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Title: The Tides and Kindred Phenomena in the Solar SystemThe Substance of Lectures Delivered in 1897 at the LowellInstitute, Boston, Massachusetts

Author: Sir George Howard Darwin

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THE TIDESAND KINDRED PHENOMENA IN THE

SOLAR SYSTEM

THE SUBSTANCE OF LECTURES DELIVERED

IN 1897 AT THE LOWELL INSTITUTE,

BOSTON, MASSACHUSETTS

BY

GEORGE HOWARD DARWINPLUMIAN PROFESSOR AND FELLOW OF TRINITY COLLEGE IN THE

UNIVERSITY OF CAMBRIDGE

BOSTON AND NEW YORK

HOUGHTON, MIFFLIN AND COMPANYT Riffirsi Pre&, Cambridge

1899


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COPYRIGHT, 1898, BY GEORGE HOWARD DARWINALL RIGHTS RESERVED


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PREFACE

In 1897 I delivered a course of lectures on the Tides atthe Lowell Institute in Boston, Massachusetts, and this bookcontains the substance of what I then said. The personalform of address appropriate to a lecture is, I think, apt to berather tiresome in a book, and I have therefore taken pains toeliminate all traces of the lecture from what I have written.

A mathematical argument is, after all, only organizedcommon sense, and it is well that men of science shouldnot always expound their work to the few behind a veil oftechnical language, but should from time to time explain toa larger public the reasoning which lies behind their mathe-matical notation. To a man unversed in popular exposition it

needs a great effort to shell away the apparatus of investiga-tion and the technical mode of speech from the thing behindt, and I owe a debt of gratitude to Mr. Lowell, trustee of

the Institute, for having afforded me the occasion for makingthat effort.

It is not unlikely that the first remark of many who see mytitle will be that so small a subject as the Tides cannot de-mand a whole volume; but, in fact, the subject branches outn so many directions that the difficulty has been to attain

to the requisite compression of my matter. Many popularworks on astronomy devote a few pages to the Tides, but, as


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PREFACE iv

far as I know, none of these books contain explanations ofthe practical methods of observing and predicting the Tides,

or give any details as to the degree of success attained bytidal predictions. If these matters are of interest, I invitemy readers not to confine their reading to this preface. Theater chapters of this book are devoted to the consideration

of several branches of speculative Astronomy, with which thetheory of the Tides has an intimate relationship. The prob-

ems involved in the origin and history of the solar and ofother celestial systems have little bearing upon our life onthe earth, yet these questions can hardly fail to be of inter-est to all those whose minds are in any degree permeated bythe scientific spirit.

I think that there are many who would like to understandthe Tides, and will make the attempt to do so provided theexposition be sufficiently simple and clear; it is to such read-ers I address this volume. It is for them to say how far Ihave succeeded in rendering these intricate subjects interest-ng and intelligible, but if I have failed it has not been for

ack of pains.The figures and diagrams have, for the most part, been

made by Mr. Edwin Wilson of Cambridge, but I have toacknowledge the courtesy of the proprietors of Harpers, theCentury, and the Atlantic Monthly magazines, in supplyingme with some important illustrations.


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PREFACE v

A considerable portion of Chapter III. on the Bore isto appear as an article in the Century Magazine for October

1898, and the reproductions of Captain Moores photographsof the Bore in the Tsien-Tang-Kiang have been preparedfor that article. The Century has also kindly furnished theblock of Dr. Isaac Robertss remarkable photograph of thegreat nebula in the constellation of Andromeda; it originallyappeared in an article on Meteorites in the number for Octo-

ber, 1890. The greater portion of the text and the whole ofthe illustrations ofChapter XX. were originally published inHarpers Magazine for June, 1889. Lastly, portions of Chap-ters XV. and XVI. appeared in the Atlantic Monthly forApril, 1898, published by Messrs. Houghton, Mifflin & Co.,who also make themselves responsible for the publication ofthe American edition of this book.

In conclusion, I wish to take this opportunity of thankingmy American audience for the cordiality of their reception,and my many friends across the Atlantic for their abundanthospitality and kindness.

G. H. DARWIN.Cambridge, August, 1898.


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CONTENTS

CHAPTER I

TIDES AND METHODS OF OBSERVATION

Definition of tide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Oceanic tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 4Methods of observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Tide-gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610Tide-curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Site for tide-gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13rregularities in tide-curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Authorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

CHAPTER II

SEICHES IN LAKES

Meaning of seiche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Uses of scientific apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18F o r e l s p l e m y r a m e t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19, 20Records of the level of the lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

nterpreta tio n o f reco rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2223Limnimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Mode of oscillation in seiches . . . . . . . . . . . . . . . . . . . . . . . . . . . 2629Wave motion in deep and in shallow water . . . . . . . . . . . . . 3033


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CONTENTS vii

Composition of waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3337Periods of seiches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38, 39

C a u s e s o f s e i c h e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40, 41Vibrations due to wind and to steamers . . . . . . . . . . . . . . . . 4248Aerial waves and their action on lakes and on the sea . . 4954A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54, 55

CHAPTER III

TIDES IN RIVERSTIDE MILLS

Definition of ebb and flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Tidal currents in rivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5759Progressive change of wave in shallow water . . . . . . . . . . . . 59, 60The bore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Captain Moores survey of the Tsien-Tang-Kiang . . . . . . . 6165Diagram of water-levels during the flow . . . . . . . . . . . . . . . . 65, 66Chinese superstition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6972Pictures of the bore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Other cases of bores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Causes of the bore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73T i d a l e n e r g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74, 75Tide mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75, 76Authorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76


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CONTENTS viii

CHAPTER IV

HISTORICAL SKETCH

Theories of the Chinese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77, 78Theories of the Arabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7880Theories of the Norsemen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80, 81Writings of Posidonius and Strabo . . . . . . . . . . . . . . . . . . . . . 8185Seleucus the Babylonian on the diurnal inequality . . . . . . 85, 86

Galileo and Kepler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Newton and his successors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8789Empirical method of tidal prediction ................... 8991Authorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

CHAPTER V

TIDE-GENERATING FORCE

nertia and centrifugal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9294Orbital motion of earth and moon . . . . . . . . . . . . . . . . . . . . . 9496T i d e - g e n e r a t i n g f o r c e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97101

Law of its dependence on the moons distance . . . . . . . 102104E a r t h s r o t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104, 105

Second explanation of tide-generating force . . . . . . . . . . 105, 106Horizontal tide-generating force . . . . . . . . . . . . . . . . . . . . . . 106, 107Successive changes of force in the course of a day . . . . 108, 110


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CONTENTS ix

A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

CHAPTER VIDEFLECTION OF THE VERTICAL

Deflection of a pendulum by horizontal tidal force . . . 111114Path pursued by a pendulum under tidal force . . . . . . . 114115Object of measuring the deflection of a pendulum . . . . 115118

Attempt to measure deflection by bifilar pendulum . . 118128Microsisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128130The microphone as a seismological instrument . . . . . . . 130133Paschwitzs work with the horizontal pendulum . . . . . . 134, 135Supposed measurement of the lunar deflection of gravity . . 135A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

CHAPTER VII

THE ELASTIC DISTORTION OF THE EARTHS SURFACE BYVARYING LOADS

Distortion of an elastic surface by superincumbent

l o a d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137140Application of the theory to the earth . . . . . . . . . . . . . . . 140, 141Effects of tidal load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141143Probable deflections at various distances from the

coast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143147


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CONTENTS x

Deflections observed by M. dAbbadie and byDr. Paschwitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147148

Effects of atmospheric pressure on the earths surface 148150A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

