture. 5th Edition, p. 204,
London, 1900.
119. Fleming, J. A. Waves and Ripples in Water, Air, and iEther, p. 11,
London, 1902.
120. White, W. H. Manual of Naval Architecture. 5th Edition, p. 205,
London, 1900.
121. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 44, Washington, 1904.
122. Ibid., pp. 97-103.
123. Airy, G. B. On Tides and Waves. Encyclopedia Metropolitana. V,
292, 1848.
124. Russell, J. Scott. Report on Waves made to the Meetings in 1842
and 1843.
Report of the British Association. XIV, 325, 1844 (1845).
125. Airy, G. B. On Tides and Waves. Encyclopedia Metropolitana.
V, 346, 1848.
126. Lagrange. Mechanique Analitique, p. 491, Paris, 1788.
127. Caligny, A. de. Oscillations do l'Eau, p. 199, Paris, 1883.
128. KrCmmel, Otto. Handbuch der Ozeanographie. II. Die Bewe-
gungsformen des Meeres, p. 29, Stuttgart, 1911.
129. Ibid., p. 152.
130. Russell, J. Scott. Report of the Committee on Waves, appointed
by the British Association at Bristol in 1836, etc.
Report of the British Association. VII, 417-496, 1837 (1838). Russell, J. Scott. Report on Waves, made to the Meetings in 1842 and 1843.
Report of the British Association. XIV, 311-390, 1844 (1845).
131. Russell, J. Scott. Report on Waves, made to the Meetings in 1842
and 1843.
Report of the British Association. XIV, 340-347, 1844 (1845).
132. Ibid., pp. 340, 354.
133. Ibid., pp. 325-328.
134. KrCmmel, Otto. Handbuch der Ozeanographie. II. Die Bewegungs-
formen des Meeres, pp. 149-150, Stuttgart, 1911. Wheeler, W. H. A Practical Manual of Tides and Waves, p. 131, London, 1906. 135 Wheeler, W. H. A Practical Manual of Tides and Waves, p. 57, London, 1906.
136. Caligny, A. de. Oscillations de PEau, pp. 191-211, Paris, 1883.
137. Hunt, A. R. On the Action of Waves on Sea-Beaches and Sea-Bot-
toms. Proc. Roy. Dublin Soc, N. S. IV, 251-259, 1884.
REFERENCES 53
138. Russell, J. Scott. Report on Waves, made to the Meetings in 1842
and 1843. Report of the British Association. XIV, 372-373, 1844 (1845).
139. Hobbs, Wm. H. Origin of Ocean Basins in the Light of the New Seis-
mology. Bull. Geol. Soc. Amer. XVIII, 242, 1907.
140. Krummel, Otto. Handbuch der Ozeanograpbie. II. Die Bewegungs-
formen des Meeres, p. 133, Stuttgart, 1911.
141. Russell, J. Scott. Report on Waves, made to the Meetings in 1842
and 1843.
Report of the British Association. XIV, 323, 1844 (1845).
142. Reid, H. F. Earthquake Sea Waves. Unpublished paper read at
Princeton Meeting of Geological Society of America. December, 1913.
143. Russell, J. Scott. Report on Waves, made to the Meetings in 1842
and 1843.
Report of the British Association. XIV, 323, 1844 (1845).
144. Weber, Ernst Heinrich and Wilhelm. Wellenlehre auf Experi-
mente Gegriindet. 433 pp., Leipzig, 1825.
145. Krummel, Otto. Handbuch der Ozeanographie. II. Die Bewegungs-
formen des Meeres. 766 pp., Stuttgart, 1911.
146. Thoulet, J. Occanographie Dynamique. 131 pp., Paris, 1896.
147. Krummel, Otto. Handbuch der Ozeanographie. II. Die Bewe-
gungsformen des Meeres, pp. 136-137, Stuttgart, 1911.
148. Ibid., p. 141.
149. Ibid., p. 148.
150. Ibid., p. 149.
Wheeler, W. H. A Practical Manual of Tides and Waves, p. 131, London, 1906.
151. Krummel, Otto. Handbuch der Ozeanographie. II. Die Bewegungs-
formen des Meeres, p. 149, Stuttgart, 1911.
152. Ibid., p. 139.
153. Ibid., p. 148.
Gaillard, D. D. Wave Action in Relation to Engineering Structures. Corps of Engineers U. S. Army, Professional Paper No. 31, p. 91, Washington, 1904.
154. Russell, J. Scott. [On local changes of tide heights and on the char-
acter of the tide-wave.] Min. Proc. Inst. Civ. Eng. VII, 364, 1848.
155. Wheeler, W. H. The Sea Coast; Destruction: Littoral Drift: Pro-
tection, p. 8, London, 1902.
156. Haupt, L. M. Discussion on the Dynamic Action of the Ocean in
Building Bars. Proc. Am. Phil. Soc. XXVI, pp. 147, 148, 155, 1889.
157. Marsh, Geo. P. The Earth as Modified by Human Action, p. 538,
New York, 1907.
158. Krummel, Otto. Handbuch der Ozeanographie. II. Die Bewegungs-
formen des Meeres, p. 158, Stuttgart, 1911.
159. Ibid., p. 166.
160. Harris, R. A. Manual of Tides, Part V. U. S. Coast Surv. Rept. for
1907. Appendix No. 6, 472-482, 1907.
54 WATER WAVES
161. Thoulet, J. Oceanographie Dynamique, pp. 71-84, Paris, 1896.
162. Dawson, W. Bell. Illustrations of Remarkable Secondary Tidal
Undulations in January, 1899, as Registered on Recording Tide Gauges in the Region of Nova Scotia. Trans. Roy. Soc. Canada. 2nd Ser., V. Sec. Ill, 24, 1899.
163. Darwin, G. R. The Tides and Kindred Phenomena in the Solar
System, pp. 16-49, London, 1898.
164. SandbtrAm, J. W. Dynamische Versuche mit Meerwasser. Annalen
der Hydrographie und Maritimen Meteorologie. XXXVI, 10, 1908.
165. Hblland-Hansen, BjGrn and Nansen, Fridtjof. The Norwegian
Sea, p. 116, Christiania, 1909.
166. Ekman, V. W. On Dead Water. The Norwegian North Polar Expedi-
tion 1893-1896, Scientific Results. V, No. 15, p. 42, Christiania, 1906.
167. Helland-Hansen, BjGrn and Nansen, Fridtjof. The Norwegian
Sea, pp. 114-117, Christiania, 1909.
168. Stokes, George Gabriel. On the Theory of Oscillatory Waves.
Mathematical and Physical Papers, I, 212-219, 1880.
CHAPTER n THE WORE OF WAVES
Advance Summary. — Water waves, whose general charac- teristics were discussed in Chapter I, possess energy capable of effecting profound changes upon the margins of the land or upon artificial structures with which they may come into con- tact. Geologist, geographer, and engineer must each concern himself with the nature and magnitude joi wave energy, and with the manner in which waves accomplish their work. The layman finds the destructive energy of waves a source of inter- est and wonder, and not unnaturally regards the meeting-place of land and sea as one of the most fascinating of Nature's lab- oratories.
