J.J. Stoker - Water waves. The mathematical theory with applications, страница 4

Such a linearizationresults from assuming that the amplitude of the waves is small. Themost famous application ofthis theoryisto the tides in the oceans(and also in the atmosphere, for that matter); strange though itseems at first sight, the oceans can be treated as shallow for thisphenomenonsince thewave lengths of the motionsare very longbecause of the large periods of the disturbances caused by the moonand the sun.

This theory, as applied to the tides, is dealt with onlyvery summarily, since an extended treatment is given by Lamb[L.3]. Instead, some problems connected with the design of floatingbreakwaters in shallow water are discussed, together with brieftreatments of the oscillations in certain lakes (the lake at Genevaexample) called seiches, and oscillations in harbors.Finally, Part III concludes with Chapter 11 on the subject ofin Switzerland, forXXINTRODUCTIONmathematical hydraulics, which is to be understood here as referringto flows and wave motions in rivers and other open channels withrough sides.

The problems of this chapter are not essentially different,as far as mathematical formulations go, from the problems treatedin the preceding Chapter 10. They differ, however, on the physicalside because of the inclusion of a force which is just as important asgravity, namely a force of resistance caused by the rough sides andbottom of the channels. This force is dealt with empirically byadding a term to the equation expressing the law of conservation ofmomentum that is proportional to the square of the velocity andwith a coefficient depending on the roughness and the so-calledhydraulic radius of the channel. The differential equations remain ofthe same type as those dealt with in Chapter 10, and the same underlying theory based on the notion of the characteristics applies.Steady motions in inclined channels are first dealt with.

In particular, a method of solving the problem of the occurrence of rollwaves in steep channels is given; this is done by constructing aprogressing wave by piecing together continuous solutions throughbores spaced at periodic intervals, This is followed by the solutionof a problem of steady motion which is typical for the propagationof a flood down a long river; in fact, data were chosen in such a wayas to approximate the case of a flood in the Ohio River.treatmentAnext given for a flood problem so formulated as to correspondapproximately to the case of a flood wave moving down the Ohioto its junction with the Mississippi, and w'th the result that disturbances are propagated both upstream and downstream in the Mississippi and a backwater effect is noticeable up the Ohio.

In theseisnecessary to solve the differential equations numerically(in contrast with most of the problems treated in Chapter 10, inwhich interesting explicit solutions could be given), and methods ofproblemsit isdoing so are explained in detail. In fact, a part of the elements ofnumerical analysis as applied to solving hyperbolic partial differentialequations by the method of finite differences is developed. The resultsof a numerical prediction of a flood over a stretch of 400 miles inthe Ohio River aswas the 1945 flooditactually exists are given.

The flood in questionand the predicone of the largest on recordmade (starting with the initial state of the river and using theknown flows into it from tributaries and local drainage) by numericaltionsintegration on a high speed digital computer (the Univac) checkquite closely with the actually observed flood. Numerical predictionsINTRODUCTIONXXIwere also made for the case of a flood (the 1947 flood in this case)coming down the Ohio and passing through its junction with theMississippi; the accuracy of the prediction was good. This is a casein which the simplified methods of the civil engineers do not workwell. These results, of course, have important implications for thepractical applications.Finally Part IV, made up of Chapter 12, closes the book- with afew solutions based on the exact nonlinear theory. One class of problemsis solved byassuming a solution in the form of power series in thetime, which implies that initial motions and motions for a short timeonly can be determined in general.

Nevertheless, some interestingcases can be dealt with, even rather easily, by using the so-calledLagrange representation, rather than the Euler representation whichisused otherwise throughout the book. The problem of the breakingof a dam, and, more generally, problems of the collapse of columnsof a liquid resting on a rigid horizontal plane can be treated in thisway. The book ends with an exposition of the theory due to LeviCivita concerning the problem of the existence of progressing waves offinite amplitude in water of infinite depth which satisfy exactly thenonlinear free surface conditions.AcknowledgmentsWithout the support of the Mathematics Branch and the MechanicsBranch of the Office of Naval Research this book would not have beenwritten. The author takes pleasure in acknowledging the help andencouragement given to him by the ONR in general, and by Dr.

Joachim Wcyl, Dr. Arthur Grad, and Dr. Philip Eiscnberg in particular.Although she is no longer working in the ONR, it is neverthelessappropriate at this place to express special thanks to Deanof the Mathematics Branch when thisMina llecs, who was headbook was begun.Among those who collaborated with the author in the preparationof the manuscript, Dr.