CHAPTER VIII

EQUILIBRIUM THEORY OF TIDES

Explanation of the figure of equilibrium .............. 152154Map of equilibrium tide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154155Tides according to the equilibrium theory ............ 155159Solar tidal force compared with lunar . . . . . . . . . . . . . . . . 159161

Composition of lunar and solar tides . . . . . . . . . . . . . . . . . 161, 162

Points of disagreement between theory and fact . . . . . . 162165A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

CHAPTER IX

DYNAMICAL THEORY OF THE TIDE WAVE

Free and forced waves in an equatorial canal ......... 166168Critical depth of canal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168170General principle as to free and forced oscillations . . . 170177nverted and direct oscillation . . . . . . . . . . . . . . . . . . . . . . . 175, 176

Canal in latitude 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177, 178


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CONTENTS xi

Tides where the planet is partitioned into canals .......... 178Removal of partitions; vortical motion of the water . . 179, 180

Critical latitude where the rise and fall vanish . . . . . . . 180, 181D i u r n a l i n e q u a l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181183A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

CHAPTER X

TIDES IN LAKESCOTIDAL CHART

The tide in a lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185188The Mediterranean Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 189Derived tide of the Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . 189191Cotidal chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191195A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

CHAPTER XI

HARMONIC ANALYSIS OF THE TIDE

Tide in actual oceans due to single equatorial satellite 195198

Substitution of ideal satellites for the moon . . . . . . . . . . 199201Partial tide due to each ideal satellite . . . . . . . . . . . . . . . . 201203Three groups of partial tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Semidiurnal group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203206Diurnal group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206208M e t e o r o l o g i c a l t i d e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208, 209


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CONTENTS xii

Shallow water tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209212A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

CHAPTER XII

REDUCTION OF TIDAL OBSERVATIONS

Method of singling out a single partial tide . . . . . . . . . . 212215

Variety of plans adopted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215218Tidal abacus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218222A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

CHAPTER XIII

TIDE TABLES

Definition of special and general tables . . . . . . . . . . . . . . . . . . . 223Reference to moons transit . . . . . . . . . . . . . . . . . . . . . . . . . . 224, 225Examples at Portsmouth and at Aden . . . . . . . . . . . . . . . 225231General inadequacy of tidal information . . . . . . . . . . . . . . . . . . 232

Method of calculating tide tables .................... 232235Tide-predicting machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235243A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243


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CONTENTS xiii

CHAPTER XIV

THE DEGREE OF ACCURACY OF TIDAL PREDICTION

Effects of wind and barometric pressure . . . . . . . . . . . . . . 244, 245E r r o r s a t P o r t s m o u t h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245, 246Errors at Aden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247252A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

CHAPTER XV

CHANDLERS NUTATIONTHE RIGIDITY OF THE EARTH

Nutation of the earth and variation of latitude . . . . . . . 254257Elasticity of the earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257, 258Tide due to variation of latitude . . . . . . . . . . . . . . . . . . . . . 258, 259Rigidity of the earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259263Transmission of earthquake shocks ................... 264, 265A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265, 266

CHAPTER XVITIDAL FRICTION

Friction retards the tide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266269Retardation of planetary rotation . . . . . . . . . . . . . . . . . . . . 269271Reaction on the satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271274


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CONTENTS xiv

Ancient eclipses of the sun . . . . . . . . . . . . . . . . . . . . . . . . . . . 274, 275Law of variation of tidal friction with moons distance 275277

Transformations of the month and of the day . . . . . . . . 277282nitial and final conditions of motion . . . . . . . . . . . . . . . . . 282, 283Genesis of the moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283287Minimum time requisite for the evolution . . . . . . . . . . . . 287, 288Rotation of the moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288, 289The month ultimately to be shorter than the day . . . . 289291

CHAPTER XVII

TIDAL FRICTION (continued)

Discovery of the Martian satellites ................... 291299Rotation of Mercury, of Venus, and of the Jovian

s a t e l l i t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299, 300Adaptation of the earths figure to changed rotation . 300303Ellipticity of the internal strata of the earth . . . . . . . . . 303305Geological evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305307Distortion of a plastic planet and trend of continents 307309

Obliquity of the ecliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309313Eccentricity of lunar orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 313, 314Eccentricity of the orbits of double stars .................. 314Plane of the lunar orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314, 315S h o r t s u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315, 316A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316


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CONTENTS xv

CHAPTER XVIII

THE FIGURES OF EQUILIBRIUM OF A ROTATING MASS OF

LIQUID

P l a t e a u s e x p e r i m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317320Stability of a celestial sphere of liquid . . . . . . . . . . . . . . . . 320322The two ellipsoids of Maclaurin and that of Jacobi . . . 322324

Transitions with change of rotation . . . . . . . . . . . . . . . . . . 324, 325Coalescence of Jacobis with Maclaurins ellipsoid . . . . 325327Poincares law of stability and coalescence . . . . . . . . . . . 327, 328Poincares pear-shaped figure . . . . . . . . . . . . . . . . . . . . . . . . 328, 329Hour-glass figure of equilibrium . . . . . . . . . . . . . . . . . . . . . . 329333Figures of planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333, 334

A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

CHAPTER XIX

THE EVOLUTION OF CELESTIAL SYSTEMS

The Nebular Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336340Nebula in Andromeda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340, 341Distribution of satellites in the solar system .......... 341344Genesis of celestial bodies by fission . . . . . . . . . . . . . . . . . . . . . . 345Dr. Sees speculations as to systems of double stars . . 345347Diversity of celestial bodies . . . . . . . . . . . . . . . . . . . . . . . . . . 347349


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CONTENTS xvi

A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

CHAPTER XXSATURNS RINGS

Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350, 351Discovery of Saturns rings . . . . . . . . . . . . . . . . . . . . . . . . . . 351355Diagram of the rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356359

Roches investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359, 360Roches limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361363The limit for the several planets . . . . . . . . . . . . . . . . . . . . . 363, 364Meteoric constitution of Saturns rings ............... 364, 365Maxwells investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365370Spectroscopic examination of the rings . . . . . . . . . . . . . . . 370372A u t h o r i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372


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LIST OF ILLUSTRATIONS

1. Well for Tide-Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72. Pipe of Tide-Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3. Indian Tide-Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4. Leges Tide-Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5. Bombay Tide-Curve from Noon, April 22, toNoon, April 30, 1884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6. Sites for a Tide-Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7. Plemyrameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8, 9. Records of Seiches at Evian . . . . . . . . . . . . . . . . . . . . 24

10. Map of Lake of Geneva . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11. Wave in Deep Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112. Wave in Shallow Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13. Simple Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Composition of Two Equal and Opposite Waves 36

14. Vibrations due to Steamers . . . . . . . . . . . . . . . . . . . . . . . 47

15. Progressive Change of a Wave in ShallowWater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

16. Chart of the Estuary of the Tsien-Tang-Kiang 63

17. Bore-Shelter on the Tsien-Tang-Kiang . . . . . . . . . . 66

18. Diagram of the Flow of the Tide on theTsien-Tang-Kiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


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LIST OF ILLUSTRATIONS xviii

19. Pictures of the Bore on the Tsien-Tang-Kiang 71

20. Earth and Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

21. Revolution of a Body without Rotation . . . . . . . 10022. Tide-generating Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

23. Horizontal Tide-generating Force . . . . . . . . . . . . . . 109

24. Deflection of a Pendulum; the Moon andObserver on the Equator . . . . . . . . . . . . . . . . . . . . . 114

25. Deflection of a Pendulum; the Moon in N.Declination 15, the Observer in N.

Latitude 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

26. Bifilar Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

27. Form of Dimple in an Elastic Surface . . . . . . . . . . 139

28. Distortion of Land and Sea-Bed by Tidal Load 14329. Chart of Equilibrium Tides . . . . . . . . . . . . . . . . . . . . . . 156