In the present chapter the nature of wave energy is first dis- cussed, and the manner of wave attack upon cliffs and sloping shores is briefly treated. It is then shown that the dynamic pressures exerted by waves may be measured with reasonable exactness, and calculated and measured pressures are shown to be in substantial agreement. Some of the most striking examples of damage done by storm waves are next passed in review, in order that the reader may visualize the magnitude of the force responsible for the modification of shore features and th^ manifold methods of its working. In order to determine which parts of a shore or what artificial structures will suffer most from wave attack, it is essential to know precisely what factors control wave energy, and these are briefly considered. A process of "wave refraction " is shown to be responsible for the concentration of wave attack upon projecting headlands and for the comparative immunity of shores about the heads of bays. In conclusion, attention is directed to the vitally im- portant question as to how far below the water surface wave action may be appreciable.
Wave Energy. — It can readily be shown that a wave transmits energy along the surface of a water body, and delivers this energy on the beach or against some artificial obstacle. When a ship is
55
56 THE WORK OF WAVES
propelled through the water, a wave is pushed up by the bow. It took a certain amount of energy to raise this mound of water, and that amount was taken away from the energy of the moving vessel, thereby causing the vesseFs motion to be retarded. The wave passes over the surface of the water; and if it finally dashes upon some beach, the energy there expended is the same energy imparted by the moving boat, less a small amount lost through friction.
The mere spreading apart of the water by a vessel's bow does not require the expenditure of energy. If it were not for other causes of resistance, a ship once started through the water would move on forever, unimpeded by the pushing apart of the water in front. The common idea that a vessel's bow is made sharp so that it may cut into the water like a wedge and more easily push it out of the way, is erroneous. No part of the resistance to a ship's motion arises directly from the pushing of water to either side by the bow1. A great deal of resistance does arise, however, from the fact that energy is used up in making waves, and one object of the naval architect is to design a vessel of such form that it will produce the fewest and smallest waves possible.
The energy of a wave depends upon its length and height, and is of two types : the kinetic energy due to the orbital move- ment of the water particles; and the potential energy due to the fact that the center of gravity of the mass of water composing a wave is raised slightly above the position it occupies when the water is at rest. It can be shown that the two types of energy are exactly equal in amount; in other words, the energy of a wave is half kinetic and half potential. Since we know that a cubic foot of sea water weighs about 64 pounds, it is easy to calculate the total energy of either shallow-water or deep-water waves in foot-tons per linear foot of wave crest. The formulae employed in such calculations are too complex for discussion here, but may be found in Gaillard's treatise2, and similar works.
During the advance of a deep-water oscillatory wave one-half of the total wave energy is transmitted forward with the wave form. The energy of shallow-water oscillatory waves is from 1 per cent to 11 per cent less than the energy of deep-water waves of equal length and height, but just as in the case of deep-water waves one-half the total wave energy is transmitted onward. In both cases it is the potential energy which is thus carried
NATURE OF WAVE ATTACK 57
forward with the wave1. In the wave of translation the energy is also partly kinetic and partly potential; but as this wave leaves still water behind it, at the original level, the entire wave energy must pass forward with the wave form. We have already seen that when oscillatory waves pass into water which shoals very gradually, they are slowly transformed into waves of translation, or at least acquire some of the characteristics of such waves. From this it follows that an oscillatory wave may, by changing into a wave of translation, deliver at the shore all, or nearly all, its energy4. This may help to explain the fact that the blows of storm waves against a cliff or sea wall often exceed in vio- lence the available energy calculated for the waves on the as- sumption that they are waves of oscillation.
Nature of Wave Attack. — The nature of the force exerted by a wave upon any obstacle, such as a cliff or beach, depends in part upon the type of wave and its condition at the moment of collision with the obstacle. If an unbroken oscillatory wave strikes a vertical wall or cliff the base of which reaches down to deep water, the wave is reflected back. At the instant of con- tact the crest of the wave rises to twice its normal height and the cliff is subjected to the hydrostatic pressure of this unusually high water column. The absence of any forward thrust of the water mass under these conditions is shown by the behavior of boats which have been observed to rise and fall with successive waves without touching the vertical wall only a few feet distant. Hagen* concludes that under such circumstances debris must accumulate at the base of the wall and that therefore the preju- dice against vertical sea walls and harbor walls, based on the fear of undermining by wave action, is ill-founded.
A wave of translation striking a vertical wall or cliff under the same circumstances is also reflected; but it delivers against the cliff a vigorous push due to the forward thrust of the whole mass of the wave, in addition to subjecting the obstruction to hydro- static pressure. Stevenson6 found that " oscillatory waves become waves of translation when they reach the unfinished part of a ver- tical sea wall, and that they then exert a force nearly 6 times greater than if they had remained waves of oscillation." If either type of wave breaks just before reaching the cliff, in such manner that the forward falling crest of the wave strikes the cliff face, the only force exerted is that due to the forward motion of the water
58 THE WORK OF WAVES
particles. This motion may exceed the velocity of the wave itself at the time of breaking, the crest shooting forward beyond the main body of the wave as it falls. When an oscillatory wave breaks a short distance out in front of the cliff, so that the forward pitching crest does not strike the cliff, but plunges into the water at its base, the regular orbital motion is destroyed and a " whirlpool turbulence " is produced, the forces of which are not easily analyzed. In a similar manner, if a wave of translation breaks just before reaching a cliff, it " becomes a surge or broken foam, a disintegrated heap of water particles, having lost all continuity." The moving waters of the surge or whirlpool turbulence may exert considerable dynamic pressure on the base of the cliff, and some hydrostatic pressure, depend- ing on the height to which the water rises. When either oscil- latory waves or waves of translation break far out from the base of the cliff, smaller waves of translation may traverse the inter- vening water and operate upon the cliff in the manner already described.
On a sloping shore of fairly steep inclination, oscillatory waves may arrive almost at the beach before losing their essential characters. When such a wave breaks the falling crest dashes down upon the water which is returning seaward from the swash of the preceding wave. The falling wave crest thus strikes a cushion of moving water which may be of considerable thickness. A zone of great confusion is thus produced, the force of the wave is largely dissipated, and part of its volume augments the sheet of water moving seaward, while a larger part starts up the beach. Almost instantly the remainder of the breaking wave over- takes the zone of disturbance, the forward oscillation under the crest checking and possibly reversing the seaward motion of the bottom water, while the landward moving water is enor- mously augmented in volume. At the same time the orbital motion of the water is largely destroyed, and in the form of a confused mass it rushes up the beach until stopped by gravity and friction, when it flows back with gradually increasing velocity to meet the next oncoming wave. Under these conditions much of the energy of the wave is consumed by friction in the turbulent waters, while another part is expended in the impact of the falling crest upon the bottom wherever the sheet of seaward moving water is effectively pierced. The beach itself is affected
NATURE OF WAVE ATTACK
60 THE WORK OF WAVES
mainly by the sheet of water which is propelled up the slope by the remnant of the wave's energy of motion, and which re- turns under the action of gravity.
