J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 76
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if the sides of the body in contactwith the water do not extend vertically below the undisturbed freesurface) it follows that <p n would then be continuous.In order to make further progress it would be necessary to specifythe properties of the immersed body.
However, we have succeeded inobtaining the equation (10.13.18) which is generally valid and of basicimportance for our theory together with the transition conditions(10.13.24) valid at the edge of immersed bodies; in particular, we havethe definitive equation for the free surface itself in the form of thelinear wave equation (10.13.19). The idea behind this method of approximation is to get rid of the depth variable by an integration overthe depth so that the problems then arc considered only in the x, zplane. As a result, the problems are no longer problems in potentialtheory in three space variables, but rather problems involving the wavecq nation with only two space variables, and hence they are more opento attack by known methods.
How this comes about will be seen inspecial cases in the following.example of the application of the above theory we conproblem of the tides in the oceans, with a view towherethis theory fits into the theory of gravity waves inindicatinggeneral, but not with the purpose of giving a detailed exposition. (Fordetails, the long Chapter VIII in Lamb [L.3] should be consulted.)To begin with, it might seem incredible at first sight that the shallowAs afirstsider briefly thewater theory could possibly be accurate for the oceans, since depths ofmore occur. However, it is the depth in relation to thewave length of the motions under consideration which is relevant.The tides are forced oscillations caused by the tide-generating attractions of the moon and the sun, and hence have the same periodsas the motions of the sun and moon relative to the earth.
These periodsare measured in hours, and consequently the tidal motions in thewater result in waves having wave lengths of hundreds of miles;*five miles orthe depth-wave length ratio is thus quite small and the shallowwater theory should be amply accurate to describe the tides. Thisequation (10.13.19), or rather, itsanalogue for the case of water lying on a rotating spheroid (with Coriolis terms put in if a coordinate system rotating with the earth ismeans,*in effect, that the differentialFor example, in water of depth 10,000 feet (perhaps a fairly reasonableaverage value for the depth of the oceans) a steady progressing wave havinga length of 10,000 feet has a period of only 44.2 sec. (cf. Lamb [L.3], p.
369).Since the wave length varies as the square of the period, the correctness of ourstatement is obvious.WATER WAVES422would serve as a basis for calculating tidal motions. Of course,would be defined in terms of the gravitational forcesdue to the attraction of the sun and moon. The variable depth of thewater in the oceans would come into play, as well as boundary conditions at the shore lines of the continents. Presumably,would beitscouldbeintoharmonicobtainedcomponents (whichanalyzedfrom astronomical data), the response to each such harmonic wouldbe calculated, and the results superimposed. Such a problem constiused),the functionWtWttutes a linear vibration problem of classical type it is essentially thesame as the problem of transverse forced oscillations of a tightlymembrane with an irregular boundary.
If itwere not for one essential difficulty, to be mentioned in a moment,such a problem would in all likelihood be solvable numerically byusing modern high speed computational equipment. The difficultymentioned was pointed out to the author in a conversation withH. Jeffreys, and it is that there are difficulties in prescribing an appropriate boundary condition in coastal regions where there is dissipation of energy in the tidal motions (in the bay of Fundy, for example, to take what is probably an extreme case). At other eoastal regions the correct boundary condition would of course often be simplythat the component of the velocity normal to the coast line vanishes.Of course, there would also be a difficulty in using a differentialequation like (10.13.19) near any shores where h = 0, since the differential equation becomes singular at such points.
Nevertheless, acomputation of the tides on a dynamical basis would seem to be aworthwhile problem perhaps it could be managed in such a way asto help, in conjunction with observations of the actual tides, inproviding information concerning the dissipation of energy in suchstretched non-uniformmotions.isThese remarks might be taken to imply that the dynamical theorynot at present used to compute the tides. This is not entirely correet,since the tide tables for predicting the tides in various parts of theworld are based on fundamental consequences of the assumption thatthe tides are indeed governed by a differential equation of the samegeneral type as (10.13.19).
The point is that the oceans are regarded asa linear vibrating system under forced oscillations due to excitationfrom the periodic forces of attraction of the sun and moon. It isassumed that all free vibrations of the oceans were long ago dampedout, and hence, as remarked above, that the tidal motions now existing in the oceans are a superposition of simple harmonic oscillationsLONG WAVES IN SHALLOW WATER423having periods which are very accurately known from astronomicalTo obtain tide tables for any given point a superpositionof oscillations of these frequencies is taken with undetermined amplitudes and phases which are then fixed by comparing them with aobservations.harmonic analysis of actualtidal observationsmadeat the point inquestion.