30. Forced Oscillations of a Pendulum . . . . . . . . . . . . . 175

31. The Tide in a Lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

32. Chart of Cotidal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

33. Tidal Abacus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

34. Curves of Intervals and Heights atPortsmouth and at Aden . . . . . . . . . . . . . . . . . . . . . . 230

35. Diagram of Tide-predicting Instrument . . . . . . . . 238

36. Frictionally Retarded Tide . . . . . . . . . . . . . . . . . . . . . 269


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LIST OF ILLUSTRATIONS xix

37. Maclaurins and Jacobis Ellipsoids ofEquilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

38. Figures of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

39. Hour-glass Figure of Equilibrium . . . . . . . . . . . . . . . 331

40. Nebula in Andromeda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

41. The Planet Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

42. Diagram of Saturn and his Rings . . . . . . . . . . . . . . . . 357

43. Roches Figure of a Satellite when elongatedto the utmost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361


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THE TIDES

CHAPTER I

TIDES AND METHODS OF OBSERVATION

The great wave caused by an earthquake is often de-scribed in the newspapers as a tidal wave, and the same

name is not unfrequently applied to such a short series ofenormous waves as is occasionally encountered by a ship inthe open sea. We must of course use our language in the man-ner which is most convenient, but as in this connection theadjective tidal implies simply greatness and uncommon-ness, the use of the term in such a sense cannot be regarded

as appropriate.The word tidal should, I think, only be used when we

are referring to regular and persistent alternations of rise andfall of sea-level. Even in this case the term may perhaps beused in too wide a sense, for in many places there is a regularalternation of the wind, which blows in-shore during the dayand out during the night with approximate regularity, andsuch breezes alternately raise and depress the sea-level, andthus produce a sort of tide. Then in the Tropics there is aregularly alternating, though small, periodicity in the pres-sure of the atmosphere, which is betrayed by an oscillation in

1


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CHAPTER I 2

the height of the barometer. Now the ocean will respond tothe atmospheric pressure, so that the sea-level will fall with

a rising barometer, and rise with a falling barometer. Thusa regularly periodic rise and fall of the sea-level must resultfrom this cause also. Again, the melting of the snows ingreat mountain ranges, and the annual variability in rainfalland evaporation, produce approximately periodic changes ofevel in the estuaries of rivers, and although the period of

these changes is very long, extending as they do over thewhole year, yet from their periodicity they partake of thetidal character.

These changes of water level are not, however, tides inthe proper sense of the term, and a true tide can only beadequately defined by reference to the causes which producet. A tide, in fact, means a rising and falling of the water of

the ocean caused by the attractions of the sun and moon.Although true tides are due to astronomical causes, yet

the effects of regularly periodic winds, variation of atmo-spheric pressure, and rainfall are so closely interlaced with

the true tide that in actual observation of the sea it is neces-sary to consider them both together. It is accordingly prac-tically convenient to speak of any regular alternation of sea-evel, due to the wind and to the other influences to which

I have referred, as a Meteorological Tide. The addition ofthe adjective meteorological justifies the use of the term


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TIDES AND METHODS OF OBSERVATION 3

tide in this connection.We live at the bottom of an immense sea of air, and if the

attractions of the sun and moon affect the ocean, they mustalso affect the air. This effect will be shown by a regular riseand fall in the height of the barometer. Although such aneffect is undoubtedly very small, yet it is measurable. Thedaily heating of the air by the sun, and its cooling at night,produce marked alternations in the atmospheric pressure

and this effect may by analogy be called an atmosphericmeteorological tide.

The attractions of the moon and sun must certainly actnot only on the sea, but also on the solid earth; and, since theearth is not perfectly rigid or stiff, they must produce an al-ternating change in its shape. Even if the earth is now so stiffthat the changes in its shape escape detection through theirminuteness, yet such changes of shape must exist. Theres much evidence to show that in the early stages of their

histories the planets consisted largely or entirely of moltenrock, which must have yielded to tidal influences. I shall

then, extend the term tide so as to include such alternat-ng deformations of a solid and elastic, or of a molten and

plastic, globe. These corporeal tides will be found to leadus on to some far-reaching astronomical speculations. Thetide, in the sense which I have attributed to the term, coversa wide field of inquiry, and forms the subject of the present


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CHAPTER I 4

volume.I now turn to the simplest and best known form of tidal

phenomena. When we are at the seashore, or on an estuary,we see that the water rises and falls nearly twice a day. Tobe more exact, the average interval from one high water tothe next is twelve hours twenty-five minutes, and so highwater falls later, according to the clock, by twice twenty-five minutes, or by fifty minutes, on each successive day

Thus if high water falls to-day at noon, it will occur to-morrow at ten minutes to one. Before proceeding, it maybe well to remark that I use high water and low water astechnical terms. In common parlance the level of water maybe called high or low, according as whether it is higher orower than usual. But when the level varies periodically

there are certain moments when it is highest and lowest,and these will be referred to as the times of high and lowwater, or of high and of low tide. In the same way I shallspeak of the heights at high and low water, as denoting thewater-level at the moments in question.

The most elementary observations would show that thetime of high water has an intimate relationship to the moonsposition. The moon, in fact, passes the meridian on the aver-age fifty minutes later on each succeeding day, so that if highwater occurs so many hours after the moon is due south onany day, it will occur on any other day about the same num-


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ber of hours after the moon was south. This rule is far frombeing exact, for it would be found that the interval from the

moons passage to high water differs considerably accordingto the age of the moon. I shall not, however, attempt toexplain at present how this rough rule as to the time of highwater must be qualified, so as to convert it into an accuratestatement.

But it is not only the hour of high water which changes

from day to day, for the height to which the water rises variesso conspicuously that the fact could not escape the notice ofeven the most casual observer. It would have been necessaryto consult a clock to discover the law by which the hour ofhigh water changes from day to day; but at the seashore itwould be impossible to avoid noticing that some rocks orshoals which are continuously covered by the sea at one partof a fortnight are laid bare at others. It is, in fact, aboutfull and new moon that the range from low to high water isgreatest, and at the moons first or third quarter that therange is least. The greater tides are called springs, and

the smaller neaps.The currents produced in the sea by tides are often very

complicated where the open sea is broken by islands andheadlands, and the knowledge of tidal currents at each places only to be gained by the practical experience of the pi-ot. Indeed, in the language of sailors, the word tide is


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CHAPTER I 6

not unfrequently used as meaning tidal current, without ref-erence to rise and fall. These currents are often of great

violence, and vary from hour to hour as the water rises andfalls, so that the pilot requires to know how the water standsn-shore in order to avail himself of his practical knowledge

of how the currents will make in each place. A tide tables then of much use, even at places where the access to a

harbor is not obstructed by a bar or shoal. It is, of course

still more important for ships to have a correct forecast ofthe tides where the entrance to the harbor is shallow.

I have now sketched in rough outline some of the pecu-iarities of the tides, and it will have become clear that the

subject is a complicated one, not to be unraveled withoutregular observation. I shall, therefore, explain how tides areobserved scientifically, and how the facts are collected uponwhich the scientific treatment of the tides is based.