It should be noted that after the oscillatory wave breaks, the confused mass of water propelled up the slope of the beach may be regarded as an irregular type of wave of translation. When a typical wave of translation breaks immediately at the foot of the beach, its falling crest must also meet the backward flowing water cast up by the preceding wave, and give rise to much the same phenomena as the breaking oscillatory wave.
On a coast bordered by water so shallow that large oscillatory waves are broken some distance out from the shoreline, waves of translation and small oscillatory waves alone may reach the beach. If the beach slopes very gradually under water, there may be a secondary line of surf a short distance out where these waves break, and the amount of wave energy which finally reaches the beach itself may be quite insignificant. On the other hand, if the water between the shoreline and the zone where the great oscillatory waves break is of fairly constant depth, and the shore rises fairly abruptly at the inner margin of the shallow, waves of translation of considerable size may deliver their whole energy upon the beach. The latter is then subjected to the static pressure due to the wave height, and the dynamic force of the rapidly moving water particles.
When a wave comes in contact with a vertical or very steep wall or cliff, a relatively small portion of the wave mass may be shot upward (Plates IV and V). It appears that under these circum- stances the energy of a large portion of the wave is suddenly communicated to the smaller water mass. The result is that the velocity of this mass may be very great, and it may deliver a blow of terrific force upon a small area. Overhanging cliffs or projec- tions from cliff faces, the roofs of sea caves, and other masses of rocks favorably situated may be subjected to blows from below which have the strength of a battering ram. The energy expended is the kinetic energy due to the swift motion of the water masses.
Masses of water shot into the air in the manner just described may encounter no obstacle in their upward flight, but may descend upon the level summit or sloping face of a cliff, the sur- face of a beach, or some artificial structure. Such falling masses
NATURE OF WAVE ATTACK 61
Piatc V
Photo vv A, At, Cromack. Water forced vertically upward by wave breaking against sea wall at Scarborough, England.
62 THE WORK OF WAVES
of water are capable of executing considerable damage because of the great energy they acquire by descending with the ever- increasing velocity due to gravitation.
Wave Dynamometer. — Stevenson has shown that the action of a wave is not at all lite the sudden impact of a hard body, but is analogous to the steady pressure of a current, because the wave acts with a continuous pressure for an appreciable length of time7. It follows from this that if waves are allowed to come against a vertical plate which has a spring back of it, and if the
FlO. 11. -y Stevenson's Wave Dynamometer. DEFD m a cast-iron cylinder, bolted to the rock by the flanges at O. AA u an iron disk against Which the waves impinge, fastened to guide rods BB, which pass through holes in the plate CC. When waves strike the disk AA, rings of leather TT are moved along the guide rods, registering the extent to which the spring is lengthened. LL is a door opened for the purpose of reading the instrument.
change in length of the spring due to the pressure against the plate isdetermined, we shall have a proper measure of the dynamic pressure exerted by the wave. Stevenson* devised such an instru- ment, called a dynamometer, and was the first man to measure the force of waves. Gaillard confirmed Stevenson's results, but pointed out that the spring dynamometer measures only the dy- namic pressure of the moving water in the wave, and gives no information as to the static pressure resulting from the weight of the water mass. This is due to the fact that static pressure is just as great on the back of the plate as on the front, and therefore produces no effect on the spring. He therefore designed a dia-
MEASUREMENTS OF WAVE ENERGY 63
phragm dynamometer having a sheet of rubber stretched over one end of a short iron cylinder, the other end being closed by an iron plate. Pressures due to the waves thus affect but one side of the instrument by pushing in the rubber diaphragm, and the magnitudes of the pressures are determined by means of a gauge attached to the cylinder. To measure the static pres- sure due to th£ column of water in the wave the instrument is placed with its face horizontal and upward at the desired depth in the water. When placed with its face vertical so as to receive the full impact of the advancing wave the instrument records both dynamic and static pressures9.
Measurements of wave force with dynamometers indicate that the static pressures exerted by waves are considerably less than their dynamic pressure?. On Lake Superior, Gaillard found the static pressure of a wave 10.5 feet high and 150 feet long, to be 3.23 lbs. per square inch, or about 450 lbs. per square foot, the dynamometer being 9 feet below the wave crest. The dy- namic pressures of waves 10 feet high and 150 feet long varied from 460 to 965 lbs. per square foot on a dynamometer placed about a foot higher than that for the measurement of static pressures10. Adequate observations of wave pressures by means of suitable dynamometers have not yet been made, those for static pressures being especially insufficient in number.
Measurements of Wave Energy. — In order to gain some con- ception of the enormous power of waves we have only to consider the theoretical pressures calculated for waves of different size, the actual pressures recorded by dynamometers on exposed coasts, or the damage to harbor works done by storm waves. Gaillard has calculated that a wave 10 feet high and 100 feet in length may strike an obstruction with a pressure of 1675 lbs. per square foot, while a wave 12 feet high and 200 feet long should exert a maximum dynamometer pressure of 2436 lbs. per square foot. The total theoretical energy of such a wave is 109 foot-tons for every linear foot of wave crest. Great ocean waves such as those which destroyed part of the breakwater at Wick, Scotland, in 1872, if we assume a height of 42 feet and a length of 500 feet, should produce a pressure of 6340 pounds per square foot11.
Dynamometer readings show that during storms on Lake Superior the waves develop a force of from 1600 to 2500 lbs. per square foot12. Stevenson found that the Atlantic Ocean
THE WORK OF WAVES
DAMAGE BY STORM WAVES 65
waves near the island of Tyree on the Scottish coast had an average force of 611 lbs. per square foot during the summer months, whereas the average for the winter months was 2086 lbs., or more than three times as great. The greatest force recorded at this point was 6083 lbs. or practically that calculated theoretically for a very large ocean wave. Another reading of 5323 lbs. was secured. On the east coast of Scotland pressures of more than 6000 lbs. per square foot were recorded13.
Damage by Storm Waves. — Such enormous pressures are cap- able of producing remarkable results. Stevenson describes an in- stance in which a block of stone weighing 1\ tons and situated 20 feet above sealevel was driven before the waves for a distance of 73 feet over rugged ledges14. At North Beach, Florida, a solid block of concrete weighing 4500 lbs. was moved 12 feet horizontally and turned over on its side, while a second block weighing 3600 lbs. and having its center at high-water level was shifted several inches by waves which were not over 4 feet high. During a severe storm on December 25, 1836, stones forming part of the break- water'at Cherbourg and weighing nearly 7000 lbs. were thrown over a wall 20 feet high which surmounts the stone embank- ment. In the harbor of Cette a block of concrete 2500 cubic feet in volume and weighing about 125 tons was shifted more than 3 feet from its original position. Perhaps the most won- derful example of wave work is that accomplished by ocean storm waves upon the breakwater at Wick in December, 1872, and described in Stevenson's treatise on " Harbors." The seaward end of this breakwater was protected by a monolithic block of cement rubble 45 feet long, 26 feet wide and 11 feet thick, and weighing more than 800 tons, resting on great blocks of stone which were bound solidly to the monolith by iron rods 3J inches in diameter running through holes in the stones and embedded in the cement rubble. The entire mass, weighing 1350 tons, was torn from its place by the waves and dropped inside the pier, where it was found unbroken after the storm subsided. A much larger mass of concrete was substituted for the one removed, the new block having a volume of 1500 cubic yards, and weighing 2600 tons. In 1877 this enormous mass was similarly carried away by the waves15.