Andreas Troesch should be singled out forspecial thanks. His careful and critical reading of the manuscript remany improvements and the uncovering and correction ofand obscurities of all kinds. Another colleague, Professor E.Isaacson, gave almost as freely of his time and attention, and alsoaided materially in revising some of the more intricate portions of thesulted inerrorsbook. To these fellow workers the author feels deeply indebted.Miss Helen Samoraj typed the entire manuscript in a most efficient(and also good-humored) way, and uncoveredmanyslipsandin-consistencies in the process.The drawings for the book were made by Mrs. Beulah Marx andMiss Lark in Joyner. The index was prepared by Dr. George Booth andDr. Walter Littman with the assistance of Mrs.

Halina Montvila.A considerable part of the material in the present book is the resultof researches carried out at the Institute of Mathematical Sciences ofNew York University as part of its work under contracts with theOffice of Naval Research of the U.S. Department of Defense, and to alesser extent under a contract with the Ohio River Division of theCorps of Engineers of the U.S. Army.

The author wishes to express histhanks generally to the Institute; the cooperative and friendly spiritof its members, and the stimulating atmosphere it has provided haveresulted in the carrying out of quite a large number of researches inthe field of water waves. A good deal of these researches and newresults have come about through the efforts of Professors K. O. FriedxxiiiACKNOWLEDGMENTSXXIVrichs, Fritzfornia),John, J. B. Keller, H.

Lewy (of the University of CaliS. Peters, together with their students or with visitorsand A.at the Institute..1.NewYork, N.Y.January, 1957.J.STOKERContentsPARTICHAPTKRPACKIntroductionixAcknowledgments1.1.11.21.33of conservation of momentum and massHelmholty/s theoremPotential flow and Bernoulli's lawThe laws910conditionsBoundarySingularities of the velocity potential121.0Notions concerning energy and energy fluxFormulation of a surface wave problem1315The Two Basic Approximate Theories192.1Theory of waves of small amplitude2.2Shallow water theory to lowest order.

Tidal theoryGas dynamics analogySystematic derivation of the shallow water theory2.4PART3.13.23.33.4in222527AWaves Simple HarmonicSimple Harmonic Oscillations19....IISubdivision4.71.52.33.31.41.72.xxiiiBasic Hydrodynamicsin theWaterofTimeConstant Depth..37Standing wavesSimple harmonic progressing wavesEnergy transmission for simple harmonic waves of small ampli-3745tude47Group51velocity. DispersionWaves Maintained by Simple Harmonic Surface PressureWater of Uniform Depth. Forced Oscillations4.1Introduction4.2Thesurface pressureisperiodic for all values ofa?in555557CONTENTSXXVIPAGECHAPTER4.8Thevariable surface pressureisconfined to a segment of the58surface4.45.Periodic progressing waves against a vertical07cliffWaves on Sloping Beaches and Past Obstacles695.1Introduction and5.25.3Two-dimensional waves over beaches sloping at anglesThree-dimensional waves against a vertical cliff5.4Waves on5.5co=n/2n7784sloping beaches.

General caseDiffraction of waves around a vertical wedge. Sommerfeld'sdiffraction5.669summary95109problemBrief discussions of additional applications and of other methodsof solutionSubdivision133BMotions Starting from Rest. Transients6.Unsteady Motions6.16.26.3149General formulation of the problem of unsteady motionsUniqueness of the unsteady motions in bounded domainsOutline of the Fourier transform techniqueMotions due to disturbances originating at the surface..149.150153...156....1636.6Application of Kelvin's method of stationary phaseDiscussion of the motion of the free surface due to disturbancesinitiated when the water is at restWaves due to a periodic impulse applied1676.76.46.5initially at rest.to the waterwhenDerivation of the radiation condition for purelyperiodic waves6.8Justification of the6.9A174method of stationary phase181time-dependent Green's function. Uniqueness of unsteadymotions in unbounded domains when obstacles are presentSubdivision.187CWaves on a Running Stream.

Ship Waves7.Two-dimensional Waves on a Running StreamUniform Depth7.1Steady motions in water offree surfaceinfinitedepth withinWaterof198p =on the199CONTENTSXXVIICHAPTER7.27.37.48.PAGESteady motions in water of infinite depth with a disturbing pressure on the free surface201Steady waves in water of constant finite depthUnsteady waves created by a disturbance on the surface of arunning stream207Waves Caused by a Moving PressureWave Patternthe8.1AnPoint. Kelvin's Theory ofcreated by a Moving Shipidealized version of the shipTheclassical shipThe Motion219wave problem.Details of the solutionof a Ship, as a Floating Rigid..Body, in a Seaway2242459.1Introduction and9.2General formulation of the problemLinearization by a formal perturbation procedureMethod of solution of the problem of pitching and heaving of a264ship in a seaway having normal incidence2789.39.4Long Wavesin245summaryPART10.219wave problem.