The rise and fall of the sea may, of course, be roughlyestimated by observing the height of the water on posts orat jetties, which jut out into moderately deep water. But

as the sea is continually disturbed by waves, observations ofthis kind are not susceptible of accuracy, and for scientificpurposes more elaborate apparatus is required. The exactheight of the water can only be observed in a place to whichthe sea has a moderately free access, but where the channels so narrow as to prevent the waves from sensibly disturbing


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Fig. 1.Well for Tide-Gauge


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CHAPTER I 8

the level of the water. This result is obtained in a consider-able variety of ways, but one of them may be described as

typical of all.A well (fig. 1) about two feet in diameter is dug to a depthof several feet below the lowest tide and in the neighborhoodof deep water. The well is lined with iron, and a two-inchron pipe runs into the well very near its bottom, and passes

down the shore to the low-water line. Here it is joined to a

flexible pipe running out into deep water, and ending witha large rose pierced with many holes, like that of a wateringcan. The rose (fig. 2) is anchored to the bottom of the sea,and is suspended by means of a buoy, so as to be clear of thebottom. The tidal water can thus enter pretty freely into thewell, but the passage is so narrow that the wave motion isnot transmitted into the well. Inside the well there floats awater-tight copper cylinder, weighted at the bottom so thatt floats upright, and counter-poised so that it only just keepsts top clear of the water. To the top of the float there is

fastened a copper tape or wire, which runs up to the top of

the well and there passes round a wheel. Thus as the waterrises and falls this wheel turns backwards and forwards.

It is hardly necessary to describe in detail the simplemechanism by which the turning of this wheel causes a pen-cil to move backwards and forwards in a straight line. Themechanism is, however, such that the pencil moves horizon-


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tally backwards and forwards by exactly the same amountas the water rises or falls in the well; or, if the rise and fall

Fig. 2.Pipe of Tide-Gauge

of the tide is considerable, the pencil only moves by half asmuch, or one third, or even one tenth as much as the water.At each place a scale of reduction is so chosen as to bringthe range of motion of the pencil within convenient limits.


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CHAPTER I 10

We thus have a pencil which will draw the rise and fall ofthe tide on the desired scale.

It remains to show how the times of the rise and fall arendicated. The end of the pencil touches a sheet of paperwhich is wrapped round a drum about five feet long andtwenty-four inches in circumference. If the drum were keptstill the pencil would simply draw a straight line to and froalong the length of the drum as the water rises and falls

But the drum is kept turning by clockwork, so that it makesexactly one revolution in twenty-four hours. Since the drums twenty-four inches round, each inch of circumference cor-

responds to one hour. If the water were at rest the pencilwould simply draw a circle round the paper, and the begin-ning and ending of the line would join, whilst if the drumremained still and the water moved, the pencil would draw astraight line along the length of the cylinder; but when bothdrum and water are in motion, the pencil draws a curve onthe cylinder from which the height of water may be read offat any time in each day and night. At the end of twenty-four

hours the pencil has returned to the same part of the paperfrom which it started, and it might be thought that therewould be risk of confusion between the tides of to-day andthose of yesterday. But since to-day the tides happen aboutthree quarters of an hour later than yesterday, it is foundthat the lines keep clear of one another, and, in fact, it is


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Fig. 3.Indian Tide-Gauge

usual to allow the drum to run for a fortnight before chang-ng the paper, and when the old sheet is unwrapped from thedrum, there is written on it a tidal record for a fortnight.

The instrument which I have described is called a tide-gauge, and the paper a tide-curve. As I have already said,tide-gauges may differ in many details, but this description

will serve as typical of all. Another form of tide-gauge isshown in fig. 4; here a continuous sheet of paper is placed overthe drum, so that there is no crossing of the curves, as in thefirst example. Yet another form, designed by Lord Kelvins shown on p. 170 of vol. iii. of his Popular Lectures.


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CHAPTER I 12

Fig. 4.Leges Tide-Gauge


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The actual record for a week is exhibited in fig. 5, on areduced scale. This tide-curve was drawn at Bombay by a

tide-gauge of the pattern first described. When the paperwas wrapped on the drum, the right edge was joined to theeft, and now that it is unwrapped the curve must be followed

out of the paper on the left and into it again on the right.The figure shows that spring tide occurred on April 26, 1884the preceding neap tide was on the 18th, and is not shown

It may be noticed that the law of the tide is conspicuouslydifferent from that which holds good on the coast of England,for the two successive high or low waters which occur onany day have very different heights. Thus, for example, onApril 26 low water occurred at 5.50 p.m., and the waterfell to 5 ft. 2 in., whereas the next low water, occurring at5.45 a.m. of the 27th, fell to 1 ft. 3 in., the heights beingn both cases measured from a certain datum. When we

come to consider the theory of the tides the nature of thisrregularity will be examined.

The position near the seashore to be chosen for the erec-

tion of the tide-gauge is a matter of much importance. Thechoice of a site is generally limited by nature, for it should benear the open sea, should be sheltered from heavy weatherand deep water must be close at hand even at low tide.

In the sketch map shown in fig. 6 a site such as a is agood one when the prevailing wind blows in the direction


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CHAPTER I 14

of the arrow. A position such as b, although well shelteredfrom heavy seas, is not so good, because it is found that tide-

curves drawn atb

would be much zigzagged. These zigzagsappear in the Bombay curves, although at Bombay they areusually very smooth ones.

These irregularities in the tide-curve are not due to tides,and as the object of the observation is to determine the na-ture of the tides it is desirable to choose a site for the gauge

where the zigzags shall not be troublesome; but it is not al-ways easy to foresee the places which will furnish smoothtide-curves.

Most of us have probably at some time or other made ascratch on the sand by the seashore, and watched the waterrise over it. We generally make our mark on the sand at thefurthest point, where the wash of a rather large wave hasbrought up the water. For perhaps five or ten minutes nowave brings the water up as far as the mark, and one be-gins to think that it was really an extraordinarily large wavewhich was marked, although it did not seem so at the time

Then a wave brings up the water far over the mark, and im-mediately all the waves submerge it. This little observationsimply points to the fact that the tide is apt to rise by jerksand it is this irregularity of rise and fall which marks thenotches in the tide-curves to which I have drawn attention.

Now in scientific matters it is well to follow up the clues


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Fig. 5.Bombay Tide-Curve from Noon, April 22, to Noon

April 30, 1884


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CHAPTER I 16

Fig. 6.Sites for a Tide-Gauge

afforded by such apparently insignificant facts as this. An

nteresting light is indeed thrown on the origin of thesenotches on tide-curves by an investigation, not very directlyconnected with tidal observation, on which I shall make adigression in the following chapter.

Authorities.

Bairds Manual for Tidal Observations (Taylor & Francis1886). Price 7s. 6d. Figs. 1, 2, 3, 6 are reproduced from thiswork.

The second form of tide-gauge shown in fig. 4 is made by


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Messrs. Lege, and is reproduced from a woodcut kindly providedby them.

Sir William Thomsons (Lord Kelvins) Popular Lectures andAddresses, vol. iii. (Macmillan, 1891).


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CHAPTER II

SEICHES IN LAKES

It has been known for nearly three centuries that thewater of the Lake of Geneva is apt to rise and fall by a fewnches, sometimes irregularly and sometimes with more oress regularity; and the same sort of oscillation has been ob-

served in other Swiss lakes. These quasi-tides, called seicheswere until recently supposed only to occur in stormy weather,but it is now known that small seiches are of almost dailyoccurrence.1

Observations were made by Vaucher in the last centuryon the oscillations of the Lake of Geneva, and he gave an

account of a celebrated seiche in the year 1600, when thewater oscillated through three or four feet; but hardly anysystematic observation had been undertaken when ProfessorForel, of Lausanne, attacked the subject, and it is his verynteresting observations which I propose to describe.

Doctor Forel is not a mathematician, but is rather a nat-

uralist of the old school, who notes any interesting fact andthen proceeds carefully to investigate its origin. His papers

1The word seiche is a purely local one. It has been alleged to bederived from seche, but I can see no reason for associating drynesswith the phenomenon.

18


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SEICHES IN LAKES 19

have a special charm in that he allows one to see all theworkings of his mind, and tells of each difficulty as it arose

and how he met it. To those who like to read of such work,almost in the form of a narrative, I can strongly recommendthese papers, which afford an admirable example of researchthoroughly carried out with simple appliances.