The terrific impact which a wave may deliver against the face of a vertical wall may be appreciated from the fact that the
66 THE WORK OF WAVES
facing stones of the Wick breakwater, having the same density as granite, were shattered by the sea in February, 1872. At Dunkirk waves from the narrow southern arm of the North Sea strike the coast with an impact which causes a trembling of the ground more than a mile inland16. That waves have the power to wrench from place objects situated well above the main body of the wave is shown by the effects of a storm upon Dhuheartach lighthouse on the west coast of Scotland, during which fourteen stones weighing 2 tons each were torn from their positions 37 feet above high tide, and dropped into deep water. Cast-iron lamp posts on the pier heads at Duluth, located 19 feet above lake level, have repeatedly been broken off by wave action.
The lifting power of waves is often illustrated by damage to har- bor works. At North Beach, Florida, a block of concrete weighing 104 tons was lifted vertically upward three inches by the wave pressure transmitted through crevices below the mass. During a storm on Lake Superior a mass of trap rock 2 cubic yards in volume and weighing about 4$ tons was raised by a wave from its place alongside an old breakwater at Duluth and deposited on the surface of the breakwater some 5 or 6 feet above its original position. A more striking example occurred at Ymuiden on the coast of Holland, when a concrete block weighing 20 tons was lifted 12 feet vertically by a wave and deposited on a pier above high-water level.
Waves deflected upward by a sloping surface may accom- plish work at high levels. The keeper of Trinidad Head light station, on the Pacific Coast, reports that during the storm of December 28, 1913, the waves repeatedly washed over Pilot Rock, 103 feet high. One unusually large wave struck the cliffs below the light and rose as a solid sea apparently to the same level at which he was standing in the lantern, 196 feet above mean high water, the spray rising 25 feet or more higher. The shock of the impact against the cliffs and tower was terrific, and stopped the revolving of the light. Lake Superior waves reached the door of a light-keeper's dwelling situated 140 feet back from the water and 60$ feet above it, carrying away a board walk and doing other minor damage. On the Bound Skerry in the Shetland Islands blocks of stone from 6 to 13 tons in weight have been forced from their places at a level which is 70 to 75 feet above the sea.
DAMAGE BY STORM WAVES 67
The destructive power of the masses of water hurled to re- markable heights by breaking waves is greater than one might suppose. At the Bell Rock lighthouse in the North Sea a ground- swell, without the aid of wind, drove water to the summit of the tower 106 feet above high tide, and broke off a ladder at an elevation of 86 feet. A bell weighing 3 cwt. was broken from its place in the Bishop Rock lighthouse, 100 feet above high water mark, during a gale in 1860; and at Unst, in the Shetland Islands, a door was broken open at a height of 195 feet above the sea. The keeper of Tillamook Rock lighthouse, on the coast of Oregon, reports that in the winter of 1902 the water of waves was thrown more than 20.0 feet above the level of the sea, de- scending upon the roof of his house in apparently solid masses. In October 1912, and again in November 1913, the panes of plate glass in the lantern of this same light, 132 feet above mean high water, were broken in by storm waves.
Great damage may be accomplished by the falling water. Ac- cording to Shield17 "it is no uncommon occurrence for storm waves, striking a vertical breakwater face, to throw heavy masses of water to a height of at least 100 feet, often very much higher. Such water in its descent on reaching the roadway of the break- water upon which it falls, will have attained a velocity of about 80 feet per second, or nearly double the velocity and four times the force of the water striking the face of the break water/ ' Dur- ing a severe gale at Buffalo in December, 1899, seventy big tim- bers, 12 X 12 inches in thickness, 12 feet long, and 10 feet between supports, were broken in two in the middle by the impact of the falling water. This same breakwater was further damaged a year later when waves breaking against it were hurled from 75 to 125 feet into the air, the falling water crushing the big timbers on which it fell as though they had been pipestems.
A part of the geological work accomplished by waves is due to the direct pressure exerted upon air and water imprisoned in crevices, and another part to the sudden expansion of air in crevices and pore spaces when the rapid retreat of a wave creates a partial vacuum outside. The effect of compressed air may be inferred from the fact that waves coming against a breakwater in Buffalo harbor produced such high pressure upon the air under the concrete shell that four circular plates of concrete, 3 feet in diam- eter, 6 inches thick and weighing 530 lbs. each, serving as covers to
68 THE WORK OF WAVES
manholes, were lifted from their places. A block weighing 7 tons in the face of the breakwater at Ymuiden was started forward out of its place during a gale, the movement being toward the waves which were coming against it. According to Gaillard this phenomenon was caused " by the stroke of a wave compressing the air in the rear of it"18, but similar results are produced by expansion due to the formation of a partial vacuum in front. In 1840 a securely fastened door in the Eddystone lighthouse was burst outward during the attack of storm waves, the circumstances leading Geikie to conclude that " by the sudden sinking of a mass of water hurled against the building, a partial vacuum was formed, and the air inside forced out the door in its efforts to restore the equilibrium"19.
Another important factor in the work of waves is the effect pro- duced by stones, logs, blocks of ice, and other objects moving with the waves. It has been well said by Playfair20 that waves thus armed become a sort of " powerful artillery " with which the ocean assails the land. A large block of ice or a log may concentrate its whole momentum upon a very small area with appropriately great results. Thus, Gaillard has suggested that an exceptionally high dynamometer reading on Lake Michigan (when the instrument showed a pressure twice as great as that recorded in the same locality for a more severe storm and greater than any record for the larger waves of Lake Superior) may possibly have been caused by ice or timber. Large stones may be hurled out of the water with high velocities. At Tilla- mook Rock on the Oregon Coast, fragments of stones are torn from the cliffs during every severe storm and thrown on the roof of the light-keeper's house, about 100 feet above sealevel. " In December, 1894, one fragment weighing 135 lbs. was thrown clear above this building, and in falling broke a hole 20 feet square through the roof, practically wrecking the interior of the building. Thirteen panes of glass in the lantern were broken during the same storm"21. As already noted, this lantern is 132 feet above mean high water. The fog-signal siren horns, about 95 feet above the sea, were partially filled with rocks during the storm of October 18, 1912. The windows of the Dunnet Head lighthouse on the north coast of Scotland, which are over 300 feet above high-water mark, are sometimes broken by stones swept up the cliffs by waves22.
