J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 77
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10.13.3. In numbering the netin this case.=+425has been assumed that only modes of oscillation that aresymmetrical with respect to the center lines parallel to the sides and toto the diagonals are sought, which, however, is not the case for thepointsitFig. 10.13.3. Finite differences fora seiche=isthe lowest frequency. The boundary condition q> nsatisfied approximately by supposing that the solution is reflectedmode havingover the boundaries in such away as to yield values whicharc equal atalso indicated in Fig.
10.13.3.the mirror imagesThe formulas used for approximating the derivatives are defined asinfollows(cf.the boundaries, asisFig. 10.13.3):ffm.n+l-"^n?,w-1dz%9m,nConsequently, the differential equation (10.13.26) 1net point (m n) by the difference equationis+ 9V n+l + Vm.n-lreplaced at each,(10.13.28)-4<p mfn +<pm n+1 +<pmSuch an equationThe,is^written for each of the net points in Fig. 10.13.3.results for points 1,6,9, for example, are:WATER WAVES426+ + + + + (dw) ^ =+ 2y + + 10 + (5m) =(<5w)Vi(10.13.29)26:4<p e9:4<p 9g? 39? 8<p 7<p 528<p 7<p9? 9-These homogeneous linear equations of course have always the solution <f = 0, i = 1, 2,., 10 unless their determinant vanishes, andthis condition is a tenth degree equation in the quantity (dm) 2 thesmallest root of which furnishes an approximation to the lowestfrequency.
The exact solution of the differential equation (10.13.26)!which satisfies the boundary condition is, in the present case, <p =A cos (knx/l) cos (jnz/l), with k and / any integers, provided that.i.:,m =n22(kin for the=2+ j22)/l.Anumerical comparison of the lowest value ofi.e. the value for k1,=mode having double symmetry1with the value computed from the determinantal equationshows the approximate value of m to be too low by 6.5 %.However, this mode corresponds to one of the higher eigenvalues,jso that the accuracy of the finite difference method is rather good.The error for the lowest mode is very much smaller, but because of thelack of symmetries the amount of calculation needed to determinethe corresponding frequency would be much greater for the presentcase. If one were to treat a long narrow lake, the calculation wouldbe simpler.
It could also be advantageous to employ the Rayleigh-Ritzmethod. In principle, similar calculations could be made in morecomplicated cases (for many examples of problems solved along theselines see the book by Southwell [S.14]).Wave motions in harbors are often of a type suitable for discussionin terms of the linear shallow water theory: they are indeed often ofthe type called seiches above. In these cases oscillations of the waterin the harbor are also commonly excited by the motion at the harbormouth, which in its turn is due, of course, to wave motions generatedin the open sea.
An experimental and theoretical investigation of suchwaves in a model has been carried out by McNown [M.7]. The modelwas in the form of a circle 3.2 meters in diameter with vertical walls.The depth of the water in this idealized harbor was 16 cm. An openingof angle n/8 radians in the harbor wall permitted a connection with alarge tank in which waves (simulating the open sea) were produced.Figures 10.13.4 and 10.13.5 are photographs of the model (taken fromthe paper by McNown), which also show two specific cases of symmetrical oscillations.(10.13.26). (ItThefree vibrations again aremight be noted thatgoverned by equationMcNown makesuse of the exactLONG WAVES IN SHALLOW WATERlinear theory ratheris that the relation2= ghm427than the shallow water theory.
The only differencebetween cr 2 and m is a 2 = gm tanh mh, instead of2as given above: the differential equation for the velocitypotential 0(x, y, z; t) in the exact linear theory treated in Part I is(T,andFig. 10.13.4.written in the formVsatisfiessought2<p+min the=2<pclosed,in= A cosh m(y + h)ea harbor modeliat(p(x, z),and(p(x, z)then0.) Solutions of the differential equation are<p(r,thatits0)= J n (mr) cos nOunder the assumption that the port isboundary is the whole circle r = R. As is wellcoordinatesi.e.Wavesform(10.13.30)in polar10.13.5.(r, 0),WATER WAVES428known, <p(r, 6) is a solution of (10.13.26 ) x only if J n (mr) is a Besselfunction of order w, and since it is reasonable to look only for solutionsthat are bounded we choose the Bessel functions of the first kindwhich are regular at theThe boundary condition=(10.13.31)andorigin.at rrequires that=R-^-==R.