People are nowadays too apt to think that science canonly be carried to perfection with elaborate appliances, and

yet it is the fact that many of the finest experiments havebeen made with cardboard, cork, and sealing-wax. The prin-cipal reason for elaborate appliances in the laboratories ofuniversities is that a teacher could not deal with a largenumber of students if he had to show each of them how tomake and set up his apparatus, and a student would notbe able to go through a large field of study if he had tospend days in preparation. Great laboratories have, indeeda rather serious defect, in that they tend to make all but thevery best students helpless, and thus to dwarf their powersof resource and inventiveness. The mass of scientific work is

undoubtedly enormously increased by these institutions, butthe number of really great investigators seems to remain al-most unaffected by them. But I must not convey the impres-sion that, in my opinion, great laboratories are not usefulIt is obvious, indeed, that without them science could notbe taught to large numbers of students, and, besides, there


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CHAPTER II 20

are many investigations in which every possible refinementof apparatus is necessary. But I do say that the number

of great investigators is but little increased by laboratoriesand that those who are interested in science, but yet havenot access to laboratories, should not give up their study indespair.

Doctor Forels object was, in the first instance, to notethe variations of the level of the lake, after obliterating the

small ripple of the waves on the surface. The instrumentused in his earlier investigations was both simple and del-cate. Its principle was founded on casual observation at

the port of Morges, where there happens to be a breakwa-ter, pierced by a large ingress for ships and a small one forrowing boats. He accidentally noticed that at the small pas-sage there was always a current setting either inwards oroutwards, and it occurred to him that such a current wouldform a very sensitive index of the rise and fall of the water inthe lake. He therefore devised an instrument, illustrated infig. 7, and called by him a plemyrameter, for noting currents

of even the most sluggish character. Near the shore he madea small tank, and he connected it with the lake by means ofan india-rubber siphon pipe of small bore. Where the pipecrossed the edge of the tank he inserted a horizontal glasstube of seven millimetres diameter, and in that tube he puta float of cork, weighted with lead so that it should be of the


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SEICHES IN LAKES 21

same density as water. At the ends of the glass tube therewere stops, so that the float could not pass out of it. When

the lake was higher than the tank, the water ran through

Fig. 7.Plemyrameter

the siphon pipe from lake to tank, and the float remainedammed in the glass tube against the stop on the side to-

wards the tank; and when the lake fell lower than the tank,the float traveled slowly to the other end and remained there.The siphon pipe being small, the only sign of the waves inthe lake was that the float moved with slight jerks, insteadof uniformly. Another consequence of the smallness of thetube was that the amount of water which could be deliverednto the tank or drawn out of it in one or two hours was


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CHAPTER II 22

so small that it might practically be neglected, so that thewater level in the tank might be considered as invariable.

This apparatus enabled Forel to note the rise and fall ofthe water, and he did not at first attempt to measure theheight of rise and fall, as it was the periodicity in which hewas principally interested.

In order to understand the record of observations, it mustbe remembered that when the float is towards the lake, the

water in the tank stands at the higher level, and when thefloat is towards the tank the lake is the higher. In the di-agrams, of which fig. 8 is an example, the straight line isdivided into a scale of hours and minutes. The zigzag linegives the record, and the lower portions represent that thewater of the lake was below the tank, and the upper line thatt was above the tank. The fact that the float only moved

slowly across from end to end of the glass tube, is indicatedby the slope of the lines, which join the lower and upper por-tions of the zigzags. Then on reading fig. 8 we see that from2 hrs. 1 min. to 2 hrs. 4 mins. the water was high and the

float was jammed against the tank end of the tube, becausethere was a current from the lake to the tank. The float thenslowly left the tank end and traveled across, so that at 2 hrs5 mins. the water was low in the lake. It continued, save fortransient changes of level, to be low until 2 hrs. 30 mins.when it rose again. Further explanation seems unnecessary


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SEICHES IN LAKES 23

as it should now be easy to read this diagram, and thatshown in fig. 9.

The sharp pinnacles indicate alternations of level so tran-sient that the float had not time to travel across from oneend of the glass tube to the other, before the current was re-versed. These pinnacles may be disregarded for the present,since we are only considering seiches of considerable period.

These two diagrams are samples of hundreds which were

obtained at various points on the shores of Geneva, and ofother lakes in Switzerland. In order to render intelligiblethe method by which Forel analyzed and interpreted theserecords, I must consider fig. 8 more closely. In this case it willbe noticed that the record shows a long high water separatedfrom a long low water by two pinnacles with flat tops. Thesepieces at the ends have an interesting significance. Whenthe water of the lake is simply oscillating with a period ofabout an hour we have a trace of the form shown in fig. 9But when there exists concurrently with this another oscil-ation, of much smaller range and of short period, the form

of the trace will be changed. When the water is high in con-sequence of the large and slow oscillation, the level of theake cannot be reduced below that of the tank by the small

short oscillation, and the water merely stands a little higheror a little lower, but always remains above the level of thetank, so that the trace continues on the higher level. But


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CHAPTER II 24

when, in course of the changes of the large oscillation, thewater has sunk to near the mean level of the lake, the short

Fig. 8, 9.Records of Seiches at Evian

oscillation will become manifest, and so it is only at the endsof the long flat pieces that we shall find evidence of the quickoscillation.

Thus, in these two figures there was in one case only onesort of wave, and in the other there were two simultaneouswaves. These records are amongst the simplest of those ob-tained by Forel, and yet even here the oscillations of thewater were sufficiently complicated. It needed, indeed, thecareful analysis of many records to disentangle the severalwaves and to determine their periods.


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SEICHES IN LAKES 25

After having studied seiches with a plemyrameter forsome time, Forel used another form of apparatus, by which

he could observe the amplitude of the waves as well as theirperiod. His apparatus was, in fact, a very delicate tide-gauge, which he called a limnimeter. The only differencebetween this instrument and the one already described as atide-gauge is that the drum turned much more rapidly, sothat five feet of paper passed over the drum in twenty-four

hours, and that the paper was comparatively narrow, therange of the oscillation being small. The curve was usuallydrawn on the full scale, but it could be quickly reduced tohalf scale when large seiches were under observation.

It would be impossible in a book of this kind to followForel in the long analysis by which he interpreted his curvesHe speaks thus of the complication of simultaneous waves:All these oscillations are embroidered one on the other andnterlace their changes of level. There is here matter to dis-

turb the calmest mind. I must have a very stout faith inthe truth of my hypothesis to persist in maintaining that, in

the midst of all these waves which cross and mingle, theres, nevertheless, a recognizable rhythm. This is, however

what I shall try to prove.1 The hypothesis to which he hererefers, and triumphantly proves, is that seiches consist of a

1Deuxieme Etude, p. 544.


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CHAPTER II 26

rocking of the whole water of the lake about fixed lines, justas by tilting a trough the water in it may be set swinging,

so that the level at the middle remains unchanged, while atthe two ends the water rises and falls alternately.In another paper he remarks: If you will follow and

study with me these movements you will find a great charmn the investigation. When I see the water rising and falling

on the shore at the end of my garden I have not before me a

simple wave which disturbs the water of the bay of Morgesbut I am observing the manifestation of a far more importantphenomenon. It is the whole water of the lake which is rock-ng. It is a gigantic impulse which moves the whole liquid

mass of Leman throughout its length, breadth, and depth. . .It is probable that the same thing would be observed in fararger basins of water, and I feel bound to recognize in the

phenomenon of seiches the grandest oscillatory movementwhich man can study on the face of our globe.1

It will now be well to consider the map of Genevan fig. 10. Although the lake somewhat resembles the arc

of a circle, the curvature of its shores will make so littledifference in the nature of the swinging of the water that wemay, in the first instance, consider it as practically straight.