DAMAGE BY STORM WAVES • 69
It is perfectly evident that waves which are armed with cobblestones and enormous boulders must accomplish great erosive work when they beat against a cliff pr artificial wall. On the other hand, one must not make the mistake of assuming that waves which are not thus armed can accomplish but little work. It is true that large storm waves may beat against a cliff without removing the barnacles which are attached to its face, and that along the shores of saline lakes calcareous tufa may form on cliffs exposed to the impact of large waves23. But this merely indicates that the pressure of the liquid mass is so evenly distributed upon all sides of the strong shell, or of the mineral deposit, that the excess of pressure on any one side is not sufficiently great nor applied with sufficient suddenness to cause rupture. The same waves will wrench great blocks of rock from their places in the cliff face, and drive air and water into joint crevices with such force as to loosen large fragments of the cliff and thus contribute to the disintegration of the whole mass. A force which exerts a pressure of thousands of pounds to the square foot will discover lines of weakness in any natural cliff. Even though all sand, boulders, and other rock fragments were speedily carried out of the zone of wave action, and waves of pure water alone attacked the coasts, shorelines would retreat under wave erosion just as surely as they do when the waves are armed with abrasive materials, although the process would certainly go on much more slowly.
British geologists have long appreciated the tremendous power of the waves in destroying land areas, and with good cause; for no part of the British Isles is far removed from the sea, the wave attack on much of the coast is remarkably vigorous, and abundant ancient records and surveys permit careful computation of the rate of cliff retreat at many points. Old maps of Yorkshire show the location of many towns and villages which have been swept out of existence by the waves, their former sites being now re- presented by sandbanks far out in the sea. In 1829 there was in the harbor of Sheringham, according to Lyell24, a depth of 20 feet of water where only forty-eight years before had stood a cliff fifty feet high with houses upon it. For over half a century the cliff at Happisburgh retreated at the rate of 7 feet per year, while the cliff between Cromer and Mundesley was cut back 330 feet in the twenty-three years previous to 1861 making an
THE WORK OF WAVES
DAMAGE BY STORM WAVES 71
annual retreat of 14 feet. Matthews25 estimates that the rate of cliff erosion on the Holderness coast of Yorkshire varies from 7 feet per year in some places to 15 feet in others, while the retreat between Cromer and Mundesley since 1861 is said to have been 19 feet annually. At Southwold the annual rate has varied from 15 to 45 feet. Shakespeare's cliff (Plate VII) near Dover is so vigorously undermined that great landslides descend from the upper part of the cliff, the debris projecting far into the sea until the waves remove it and renew their attack on the cliff base. Such a landslide in 1810 caused a marked earth- quake at Dover. Detailed accounts of the rates of cliff erosion about the British Isles will be found in LyelFs " Principles of Geology "*, while Matthew's " Coast Erosion and Protection "v gives more recent data on this question. Further details are abundantly set forth in the reports of the Royal Commission on Coast Erosion of Great Britain28.
The large blocks of rock dislodged from cliff faces, as well as smaller fragments, are churned together by the waves so long as they remain within reach, either upon the beach slope or in shallow water. The surf zone has been likened by Shaler29 to a great mill in which angular fragments are quickly rounded and everything in course of time is reduced to the size of sand or fine silt and swept out to sea. How effective is this mill may be inferred from the fact that angular fragments of granite from quarries on Cape Ann, Massachusetts, become fairly well rounded by wave action in a single year, while under favorable circum- stances " the wear upon the pebbles amounts on the average to several inches per annum ,,ao. On a stormy day the roar of grinding masses of boulders often rises above the sullen thunder- ing of the surf.
A vivid picture of the working of the " sea mill," which grinds great boulders to sand and fine mud, is given by Henwood31 in an account of the visit made by him to a mine in southwest England which extended out under the sea: " When standing beneath the base of the cliff, and in that part of the mine where but nine feet of rock stood between us and the ocean, the heavy roll of the larger boulders, the ceaseless grinding of the pebbles, the fierce thundering of the billows, with the crackling and boiling as they rebounded, placed a tempest in its most appalling form too vividly before me to be ever forgotten. More than once
72 'HE WORK OF WAVES
doubting the protection of our rocky shield we retreated in affright; and it was only after repeated trials that we had con- fidence to pursue our investigations."
Conditions Affecting Wave Energy. — We have already seen that the dimensions of waves vary with differences in depth of water, strength and duration .of wind, and length of fetch of open water. It follows from this that the amount of wave energy delivered against a shore will vary with these same factors. A coast bordered by off-shore shallows escapes the most powerful wave attack, because large waves cannot traverse the shallow water. Other things being equal, that part of a shoreline facing the greatest stretch of open water will receive the largest amount of wave energy. But it must be remembered that the prevailing winds may come across a shorter stretch of open water, with the result that what appear to be the less exposed parts of a shore may really suffer the more vigorous attack. Account must also be taken of the fact that the prevailing wind may not be the dominant wind; for a few great storms from one direction may more than offset the effect of long-continued wave attack from the direction of the prevailing wind. It is, therefore, not always a simple matter to determine which parts of a shore will suffer most from the energy of waves. The observer must carefully consider the inclination of the off-shore slope; the depth of water both near the shore and farther out; the pres- ence or absence of shallows; the directions of the greatest stretch of open water, of the prevailing winds and of the greatest storm winds; and a number of other factors which may enter into the case; and must skilfully weigh the relative importance of each factor in a given case before he can reach a safe conclusion.
Among the factors affecting the energy with which waves attack a shore are two not previously mentioned. These are tidal currents and the angle at which the waves meet the shore- line. Whei* a wave encounters an opposing tidal current, the velocity and length of the wave are decreased, the height is increased, and the wave may break much as it would on a shel- ving beach. A swiftly moving tidal current may thus be quite as effective as a shallow in causing large waves to break before reaching the shore. The south coast of Shetland is protected from the waves of a southwest storm so long as a rapid tidal current off the coast is running, no matter how rough may be the
CONDITIONS AFFECTING WAVE ENERGY 73
74 THE WORK OF WAVES
outside sea; but as soon as the current ceases the surf breaks on the shore with great force32. On the other hand, if the current is located immediately at the shore, and especially if it flows in the direction of wave advance, the destructive power of the waves may be augmented. Stevenson is of the opinion that the violence of the surf at Whalsey and Wick in northern Scot- land is in part due to the action of strong tidal currents; and at certain other places the surf seems to be most destructive when the tidal currents are strongest33.
The angle at which the waves meet the shoreline has an im- portant effect upon the energy of wave attack. Waves are most destructive when they come in at right angles to the shoreline, and a very slight amount of obliquity materially decreases their power. Those portions of the breakwater at Wick which are assailed by waves coming " dead-on " have suffered much greater damage than other portions where the waves arrive at a slightly oblique angle34. It is therefore evident that where the direction of greatest fetch of open water makes an oblique angle with the shoreline, waves from that direction may be less destructive than waves developed on a shorter stretch of open water but approach- ing the land at right angles to the shore. It must not be sup- posed, however, that waves approaching a coast from a given direction maintain that direction until they break upon the beach. On the contrary, there is a very marked tendency for every wave to change its direction in such manner as to make its crest parallel with, and its direction of advance at right angles to the shoreline. Inasmuch as this tendency has an important effect upon the development of shorelines, we must give it some further consideration.
