J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 13
Текст из файла (страница 13)
In addition, the "groups"of waves thus defined in other words the configuration representedWATER WAVES52advance with the velocityin general be a functionwillaproblemby the dashed curves of Figureda/dmin the 0-direction. In our8.4.1.WaveFig. 3.4.1.mofbycso that the velocityda/dm 9 or, in terms of the= a/m,Ugroupsof the groupwave length Ais=given approximately27t/mand wave velocitybyU=(3.4.3)The matter canSommerfeldd(mc).dmdcdkalso beapproached in the following waywhich comes closer to the more usual[S.13]),(cf.cir-cumstances. Instead of considering the superposition of only twoprogressing waves, consider rather the superposition, by means ofan integral, of infinitely many waves with amplitudes and wavelengths which vary over a small range:+=(3.4.4)r* *A(m) expeJ(i(mxWQThe quantity mxmx(3.4.5)From(3.4.4)=atcan be written in the formaQtm#c=C expC=(m+Cr* *A(m) exp{i[(mJ WQa{i(m<pwhich the amplitude factor(8.4.7)+m(a)xaQ )t.one then finds(3.4.6)inat- at)} dm.ist)}>given by- mQ )x -(a-aWe are interested here in seeking out those places)*]}dm.and times(ifany)Crepresents a wave progressing with little changein form, since (3.4.6) will then furnish what we call a group of waves.Since x and t occur only in the exponential term in (8.4.7), it followswhere the function'that the values of interest are those for which this termnearly constant,i.e.those for which(mw)#(amust bea )t~ const.SIMPLE HARMONIC OSCILLATIONS58It follows that the propagation speed of(a(7)/(rara),andif(mw)issuch a group is given bysmall enough we obtain againthe formula (3.4.3).Evidently, it is important for this discussion of the notion of groupvelocity that the motion considered should consist of a superpositionof waves differing only slightly in frequency and amplitude.
Inpractice, the motions obtained in most casesthrough use of theFourier integral technique, for example,are the result of superposition of waves whose frequencies vary from zero to infinity andwhose amplitudes also vary widely. However, as we shall see inChapter 6, it happens very frequently that the motion at certainplaces and times is approximated with good accuracy by integralsof the type given in (3.4.4) with e arbitrarily small. (This is, indeed,the sense of the principle of stationary phase, to be treated in Chapter 6.) In such cases, then, groups ofcussion aboveiswaves doexistand thedis-pertinent.In our problems the relation between wave speed and wave lengthis given by (3.2.2) and consequently the velocity U of a group isreadily found, from (3.4.3), to be2Weobserve that the group velocity has the same value as was givenpropagation of energywave length as thoseof the group.
In other words, the rate at which energy is propagatedis given by the group velocity and not the phase velocity. This isoften considered as the salient fact with respect to the notion ofgroup velocity. As indicated already in the preceding section, thein the preceding section for the average rate ofin a uniform train of waves having the sameauthor does not share this view, butfeels rather thatthe kinematicconcept of group velocity is of primary significance, while the notionof velocity of propagation of energy might better be discarded. Ittrue that the two velocities, in spite of the fact that one is derivedfrom dynamics while the other is of purely kinematic origin, turnout to be the same not only in this case, but in many others aswell* but it is also true that they are not always the same forexample, the two velocities are not the same if there is dissipationof energy in the medium. In addition, we have seen in the precedingsection that the notion of velocity of energy can be derived when noisAgeneral analysis of the reason for this has been givenby Broer[B.I 8].54WATER WAVESwave group exists at all we in fact derived this velocity for the caseof a wave having but one harmonic component.In Chapter 6 we shall have occasion to see how illuminating thekinematic concept of a group and its velocity can be in interpretingand understanding the complicated unsteady wave motions whicharise when local disturbances propagate into still water.CHAPTER4Waves Maintained by Simple Harmonic Surface Pressurein4.1.Water of Uniform Depth* ForeedOscillationsIntroductionIn our previous discussions we have considered always that thepressure at the free surface was constant (usually zero) in both spaceand time.
In other words, only the free oscillations were treated andthe problems were, correspondingly, linear and homogeneous bounary value problems. Here we wish to consider two problems in whichthe surface pressure p is simple harmonic in the time and the resultingmotions are thus forced oscillations; the problems then also have anonhomogeneous boundary condition. In the first such problem weassume that the motion is two-dimensional and that the surface pressure is a periodic function of the space coordinate x over the entire#-axis; in the second problem the surface pressure is assumed to bezero except over a segment of finite length of the #-axis. In theseproblems the depth of the water is assumed to be everywhere infinite,but the corresponding problems in water of constant finite depthcan be, and have been, solved by much the same methods.The formulation of the first two problems is as follows.