Forels analysis of seiches led him to conclude that the

1Les Seiches, Vagues dOscillation, p. 11.


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SEICHES IN LAKES 27

oscillations were of two kinds, the longitudinal and the trans-verse. In the longitudinal seiche the water rocks about a line

drawn across the lake nearly through Morges, and the wa-ter at the east end of the lake rises when that at the westfalls, and vice versa. The line about which the water rockss called a node, so that in this case there is one node at the

middle of the lake. This sort of seiche is therefore called auninodal longitudinal seiche. The period of the oscillation

Fig. 10.Map of Lake of Geneva

s the time between two successive high waters at any place,and it was found to be seventy-three minutes, but the range


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CHAPTER II 28

of rise and fall was very variable. There are also longitudinalseiches in which there are two nodes, dividing the lake into

three parts, of which the central one is twice as long as theextreme parts; such an oscillation is called a binodal longi-tudinal seiche. In this mode the water at the middle of theake is high when that at the two ends is low, and vice versa;

the period is thirty-five minutes.Other seiches of various periods were observed, some of

which were no doubt multinodal. Thus in a trinodal seiche,the nodes divide the lake into four parts, of which the twocentral ones are each twice as long as the extreme parts. Ifthere are any number of nodes, their positions are such thatthe central portion of the lake is divided into equal lengths,and the terminal parts are each of half the length of thecentral part or parts. This condition is necessary in orderthat the ends of the lake may fall at places where there is nohorizontal current. In all such modes of oscillation the placeswhere the horizontal current is evanescent are called loops,and these are always halfway between the nodes, where there

s no rise and fall.A trinodal seiche should have a period of about twenty-

four minutes, and a quadrinodal seiche should oscillate inabout eighteen minutes. The periods of these quicker se-ches would, no doubt, be affected by the irregularity in

the form and depth of the lake, and it is worthy of notice


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SEICHES IN LAKES 29

that Forel observed at Morges seiches with periods of abouttwenty minutes and thirty minutes, which he conjectured to

be multinodal.The second group of seiches were transverse, being ob-servable at Morges and Evian. It was clear that these os-cillations, of which the period was about ten minutes, weretransversal, because at the moment when the water was high-est at Morges it was lowest at Evian, and vice versa. As in

the case of the longitudinal seiches, the principal oscillationof this class was uninodal, but the node was, of course, nowongitudinal to the lake. The irregularity in the width and

depth of the lake must lead to great diversity of period inthe transverse seiches appropriate to the various parts of theake. The transverse seiches at one part of the lake must

also be transmitted elsewhere, and must confuse the seichesappropriate to other parts. Accordingly there is abundantreason to expect oscillations of such complexity as to eludecomplete explanation.

The great difficulty of applying deductive reasoning to

the oscillations of a sheet of water of irregular outline anddepth led Forel to construct a model of the lake. By study-ng the waves in his model he was able to recognize many of

the oscillations occurring in the real lake, and so obtained anexperimental confirmation of his theories, although the peri-ods of oscillation in the model of course differed enormously


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CHAPTER II 30

from those observed in actuality.The theory of seiches cannot be considered as demon-

strated, unless we can show that the water of such a basin asthat of Geneva is capable of swinging at the rates observed.I must, therefore, now explain how it may be proved thatthe periods of the actual oscillations agree with the facts ofthe case.

As a preliminary let us consider the nature of wave mo-

tion. There are two very distinct cases of the undulatory mo-tion of water, which nevertheless graduate into one anotherThe distinction lies in the depth of the water compared withthe length of the wave, measured from crest to crest, in thedirection of wave propagation. The wave-length may be usedas a measuring rod, and if the depth of the water is a smallfraction of the wave-length, it must be considered shallow,but if its depth is a multiple of the wave-length, it will bedeep. The two extremes of course graduate into one another

In a wave in deep water the motion dies out pretty rapidlyas we go below the surface, so that when we have gone

down half a wave-length below the surface, the motion isvery small. In shallow water, on the other hand, the motionextends quite to the bottom, and in water which is neitherdeep nor shallow, the condition of affairs is intermediateThe two figures, 11 and 12, show the nature of the move-ment in the two classes of waves. In both cases the dotted


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SEICHES IN LAKES 31

ines show the position of the water when at rest, and thefull lines show the shapes assumed by the rectangular blocks

marked out by the dotted lines, when wave motion is disturb-ng the water. It will be observed that in the deep water, asshown in fig. 11, the rectangular blocks change their shaperise and fall, and move to and fro. Taking the topmost rowof rectangles, each block of water passes successively in timethrough all the forms and positions shown by the top row of

quasi-parallelograms. So also the successive changes of thesecond row of blocks are indicated by the second strip, andthe third and the fourth indicate the same. The changes inthe bottom row are relatively very small both as to shape

Fig. 11.Wave in Deep Water

and as to displacement, so that it did not seem worth while


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CHAPTER II 32

to extend the figure to a greater depth.Turning now to the wave in shallow water in fig. 12, we

see that each of the blocks is simply displaced sideways andgets thinner or more squat as the wave passes along. NowI say that we may roughly classify the water as being deepwith respect to wave motion when its depth is more than halfa wave-length, and as being shallow when it is less. Thus thesame water may be shallow for long waves and deep for short

ones. For example, the sea is very shallow for the great wave

Fig. 12.Wave in Shallow Water

of the oceanic tide, but it is very deep even for the largestwaves of other kinds. Deepness and shallowness are thusmerely relative to wave-length.

The rate at which a wave moves can be exactly calculatedfrom mathematical formul, from which it appears that inthe deep sea a wave 63 metres in length travels at 36 kilo-metres per hour, or, in British measure, a wave of 68 yardsn length travels 221

2miles an hour. Then, the rule for other


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SEICHES IN LAKES 33

waves is that the speed varies as the square root of the wave-ength, so that a wave 16 metres longthat is, one quarter

of63

metrestravels at18

kilometres an hour, which is halfof 36 kilometres an hour. Or if its length were 7 metres, orone ninth as long, it would travel at 12 kilometres an houror one third as quick.

Although the speed of waves in deep water depends onwave-length, yet in shallow water the speed is identical for

waves of all lengths, and depends only on the depth of thewater. In water 10 metres deep, the calculated velocity of awave is 36 kilometres an hour; or if the water were 21

2metres

deep (quarter of 10 metres), it would travel 18 kilometres(half of 36 kilometres) an hour; the law of variation beingthat the speed of the wave varies as the square root of thedepth. For water that is neither deep nor shallow, the rateof wave propagation depends both on depth and on wave-ength, according to a law which is somewhat complicated.

In the case of seiches, the waves are very long comparedwith the depth, so that the water is to be considered as

shallow; and here we know that the speed of propagation ofthe wave depends only on depth. The average depth of theLake of Geneva may be taken as about 150 metres, and itfollows that the speed of a long wave in the lake is about120 kilometres an hour.

In order to apply this conclusion to the study of seiches,


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CHAPTER II 34

we have to consider what is meant by the composition of twowaves. If I take the series of numbers

&c. 100 71 0 71 100 71 0 71 100 &c

and plot out, at equal distances, a figure of heights pro-portional to these numbers, setting off the positive numbersabove and the negative numbers below a horizontal line, I

get the simple wave line shown in fig. 13. Now, if this waves traveling to the right, the same series of numbers will rep-resent the wave at a later time, when they are all displacedtowards the right, as in the dotted line.