Wave Refraction. — When a wave {ad, Fig. 12) advances toward a coast, the direction of advance is always at right angles to the wave crest. Nearing the coast, the wave encounters shallower water off the headlands than opposite the bays; and since the velocity of shallow-water waves decreases with decreasing depth, those parts of a wave opposite headlands will lag behind the parts opposite bays, and the wave crest will begin to curve {axd}) in conformity with the curves of the shoreline. If the headlands and bays are not too pronounced, and if the shallowing of the water is not too abrupt, by the time the wave has reached the position a*<P it will have so adjusted itself as to bring its crest
WAVE REFRACTION
75
at all points nearly parallel to the shoreline. Or, as Harrison*6 has expressed it, " the velocity of the part which first reaches the shallow being lessened, the whole wave wheels round, and breaks nearly at right angles on the beach.' ' This process of " wave refraction/' as Davis has called it, accounts for the fact that swells from distant storms ordinarily are nearly parallel to
Fio. 12. — Diagram to illustrate the process of wave refraction, whereby wave attack is concentrated on headlands. (After Davis.)
the shore when they break, no matter what may have been the direction of the storm. Even within the narrow limits of a single curved beach an observer may note the tendency of the surf to break directly on shore throughout its length, although the beach may describe an arc of 90 degrees or more. ' An important consequence of wave refraction is the concen- tration of wave energy upon headlands. Since the direction
76 THE WORK OF WAVES
of wave advance is always at right angles to the crestline, and the latter becomes curved to conform with the curvature of the shoreline, it follows that a large proportion of the waves will be refracted toward the headlands. In Fig. 12 it is apparent that all that portion of the wave between a and b will be concentrated upon the short stretch of shoreline, AB, on the headland; whereas the part of the wave between b and c will be distributed over the great stretch of the bay shore, BC. In other words, wave refraction causes an enormous concentration of wave energy upon headlands and a dissipation of energy in bays. The ob- server who wishes to witness the most sublime manifestations of the power of the sea must seek the exposed headlands of the coast; while the mariner finds comparative safety within the limits of the bays, even where these are broadly open to the sea.
Waves do not always break parallel to the shore. In the first place, no wave can be refracted with sufficient abruptness to render its crest parallel to the sharp and complex irregularities of some shores. In the second place, the water is very deep close to some shores, and wave refraction does not begin t6 take place until the wave has practically reached the headlands. The wave then breaks against these projecting points of the coast first, and its remaining portions, being imperfectly re- fracted, sweep upon the shore from the headlands inward at an oblique angle. Furthermore "forced waves," or those which are still being driven forward by the wind which formed them, are not so readily refracted as " free waves " which have passed beyond the limits of the storm. It is for this reason that storm waves are more apt to strike the shore at an oblique angle than are the groundswells which arrive during calm weather. The more perfect refraction of the groundswells is due not alone to the absence of the wind's impelling force, but probably also to the fact that they extend to greater depths and hence are the sooner affected by the refracting influence of a shallowing bottom.
Depth of Wave Action. — The depth to which the ocean waters are affected by waves is a matter of much importance to all students of shore processes. We have already seen that at the depth of one wave length below the surface the water particles of oscillatory waves are moving in orbits whose di- ameters are only jj5 as great as the diameters of the orbits at
DEPTH OF WAVE ACTION 77
the surface. Since the period of the lower orbits is identical with that of the larger surface orbits, it follows that the velocity of the water particles decreases in the same proportion as the di- ameters of the orbits. In other words, the water particles at the depth of one wave length below the surface move with y^y of the velocity of the surface particles. The ability of oscil- latory waves to erode the bottom and to transport debris there- fore diminishes rapidly with increasing depth, and soon becomes negligible. In the case of waves of translation the motion of the water particles is theoretically the same from the surface to the bottom, except that the velocity near the bottom should be somewhat less than near the surface, owing to the fact that the water particles there pass through shorter, more nearly hori- zontal paths in the same length of time. With these theo- retical points in mind it will be interesting to inquire into the results obtained by different students of this phase of wave activity, and to review their opinions as to the maximum depth of efficient wave action in nature. Unfortunately, few writers distinguish between the effects of oscillatory waves and waves of translation.
Captain E. K. Calver, R. N., has observed waves which changed their color upon passing into water from 40 to 50 feet deep be- cause of their abrasive action upon the bottom36. Sir John Coode studied the movement of shingle in the vicinity of the Chesil Bank on the south coast of England, by descending to a depth of 60 or 65 feet below the surface of the sea in diving dress. He found that after a heavy storm the shingle, which was pre- viously covered with barnacles, was quite free from these shells, proving a movement of the coarse material at a depth of nearly 50 feet37. According to Hermann Fol, whose " Impressions d'un Scaphandrier '' are vividly recorded in the Revue Scienti- fique for 189088, a diver at a depth of 100 feet is tossed back and forth by the vigorous oscillatory movement of the bottom water whenever groundswells are running on the surface. Hunt quotes the testimony of pilots and masters to the effect that after a wave has broken over a vessel, sand is frequently left on the decks even when the water has a depth of 75 or 80 feet39, and describes a jar brought up in a trawl from a depth of 220 feet into which gravel the size of a hazelnut had been washed by wave agitation. Robert Stevenson states that fish disappear
THE WORK OF WAVES
DEPTH OF WAVE ACTION 79
from the fishing grounds in the North Sea during storms, due to the agitation of the water by wave action to a depth of 200 feet or more40. The same authority notes that at the Bell Rock lighthouse, off the east coast of Scotland, large stones, contain- ing upwards of 30 cubic feet and weighing two tons or more, are often thrown upon the rock from " deep water " by the waves41. Thomas Stevenson has made a very interesting com- parison between the depths at which mud reposes on the floor of different parts of the North Sea, and the vigor of wave ac- tion in those places. He finds that there is a direct relation between these two phenomena, the depth of the level at which mud accumulates increasing in much the same proportion as the violence of the waves. From this we may infer that the upper limit of mud accumulation is a measure of the maximum depth of wave disturbance in a given locality. Applying this rule to the North Sea, we find that in protected areas, as the inner parts of the Moray Firth and the Firth of Forth, and along the Holland coast in the narrow southern part of the sea, wave action reaches to a depth of 25, 50, or 100 feet; while in exposed places the disturbance is appreciable to a depth of from 300 to 500 feet or more42. According to J. N. Douglas, the fishermen off Land's End bring up stones one pound in weight, which have been washed into their lobster pots at a depth of 180 feet by the action of the ground-swell, while coarse sand is often washed from a depth of 150 feet by storm waves and hurled to the lan- tern gallery of the Bishop Rock lighthouse, 120 feet above low- water48. Kinahan reports the moving of stones weighing several hundred pounds by wave action in water from 90 to 120 feet deep on the coast of Galway44.