Foreach n this transcendental equation has infinitely many rootseach corresponding to a mode of oscillation with various nodal diathis in turn leads to the conditiondJJdrfor rm^meters anda2= gm^and with a definite frequency which ismore accurately, by a 2gm^ tanh hm ^circles,fixed((or,).,byFigureby McNown, shows a comparison of observed andcalculated amplitudes for two modes of oscillation; the upper curveis drawn for a motion having no diametral nodes and two nodal circles,while the lower is for a motion having two nodal diameters and one10,13.6, obtainedY_(kr)-2-3-4oentranceFig. 10.13.6.Comparison ofObservedTheoreticalcenterresults of experiment and theory for resonantin a circular portmentsmove-LONG WAVES IN SHALLOW WATER429circle.
The motions were excited by making waves in the tank,and providing an opening for communication with the harbor, asnoted above. The figures were drawn assuming that the amplitudeswould agree at the entrance to the harbor the experimental checknodal10.18.7.Fig. 10.13.8.of aModel of a harbor with breakwaterthus applies only to the shapes of the curves. As one sees, the exvalues are remarkably close.
The ampliperimental and theoreticaltudes used were large enough so that nonlinear effects were observed:the troughs are flatter than the crests by measurable amounts. Ofcourse, having an opening in the harbor wall violates the boundarycondition assumed, but this effect apparently is slight: changing themouth had practically no effect onangle of the opening at the harborWATER WAVES430the waves produced, and, in addition,itwas found that verylittleradiates outward through the harbor entrance.ofharbor design, involving construction of breakwaters,Problemswave energylocation of docks, etc.
are commonly studied by constructing models.of a model of a harFigs. 10.13.7 and 10.13.8 show two photographsbor,* the first before a breakwater was constructed, the secondafterward. As one sees, the breakwater has a quite noticeable effect.waves approaching theFig.
10.13.9. shows the same model, with theharbor mouth at a different angle, however; as one sees the break-Fig. 10.13.9.Model of a harbor with breakwaterwater seems to be on the wholeless effectivewhenthewavefrontsare less oblique to the breakwater. The diamond-shaped pattern, dueto reflection, of the waves on the sea side of the breakwater is inter-Y////////7* /////////AFig. 10.13.10. Floating plane slab* Thesephotographs were given to the authortory at California Institute of Technology.by the Hydrodynamics Labora-LONG WAVES IN SHALLOW WATER481Model studies are rather expensive, and consequently it mightwell be reasonable to explore the possibilities of numerical solutionof the problems, perhaps by using appropriate modifications of theesting.method offinite differences outlined above for asimple case.turn next to a discussion of the effect of floating bodies onwaves in shallow water, on the basis of the theory presented in thissection.
Only two-dimensional motions will be considered (so that allquantities are independent of the variable z). The first case to bestudied is that of the motion of a floating rigid body in the form of aWethin plane slabproblems havearc atjcFig. 10.13.10) in water of uniform depth. Suchbeen treated by F. John [J.5]. The ends of the slab(cf.i a.In accordance with the theory presented above wesurface value 3>(tT, t) and the displacement r](x t)must determine the9of the board from the differential equationswith rj0, and dropping /^ and /2 ):(cf.=(10.13.32)#,,-^0,,,= - h& xx|*|(10.13.19), (10.13.15)>a< a.We have dropped the bar over the quantity 0.
We have also assumedthat fj(x) for x < a, the rest position of equilibrium of the board, is(10.13.33)rj t*,\\\\an approximation that is justified because we assumethat the board is so light that it does not sink appreciably below thewater surface when in equilibrium. (This assumption is by no meansnecessary it would not be difficult to deal with the problem if thissimplifying assumption were not made.)zero; thisisSince fj is zero, it follows (cf.
(10.13.24)) that the transition conditions at the ends of the board are(10.13.34)X,tcontinuous at xa.We are interested in the problem of the effectiveness of the floatingboard as a barrier to a train of waves coming from the right (x+00).The equation(10.13.32) has as-its=general solutionc = VgA+ G(x + ct),in terms of two arbitrary functions F and G (as one can readily verify0(x,1)=F(xct))which clearly represent a superposition of two progressing wavesmoving to the right and to the left, respectively, with the speed Vgh.It is natural, in our present problem, to expect that for x > a thereexist in general both an incoming and an outgoing wave bea we would precause of reflection from the barrier, while for xwould<WATER WAVES432scribe only awave going outward(i.e.Weto the left).shall see thatthese qualitative requirements lead to a unique solution of our problem.consider only simple harmonic waves; it is thus natural to writeWe&(x,(10.13.35)(10.13.36)= (p(x)e= v(x)et)ri(x, t)iat,ia\x-\\a> a,< x < a,with the stipulation that the real part is to be taken at the end.
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