Thetide predictions are then made by using the result ofsuch a calculation to prepare tables for future times. The dynamicalthus used only in a qualitative way. An interesting additionpoint might be mentioned, i.e. that tides of observable amplitudesare sometimes measured which have as frequency the sums or differences (or also other linear combinations with integers) of certain oftheoryisalthe astronomical frequencies, which means from the point of view ofvibration theory that observable nonlinear effects must be present.Another type of phenomenon in nature which can be treated bythe theory derived here concerns periodic motions of rather longperiod, called seiches, which occur in lakes in various parts of theworld.Thefirstobservations of this kind seem to have beenmade byForel [F.7] in the lake at Geneva in Switzerland, in which oscillationshaving a period of the order of an hour and amplitudes of up to sixhave been observed.
In larger lakestion arc observedabout fifteen hoursfeetlarger periods of oscillain Lake Erie, for example.stillA rather destructive oscillation, generally supposed to be of the typeof a seiche, occurred in Lake Michigan in June 1054; a wave with anamplitude of the order of ten feet occurred and swept away a numberof peoplewho weremechanismisfishing from piers and breakwaters. What therise to seiches in lakes has been the object ofthat givesseems rather clear that the motionsrepresent free vibrations of the water in a lake which are excited byexternal forces of an impulsive character, the most likely type arisingfrom sudden differences in atmospheric pressure over various portionsof the water surface. Bouasse [B.I 5, p.
158] reports, however, thatthe Lisbon earthquake of 1755 caused oscillations in Loch Lomondwith a period of about 5 minutes and amplitudes of several feet.In any case, the periods observed seem to correspond to those calculated on the basis of the linear shallow water theory, which should bequite accurate for the study of seiches because of their long periods andconsiderable discussion, butitsmall amplitudes.
It follows, therefore, that the differential equationt(since tidal forces(10.13.18) is applicable; we suppose thatW ==since there are no immersedplay no role in this case), and also set fjbodies to be considered. The differential equation for 0(x, z\ t ) is thusWATER WAVES424(10.18.25)er&Thept=free natural vibrations of the lake are investigated by settingin (10.13.25) with the resultand 0(x, z; t)99(01, *)*"=2(10.13.26)(hp x ) xa+ (%) + -<p =20.As boundary condition along the shore of the lake we would have(10.13.27)(p n=0.The problem thus posedis one of the classical eigenvalue problems ofmathematical physics. Solutions <p other than the trivial solutionof (10.13.26) under the homogeneous boundary conditionq>(10.13.27) are wanted; such solutions exist only for special values ofthe circular frequency a, and these values yield the natural frequenciescorresponding to the natural modes y(x, z) which are correlated withthem.
In general, an infinite set of such natural frequencies occurs.For particular shapes and depths h rectangular or circular lakes ofconstant depth, for example it is possible to solve such problemsmore or less explicitly. In practice however, lakes have such irregular=and depths that the determination of the natural frequenciesand modes requires numerical computation. A reasonable and generally applicable method of carrying out such computations is furnishedhere, as in other instances in this and the subsequent chapter, bythe method of finite differences.* In this method, the derivatives inthe differential equation and boundary conditions are replaced bydifference quotients defined by means of the values of the functionat the discrete points of a net in the domain of the independent variables.
The resulting finite equations are then solved to yield approximate values for the unknown function at the net points. The differenceapproximation will be more accurate for a closer spacing of the netpoints. We proceed to illustrate the method for the case of a lake ofconstant depth in the form of a square of length / on each side, withoutlinesa view to comparing the result with the exact solution which is easydown in this case. The differential equation (10.13.26) canbe written in the formto write* A different method was usedby Chrystal [C.2] to calculate the periods ofthe free oscillations of Loch Earn; he found good agreement with the observations for the first six modes of oscillation.LONG WAVES IN SHALLOW WATER(10.13.26)!(p xx8J/7 is<pgz+m2<p=m =20,a2 /ghAdivision of the square in a mesh with mesh widthtaken, as indicated in Fig.
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