In contrast to the above records of significant wave action at great depths, may be mentioned a few instances of the ineffi- ciency of wave action a short distance below the surface. At the Cherbourg breakwater blocks of rubble stone 23 to 26 feet below low-water are reported by Wheeler as remaining unmoved in the roughest sea. According to the same authority, the rubble mound upon which the Alderney breakwater was later erected remained three years undisturbed by winter storms below the level of 15 feet below low-water45. Indeed, Wheeler goes to the extreme of limiting " the disturbance caused by the for- mation of waves ... to a distance below the surface about
80 THE WORK OF WAVES
equal to the height of the wave"46. At Port Elizabeth, in South sAfrica, Mr. Shield found that blocks of rubble stone, weighing from 1 to 1§ cwt. remained unmoved at a depth of 22 feet when the waves were 15 to 20 feet high47. Coode reports that in the same locality the movement of sand on the sea-bottom ceases 20 feet below the surface48. Delesse states that submarine portions of engineering structures are seldom disturbed below a depth of 16 feet in the Mediterranean, and 26 feet in the Atlantic49.
Too much importance must not be attached to the negative results just mentioned. In some of the cases we are not in possession of sufficient information regarding the degree of ex- posure of the localities in question, or of the size of the waves there generated. Engineering structures and masses of rubble stones may be so keyed together, or may have such external forms, as to receive the shock of vigorous waves without harm, while loose materials on the bottom are at the same time ma- terially affected. High waves of short length will not affect the water to as great a depth as lower waves of greater length. The positive evidence of wave disturbance at depths of several hun- dred feet is sufficient to prove that however ineffective some waves may be, other waves under favorable conditions will pro- duce an effect in deep water. W,e may take 600 feet as the limit- ing depth of ordinary wave disturbance, although Cornish sets 900 feet as the limit for the largest recorded waves50. Geikie states that ripple marks are sometimes produced (by waves) in fine sand at a depth of 600 feet, and Airy apparently attributes the breaking of groundswells in water of the same depth to interference with the bottom51. Agassiz seems to recognize the possibility of wave action off the coast of Florida to a depth of 600 feet52. More definite figures are given by Cialdi, who asserts that large waves will erode the bottom to a depth of 40 meters in the English Channel and Adriatic Sea, 50 meters in the Mediterranean Sea, and 200 meters, or about 650 feet, in the open ocean; and that at such depths the waves will put debris in motion and grind it together53. Still more convincing are the results of experiments made by Siau54 near Saint-Gilles on the Isle of Bourbon, off the coast of Madagascar. This in- genious investigator found that by sounding with a weight well coated with tallow he could determine the presence of ripple marks on the sea-bottom not only because the impression of
DEPTH OF WAVE ACTION 81
the ripples was imprinted upon the tallow surface, but also be- cause the heavy particles concentrated in the troughs and the light particles collected on the crests of the ripples adhered to the tallow in parallel bands. In this way Siau was able to prove the existence of wave-formed ripple marks, and hence of wave action, at a depth of 617 feet. In a letter to Nansen56 Sir John Murray states that great storms off the north coast of Scotland agitate fine mud at a depth of 600 feet. Murray also quotes Vionnois as authority for the statement that in the Bay of St. Jean de Luz the bottom is agitated during storms at a depth of 300 meters, or nearly 1000 feet56. Unfortunately, while these authors evidently refer to oscillatory waves in their discussions, they do not definitely exclude the possibility that waves of translation may be responsible for the deep-water movements. Nor can we be certain, in some of the cases cited, that currents may not have produced the ripple marks and other phenomena attributed to wave action.
There is no theoretical reason, however, why we should doubt the possibility of appreciable oscillatory wave action down to a depth of 600 feet. Observations with the naked eye and with the microscope convinced the Weber brothers that during the passage of oscillatory waves there is some slight mov ment of the water particles to a depth below the surface equal to 350 times the height of the waves57. Accordingly a wave 40 ^eet high should affect water particles 14,000 feet below the surface. At a depth of but 600 feet this movement must be quite pronounced, de- spite the rapid decrease in amplitude of oscillation from the surface downward, and notwithstanding that the maximum theoretical depth of wave disturbance may not ordinarily be attained in the ocean because of the long time required for the downward transmission of surface oscillations, which latter may cease or change direction before the lowest water strata are set in motion58. Assuming a groundswell with a length of 1350 feet and a height of 16 feet, which is well within the possible limits, the water particles at a depth of 600 feet (f of the wave length) would move in orbits having a diameter of 1 foot. The period of such a wave is about 16 seconds; hence the water particles at the depth indicated would oscillate with a maximum velocity of 1 foot in 5 seconds, or .06 meter per second. If at the bottom the path of oscillation were reduced to a straight
82 THE WORK OF WAVES
line 1 foot in length, the velocity for a wave of the same period would be about .04 meter per second. Such an oscillation would disturb clay, fine mud, and probably the very finest sands. Forbes has shown that fresh water moving in a shallow trough with a velocity of .077 meter per second will stir up moist brick clay59, while Sorby recently found that a current of 6 inches (.15 meters) per second would drift along common sand grains one hundredth of an inch in diameter and that " the very fine Alum-Bay sand ^fa inch in diameter " would be moved by a velocity as low as .04 meter60. According to de Lapparent a river with a bottom velocity of .15 meter per second will trans- port coarse mud61, whereas Lyell says this same velocity will move fine sand, and Hunt puts the lower limit for ordinary fine aand at .10 meter per second. The foregoing figures are based on observations in shallow fresh water. If we consider the conditions of temperature, pressure, salinity and viscosity which would exist at a depth of 600 feet in the sea, we find that sand particles of a given diameter ought to be moved by a slightly lower current velocity than in the cases cited. It seems reason- ably certain, therefore, that with an orbital diameter of 1 foot and a period of 16 seconds there would be appreciable distur- bance of the finer deposits on the sea floor. Even were the or- bital diameters as small as 1 inch and the period from 10 to 20 seconds, Cornish is of the opinion that the motion of the water would still be sufficient to hinder the deposition of the finest kinds of mud62. When associated with slow-moving tidal or other currents, a very gentle oscillatory movement of the water due to wave action may produce an important effect.
In the words of Cornish, " We may say with confidence, as a theoretical inference, that the agitation of wind-formed waves affects the bottom of the sea as far as the edge of the continental platform to such an extent as (in co-operation with tidal and other currents) to keep very fine mud moving about until it has an opportunity of subsiding over the edge of the continental shelf ,,fi3. On the other hand, it is evident that only the finest material will be affected at such depths, and that erosion of the bottom will be almost imperceptibly slight, so long as oscillatory waves alone disturb the water. Waves of translation are not as common in such deep water as nearer shore, but whenever they do occur we should expect, on theoretical grounds, a velocity of
REFERENCES 83
the bottom water comparable to that at the surface, and there- fore capable of effecting noteworthy erosion and transportation. A sufficient body of observed facts to establish this theory is not available. We are reasonably sure, however, that on exposed coasts the sea-bottom is not wholly free from some kind of wave agitation down to a depth of 600 feet at least.