A velocitypotential 0(x, y;t) is to be determined which is simple harmonic inthe time t and satisfiesV2(4.1.1)Thesurface pressure p(x;=t)p(x;(4.1.2)for=p(x) sinand the boundary conditions at thecondition(4.1.3)(cf.(2.1.20)!)r]=< 0.given byist)y-&t&55at,free surface are thedynamicalWATER WAVES50and the kinematic condition(4.1.4)=rity.means that no kinematic constraint is imposedcan deform freely subject to the given pressuredistribution. In addition, we require that <P t andshould beyThelast conditionon the surfaceitoo. This means effectively that the verticalvertical velocity components are bounded.
Insection 4.3, the amplitude p(x) of the surface pressure p will haveuniformly bounded atdisplacement anddiscontinuities atditionsontwo points and weat these pointsimpose appropriate conconsider this case.shallwhen weWe seek the most general simple harmonic solutions of our problem;they have the form= <p(x, y) cos at + y(x, y) sin at.(4.1.5)The functions <p andplane.
The conditionsy;<pm<p(p yma=p(x)yfory=0.Om=a2/g;they yield the condition(4.1.8)The phase__defined by(4.1.7)while for(4.1.4) arc easily seen tothe boundary conditionCiwith the constantand(4.1.2), (4.1.3),yield for the function(4.1.6)are of course harmonic in the lower half\p ysin atassumedmy =forpforyin (4.1.2) has the effect that^satisfiesthe homogeneous free surface condition, as one sees.We know from the first section of the preceding chapter that theonly bounded and regular harmonic functionscondition (4.1.8) are given by.V(%,y)satisfy themx1-sin mx\f(4.1.9)y whichcosIn the next two sections we shall determine the function (p(x, y),which has the phase cos at, in accordancethat part ofwith two different choices for the amplitude p(x) of the surfacei.e.pressure p.SIMPLE HARMONIC SURFACE PRESSURE4.2.The surface pressureWe consider nowinperiodic for all values ofis(4.1.2)=and(4.1.6)ooPsinAtf,is<x <p(x)Oneonce that the following function(p(x,y)=aPQgisisperiodicgiven by(4.2.1)(4.2.2)xthe case in which the surface pressurex such that p(x) inverifies at57oo.(p(x, y):e* vsinmfaAa harmonic function which satisfies the free surface boundarycondition (4.1.0) imposed in the present case.
Since the difference #of two solutions q> l9 q> 2 both satisfying all of our conditions would0, it followshomogeneous boundary condition % vm%ofourvaluecanbeobtainedboundaryproblemby<psolutionsolutionoftothetheaddinggiven by (4.2.2) anyspecialhomogeneous problem, and these latter solutions are the functionsy given by (4.1.9) since jj satisfies the same conditions as y. Thereforethe most general simple harmonic solutions of the type (4.1.5) aresatisfy thethatallsolutionsgiven in the present case by(4.2.3)0(<r,?/;t)-eXvVaP[_Qg+ Ae*sin famAfcos0M?]l\.[sinmx ]sin mx{cos\mx\\ cos atjjsin at,JABconstants which are at our disposal.
In other words,the resulting motions are, as usual in linear vibrating systems, alinear combination of the forced oscillation and the free oscillations.withandThese solutionsgivenWefirstwithout the uniqueness proofby Lambseem to have been[L.2].mmust be excluded, and that if Aofthesurfacewaves arc to be expected.near tolarge amplitudesThis means physically, as one sees immediately, that waves of largeamplitude are created if the periodic surface pressure distributionhas nearly the wave length which belongs to a surface wave of theisobserve that the case Amthat is, the wavefor pressure zero at the surfacefreeoscillation.ofthecorrespondinglengthIf instead of (4.1.2) we take the surface pressure as a progressingsame frequencywave of the form(4.2.4)p(x\t)= H sin(atfa)WATER WAVES58it isreadily found that progressing surfacegiven by0(x, y;t)(4.2.5)--= HaQgTothiswavese^m-cos (at-resultwhich arefa)./.one may, of course, add any of the wave solutions whichoccur under zero surface pressure.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