Now turn to the following schedule of numbers, and con-sider those which are written in the top row of each succes-

sive group of three rows. The columns represent equidistantspaces, and the rows equidistant times. The first set of num-bers, 100, 71, 0, &c., are those which were plotted outas a wave in fig. 13; in the top row of the second group they

Fig. 13.Simple Wave


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SEICHES IN LAKES 35

are the same, but moved one space to the right, so that theyrepresent the movement of the wave to the right in one in-

terval of time. In the top row of each successive group thenumbers are the same, but always displaced one more spaceto the right; they thus represent the successive positions of awave moving to the right. The table ends in the same way

as it begins, so that in eight of these intervals of time thewave has advanced through a space equal to its own length.

If we were to invert these upper figures, so that the num-bers on the right are exchanged with those on the left, weshould have a series of numbers representing a wave travel-ng to the left. Such numbers are shown in the second rown each group.

When these two waves coexist, the numbers must be com-pounded together by addition, and then the result is the se-ries of numbers written in the third rows. These numbersrepresent the resultant of a wave traveling to the right, andof an equal wave traveling simultaneously to the left.

It may be well to repeat that the first row of each group

represents a wave moving to the right, the second row rep-resents a wave moving to the left, and the third representsthe resultant of the two. Now let us consider the nature ofthis resultant motion; the third and the seventh columns offigures are always zero, and therefore at these two places thewater neither rises nor falls,they are, in fact, nodes. If


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CHAPTER II 36

100 71 0 71 100 71 0 71 100

100 71 0 71 100 71 0 71 100

200 142 0 142 200 142 0 142 200

71

100

71 0 71 100 71 0

71

71 0 71 100 71 0 71 100 71

142 100 0 100 142 100 0 100 142

0 71 100 71 0 71 100 71 0

0 71 100 71 0 71 100 71 0

0 0 0 0 0 0 0 0 0

71 0 71 100 71 0 71 100 71

71 100 71 0 71 100 71 0 71

142 100 0 100 142 100 0 100 142

100 71 0 71 100 71 0 71 100

100 71 0 71 100 71 0 71 100

200 142 0 142 200 142 0 142 200

71 100 71 0 71 100 71 0 71

71 0

71

100

71 0 71 100 71

142 100 0 100 142 100 0 100 142

0 71 100 71 0 71 100 71 0

0 71 100 71 0 71 100 71 0

0 0 0 0 0 0 0 0 0

71 0 71 100 71 0 71 100 71

71 100 71 0 71 100 71 0 71

142 100 0 100 142 100 0 100 142

100 71 0 71 100 71 0 71 100

100 71 0 71 100 71 0 71 100

200 142 0 142 200 142 0 142 200

Composition of Two Equal and Opposite Waves


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SEICHES IN LAKES 37

the schedule were extended indefinitely both ways, exactlyhalfway between any pairs of nodes there would be a loop,

or line across which there is no horizontal motion. In theschedule, as it stands, the first, fifth, and ninth columns areoops.

At the extreme right and at the extreme left the resultantnumbers are the same, and represent a rise of the water from200 to +200, and a subsequent fall to 200 again. If these

nine columns represent the length of the lake, the motion isthat which was described as binodal, for there are two nodesdividing the lake into three parts, there is a loop at each end,and when the water is high in the middle it is low at theends, and vice versa. It follows that two equal waves, eachas long as the lake, traveling in opposite directions, whencompounded together give the motion which is described asthe binodal longitudinal seiche.

Now let us suppose that only five columns of the tablerepresent the length of the lake. The resultant numberswhich again terminate at each end with a loop, are:

200 142 0 142 200

142 100 0 100 142

0 0 0 0 0

142 100 0 100 142


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CHAPTER II 38

200 142 0 142 200

142 100 0 100 142

0 0 0 0 0

142 100 0 100 142

200 142 0 142 200

Since the middle column consists of zero throughout, the

water neither rises nor falls there, and there is a node at themiddle. Again, since the numbers at one end are just thesame as those at the other, but reversed as to positive andnegative, when the water is high at one end it is low at theother. The motion is, in fact, a simple rocking about the cen-tral line, and is that described as the uninodal longitudinal

seiche.The motion is here again the resultant of two equal waves

moving in opposite directions, and the period of the oscilla-tion is equal to the time which either simple wave takes totravel through its own length. But the length of the waves now twice that of the lake. Hence it follows that the pe-

riod of the rocking motion is the time occupied by a waven traveling twice the length of the lake. We have already

seen that in shallow water the rate at which a wave moves isndependent of its length and depends only on the depth of

the water, and that in water of the same depth as the Lake


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of Geneva the wave travels 120 kilometres an hour. TheLake of Geneva is 70 kilometres long, so that the two waves,

whose composition produces a simple rocking of the water,must each of them have a length of 140 kilometres. Hence itfollows that the period of a simple rocking motion, with onenode in the middle of the Lake of Geneva, will be almost ex-actly 140

120of an hour, or 70 minutes. Forel, in fact, found the

period to be 73 minutes. He expresses this result by saying

that a uninodal longitudinal seiche in the Lake of Genevahas a period of 73 minutes. His observations also showedhim that the period of a binodal seiche was 35 minutes. Itfollows from the previous discussion that when there are twonodes the period of the oscillation should be half as long aswhen there is one node. Hence, we should expect that theperiod would be about 36 or 37 minutes, and the discrep-ancy between these two results may be due to the fact thatthe formula by which we calculate the period of a binodalseiche would require some correction, because the depth ofthe lake is not so very small compared with the length of

these shorter waves.It is proper to remark that the agreement between the

theoretical and observed periods is suspiciously exact. Theake differs much in depth in different parts, and it is not

quite certain what is the proper method of computing theaverage depth for the determination of the period of a se-


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che. It is pretty clear, in fact, that the extreme closenessof the agreement is accidentally due to the assumption of a

round number of metres as the average depth of the lake.The concordance between theory and observation must not,however, be depreciated too much, for it is certain that thefacts of the case agree well with what is known of the depthof the lake.

The height of the waves called seiches is very various

I have mentioned an historical seiche which had a range of asmuch as four feet, and Forel was able by his delicate instru-ments still to detect them when they were only a millimetreor a twenty-fifth of an inch in height. It is obvious, there-fore, that whatever be the cause of seiches, that cause mustvary widely in intensity. According to Forel, seiches arisefrom several causes. It is clear that anything which heapsup the water at one end of the lake, and then ceases to act,must tend to produce an oscillation of the whole. Now, arise of water level at one end or at one side of the lake maybe produced in various ways. Some, and perhaps many, se-

ches are due to the tilting of the whole lake bed by minuteearthquakes. Modern investigations seem to show that thiss a more fertile cause than Forel was disposed to allow, andt would therefore be interesting to see the investigation of

seiches repeated with the aid of delicate instruments for thestudy of earthquakes, some of which will be described in


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SEICHES IN LAKES 41

Chapter VI. I suspect that seiches would be observed at timeswhen the surface of the earth is much disturbed.

The wind is doubtless another cause of seiches. Whent blows along the lake for many hours in one direction, itproduces a superficial current, and heaps up the water atthe end towards which it is blowing. If such a wind ceasessomewhat suddenly, a seiche will certainly be started, andwill continue for hours until it dies out from the effects of

the friction of the water on the lake bottom. Again, theheight of the barometer will often differ slightly at differentparts of the lake, and the water will respond, just as doesthe mercury, to variations of atmospheric pressure. Abouta foot of rise of water should correspond to an inch of dif-ference in the height of barometer. The barometric pressurecannot be quite uniform all over the Lake of Geneva, and al-though the differences must always be exceedingly small, yett is impossible to doubt that this cause, combined probably

with wind, will produce many seiches. I shall return laterto the consideration of an interesting speculation as to the

effects of barometric pressure on the oscillation of lakes andof the sea. Lastly, Forel was of opinion that sudden squallsor local storms were the most frequent causes of seiches. Ithink that he much overestimated the efficiency of this cause,because his theory of the path of the wind in sudden and lo-cal storms is one that would hardly be acceptable to most


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CHAPTER II 42

meteorologists.Although, then, it is possible to indicate causes compe-

tent to produce seiches, yet we cannot as yet point out theparticular cause for any individual seiche. The complica-tion of causes is so great that this degree of uncertainty willprobably never be entirely removed.