R£SUMfi
We have inquired into the origin and character of the energy developed by waves, and have gained some idea of the tremendous power which they may exercise under favorable conditions. It has been seen that natural shores, as well as arti- ficial structures, must suffer severely from wave attack. The conditions affecting the vigor of wave action at the shore have briefly been discussed, and the effects of wave refraction con- sidered more fully. An inquiry as to the depth of wave action has resulted in the conclusion that the sea-bottom is affected by waves to the edge of the continental shelf, or approximately to a depth of 600 feet.
But waves are not the only forces of nature which expend their energy upon shores. Currents of various types play an important r61e in modelling shore forms, and must therefore receive our attention before we proceed to a study of the evolution of shorelines under the combined influence of waves and currents.
REFERENCES
1. Ekman, V. W. On Dead Water. The Norwegian North Polar Expe-
dition, 1893-1896, Scientific Results. V, No. 15, p. 33, Christiania, 1906. Fleming, J. A. Waves and Ripples in Water, Air, and iEther, p. 68, London, 1902.
2. Gaillard, D. D. Wave Action in Relation to Engineering Structures.
Corps of Engineers U. S. Army, Professional Paper No. 31, pp. 40, 46, Washington, 1904.
3. Ibid., pp. 40, 45, 47, 50.
4. Ibid., p. 51.
5. Hagen, G. Handbuch der Wasserbaukunst. 3. Teil. Das Meer. I.
Band, p. i97, Berlin, 1863.
6. Stevenson, Thomas. The Design and Construction of Harbours.
3rd Edition, p. 98, Edinburgh, 1886.
7. Ibid., pp. 61-62.
84 THE WORK OF WAVES
8. Stevenson, Thomas. Account of Experiments upon the Force of the Waves of the Atlantic and German Oceans. Trans. Roy. Soc.Edin. XVI, 23-32, 1849. [ 9. Gaillard, D. D. Wave Action in Relation to Engineering Structures. Corps of Engineers U. S. Army, Professional Paper No. 31, pp. 161-171, Washington, 1904.
10. Ibid., pp. 164, 167.
11. Ibid., pp. 194-211.
12. Ibid., p. 266.
13. Stevenson, Thomas. Account of Experiments upon the Force of the
Waves of the Atlantic and German Oceans. Trans. Roy. Soc. Edin. XVI, 25, 1849. Stevenson, Thomas. The Design and Construction of Harbours. 3rd Edition, pp. 55-57, Edinburgh, 1886.
14. Stevenson, Thoma . The Design and Construction of Harbours.
3rd Edition, p. 47, Edinburgh, 1886.
15. Ibid., pp. 49-52.
16. Meunier, Stanislas. La Geologie Experimental, p. 86, Paris, 1899.
17. Shield, William. Principles and Practice of Harbor Construction,
p. 81, London, 1895.
18. Gaillard, D. D. Wave Action in Relation to Engineering Structures,
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 126, Washington, 1904.
19. Geike, A. Textbook of Geology. 4th Edition, I, 568, London, 1903.
20. Playfair, John. Illustrations of the Huttonian Theory of the Earth,
p. 101, Edinburgh, 1802.
21. Gaillard, D. D. Wave Action in Relation to Engineering Structures,
Corps of Engineers U. S. Army, Professional Paper No. 31, p. 128, Washington, 1904.
22. Geikie, A. Textbook of Geology. 4th Edition, I, 561, 1903.
23. Gilbert, G. K. The Topographic Features of Lake Shores. U. S.
Geological Survey. 5th Ann. Report, p. 81, 1885.
24. Lyell, Charles. Principles of Geology. 11th Edition, I, p. 517,
New York, 1873.
25. Matthews, E. R. Coast Erosion and Protection, p. 11, London.
1913.
26. Lyell, Charles. Principles of Geology. 1 1th Edition, I, 671 pp., New
York, 1873.
27. Matthews, E. R. Coast Erosion and Protection. 147 pp., London,
1913.
28. Royal Commission on Coast Erosion. Minutes of Evidence. Reports
of the Commission. I, Part 2, 1-504, 1907.
29. Shaler, N. S. Beaches and Tidal Marshes of the Atlantic Coast.
National Geographic Monograph, I, 143, 1895.
30. Shaler, N. S. The Geology of Cape Ann, Massachusetts. U. S. Geo-
logical Survey, 9th Ann Report, p. 565, 1889.
31. Henwood, W. J. On the Metalliferous Deposits of Cornwall and
Devon. Trans. Geol. Soc. Cornwall. V, 11, 1843.
REFERENCES 85
32. Stevenson, Thomas. The Design and Construction of Harbours.
3rd Edition, p. 64, Edinburgh, 1886.
33. Ibid., pp. 70-72.
34. Ibid., pp. 35-37, 72.
35. Harrison, J. T. Observations on the Causes that are in Constant
Operation Tending to Alter the Outline of the English Coast, to Affect the Entrances of the Rivers and Harbours, and to Form Shoals and Deeps in the Bed of the Sea. Min. Proc. Inst. Civ. Eng., VII, 343, 1848.
36. Stevenson, Thomas. The Design and Construction of Harbours.
3rd Edition, p. 20, Edinburgh, 1886.
37. Coode, John. Description of the Chesil Bank, with Remarks upon
its Origin, the Causes which have Contributed to its Formation, and upon the Movement of Shingle generally. Min. Proc. Inst. Civ. Eng., XII, 534, 1853.
38. Fol, Hermann. Les Impressions d'un Scaphandrier. Revue Scien-
tifique, XLV, 715, 1890.
39. Hunt, A. R. On the Formation of Ripplemark. Proc. Roy. Soc.
London. XXXIV, pp. 9, 15, 1882.
40. Stevenson, Robert. On the Bed of the German Ocean, or North
Sea. Memoirs Wcrnerian Nat. Hist. Soc. Trans., Ill, 332, 1821.
41. Ibid., p. 332.
42. Stevenson, Thomas. The Design and Construction of Harbours. 3rd
Edition, pp. 21-25, Edinburgh, 1886.
43. Douglas, J. N. [On the depth of wave action.] Min. Proc. Inst. Civ.
Engineers. XL, 103, 1875.
44. Kinahan, G. H. The Travelling of Sea Beaches. Min. Proc. Inst. Civ.
Eng., LVIII, 284, 1879.
45. Wheeler, W. H. A Practical Manual of Tides and Waves, p. 119,
London, 1906.
46. Iltid., p. 118.
47. Ibid., p. 119.
48. Coode, John. [On the depth of wave action.] Min. Proc. Inst. Civ.
Eng., LXX, 45, 1882.
49. Delesse, M. Lithologie des Mers de France, p. 110, Paris, 1872.
50. Cornish, Vaughan. On Sea Beaches and Sand Banks. Geog. Jour.,
XI, 531, London, 1898.
51. Geikie, A.