But I have not yet referred to the point which justifies thisong digression on seiches in a book on the tides. The subject

was introduced by the irregularities in the line traced by thetide-gauge at Bombay, which indicated that there are oscilla-tions of the water with periods ranging from two minutes toa quarter of an hour or somewhat longer. Now these zigzagsare not found in the sea alone, for Forel observed on the lakeoscillations of short period, which resembled seiches in allbut the fact of their more rapid alternations. Some of thesewaves are perhaps multinodal seiches, but it seems that theyare usually too local to be true seiches affecting the wholebody of the lake at one time. Forel calls these shorter oscil-ations vibrations, thus distinguishing them from proper

seiches. A complete theory of the so-called vibrations hasnot yet been formulated, although, as I shall show below, atheory is now under trial which serves to explain, at least inpart, the origin of vibrations.

Forel observed with his limnimeter or tide-gauge thatwhen there is much wind, especially from certain quarters,


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vibrations arise which are quite distinct from the ordinaryvisible wave motion. The period of the visible waves on

the Lake of Geneva is from4

to5

seconds,1

whereas vibra-tions have periods ranging from 45 seconds to 4 minutesThus there is a clear line separating waves from vibrations.Forel was unable to determine what proportion of the areaof the lake is disturbed by vibrations at any one time, andalthough their velocity was not directly observed, there can

be no doubt that these waves are propagated at a rate whichcorresponds to their length and to the depth of the waterI have little doubt but that the inequalities which producenotches in a tide-curve have the same origin as vibrations onakes.

It is difficult to understand how a wind, whose only visi-ble effect is short waves, can be responsible for raising wavesof a length as great as a thousand yards or a mile, and yetwe are driven to believe that this is the case. But Forel alsofound that steamers produce vibrations exactly like thosedue to wind. The resemblance was indeed so exact that vi-

brations due to wind could only be studied at night, whent was known that no steamers were traveling on the lake,

and, further, the vibrations due to steamers could only be

1I observed when it was blowing half a gale on Ullswater, in Cum-berland, that the waves had a period of about a second.


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CHAPTER II 44

studied when there was no wind.His observations on the steamer vibrations are amongst

the most curious of all his results. When a boat arrives atthe pier at Morges, the water rises slowly by about 5 to8 millimetres, and then falls in about 20 to 30 seconds. Theamount and the rapidity of the rise and fall vary with thetonnage of the boat and with the rate of her approach. Afterthe boat has passed, the trace of the limnimeter shows irreg-

ularities with sharp points, the variations of height rangingfrom about two to five millimetres, with a period of abouttwo minutes. These vibrations continue to be visible dur-ng two to three hours after the boat has passed. As these

boats travel at a speed of 20 kilometres an hour, the vibra-tions persist for a long time after any renewal of them by theboat has ceased. These vibrations are called by Forel thesubsequent steamer vibrations.

That the agitation of the water should continue for morethan two hours is very remarkable, and shows the delicacyof the method of observation. But it seems yet more strange

that, when a boat is approaching Morges, the vibrationsshould be visible during 25 minutes before she reaches thepier. These he calls antecedent steamer vibrations. Theyare more rapid than the subsequent ones, having a period of aminute to a minute and a quarter. Their height is sometimestwo millimetres (a twelfth of an inch), but they are easily de-


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SEICHES IN LAKES 45

tected when less than one millimetre in height. It appearsthat these antecedent vibrations are first noticeable when

the steamer rounds the mole of Ouchy, when she is still at adistance of 10 kilometres. As far as one can judge from thespeed at which waves are transmitted in the Lake of Genevathe antecedent vibrations, which are noticed 25 minutes be-fore the arrival of the boat, must have been generated whenshe was at a distance of 12 kilometres from Morges. Fig. 14

gives an admirable tracing of these steamer vibrations.1In this figure the line a a was traced between two and

three oclock in the morning, and shows scarcely any sign ofperturbation. Between three and eight oclock in the morn-ng no observations were taken, but the record begins again

at eight oclock. The portion marked b b shows weak vibra-tions, probably due to steamers passing along the coast ofSavoy. The antecedent vibrations, produced by a steamerapproaching Morges, began about the time of its departurefrom Ouchy, and are shown at c c. The point d shows thearrival of this boat at Morges, and d shows the effect of an-

other boat coming from Geneva. The portion marked e e eshows the subsequent steamer vibrations, which were veryclear during more than two hours after the boats had passed

Dr. Forel was aware that similar vibrations occur in the

1From Les Seiches, Vagues dOscillation fixe des Lacs, 1876.


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CHAPTER II 46

sea, for he says: What are these oscillations with periodsof 5, 10, 20, or 100 minutes, which are sometimes irregular?

Are they analogous to our seiches? Not if we define seiches asuninodal oscillations, for it is clear that if, in a closed basinof 70 kilometres in length, uninodal seiches have a period of73 minutes, in the far greater basin of the Mediterranean,or of the ocean, a uninodal wave of oscillation must havea much longer period. They resemble much more closely

what I have called vibrations, and, provisionally, I shall callthem by the name of vibrations of the sea. I venture tonvite men of science who live on the seacoast to follow this

study. It presents a fine subject for research, either in thenterpretation of the phenomenon or in the establishment of

the relations between these movements and meteorologicalconditions.1

These vibrations are obviously due to the wind or tosteamers, but it is a matter of no little surprise that suchnsignificant causes should produce even very small waves of

half a mile to a mile in length.

The manner in which this is brought about is undoubt-edly obscure, yet it is possible to obtain some sort of insightnto the way in which these long waves arise. When a stone

falls into calm water waves of all sorts of lengths are instanta-

1Seiches et Vibrations des Lacs et de la Mer, 1879, p. 5.


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SEICHES IN LAKES 47

Fig. 14.Vibrations due to Steamers


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CHAPTER II 48

neously generated, and the same is true of any other isolateddisturbance. Out of all these waves the very long ones and

the very short ones are very small in height. Theoreticallywaves of infinitely great and of infinitely small lengths, yetn both cases of infinitely small heights, are generated at thenstant of the impulse, but the waves of enormous length and

those of very small length are of no practical importance, andwe need only consider the moderate waves. For the shorter

of these the water is virtually deep, and so they will eachtravel outwards at a pace dependent on length, the longerones outstripping the shorter ones. But for the longer wavesthe water will be shallow, and they will all travel together.Thus the general effect at a distance is the arrival of a longwave first, followed by an agitated rippling. The point whichwe have to note is that an isolated disturbance will generateong waves and that they will run ahead of the small ones.

It is important also to observe that the friction of the waterannuls the oscillation in the shorter waves more rapidly thant does that of the longer ones, and therefore the long waves

are more persistent. Now we may look at the disturbancedue to a steamer or to the wind as consisting of a succes-sion of isolated disturbances, each of which will create longwaves outstripping the shorter ones. These considerationsafford a sort of explanation of what is observed, but I do notunderstand how it is that the separation of the long from


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the short waves is so complete, nor what governs the lengthof the waves, nor have I made any attempt to evaluate the

greater rapidity of decrease of short waves than long ones.1

It must then be left to future investigators to elucidate thesepoints.

The subject of seiches and vibrations clearly affords annteresting field for further research. The seiches of Lake

George in New South Wales have been observed by Mr. Rus-sell, the government astronomer at Sydney; but until lastyear they do not seem to have been much studied on anyakes outside of Switzerlan

ebook of the tides and kindred phenomena in the

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