J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 12
Текст из файла (страница 12)
Thisfact is in accord with what resulted in Chapter 2 upon linearizingthe shallow water theory (cf. (2.2.16)) and the sentence immediately=Vghfollowing), which led to the linear wave equation and to cas the propagation speed for disturbances.can gain at least aWerough idea of the limits of accuracy of the linear shallow water2theory by comparing the values of c given by cgh with thosegiven by the exact formula=c*(3.2.7)=tanh2nAOne finds that c as given by% if the wave length is ten times theif the wave length is twenty times thefor various values of the ratio A/A.Vghisin errordepth and bydepth. Theby about 6lessthan 2%error of course increases or decreases with increase ordecrease in A/A.In water of infinite depth, on the other hand,observed (cf.
(3.2.2)) that(3.2.8)c2we have already= gA/2jr.computed by the formula c = gA/2rc islessthanifalready1/2 %A/A > |. One might therefore feel justifiedin concluding that variations in the bottom elevation will have butslight effect on a progressing wave provided that they do not resultin depths which are less than half of the wave length, and observationsseem to bear this out. In other words, the wave would not "feel" thebottom until the depth becomes less than about half a wave length.Actually, the error in c as2of interest to determine the paths of the individual waterparticles as the result of the passage of a progressing wave.
As in theItispreceding section we take dx and dy to represent the displacementsof a particle from its average position, and determine those displacements from the equationsSIMPLE HARMONIC OSCILLATIONS=X=y= Am sinh m(y + ^) cosAm cosh m(y + h) sin(mx47-\-at-\- a),dtddy-.-.-+ at + a),(mx(zrsinceisgiven by (3.2.1) in the present case. Integration of theseequations yieldsdx=-cosh m(y+/i)cos(rna:+ at + a),(3.2.9)dy--- sinhm(ya+ ^) s n m# +iso that the path of a particle at depthdx(yisan(rf+ a),ellipse22,fy_~~la?with semi-axes a and 6 given bya=.ab=cosh m(y+ h)sinh+ h).aOnm(ythe ellipse degenerates into a horizontalas one would expect.
Both axes of the ellipse shortenthe bottom, yh,straight line,with increase in the depth. For experimental verification of theseresults, the discussion with reference to Fig. 3.1.1 should be con-water of infinite depth the particle paths would be circles,as one can readily verify. The fact that the displacement of theparticles dies out exponentially in the depth explains why a submarineneed only submerge a slight distance below the surface a half wavein order to remain practically unaffected even by severelength, saysulted. Instorms.3.3.Energy transmissionforsimpleharmonic waves of smallamplitudeIn Chapterfluidandderiveditsforthe general formulas for the energy E stored in aflux or rate of transfer F across given surfaces were1the most general types of motion.In this sectionWATER WAVES48we applythese formulas to the special motions considered in thepresent chapter, that is, under the assumption that the free surfacestored in aconditions are linearized.
The formula for the energyERregionis(1.6.1)):(cf.E=(3.3.1)2[J(<^e+ 0; +2<*>* )+ gy]dxdydz;Rwhile the flux of energyspaceisgiven by(cf.Fin a timeTacross a surfaceSGfixed in(1.6.5)):F-(8.3.2)Weofsuppose first that the motion considered is the superpositiontwo standing waves which are simple harmonic in the time, asfollows:(3.3.3)= 9^(0?, y, z) cos at +(3.3.2) with T = 2jc/a<P(x, y, z; t)Insertion of this in99,9(#,i.e.j/,z) sin at.for a time intervalequal to the period of the oscillation, leads at once to the followingthrough SGexpression for the energy fluxFOne observes that the energy:flux over a periodiszeroifeither(p lthe motion is a standing wave: a fact which9? 2 vanishes,is not surprising since one expects an actual transport of energyonly if the motion has the character of a progressing wave.
Stillanother fact can be verified from (3.3.4) in our present cases, in whichand (p 2 are, as we know, harmonic functions: if S G is a fixed closed(p lsurface in the fluid enclosing a region R Green's formula states thatori.e. ifprovided that 9^ and <p 2 have no singularities sources or sinks forexample in R. In this case the energy flux F clearly vanishes sinceand <p 2 are harmonic. Also one sees by a similar reasoning that the9?!flux F over a period remains constant if SG is deformed withoutpassing over singularities. In particular, the energy flux through avertical plane passingfrom the bottom to the free surface of the waterSIMPLE HARMONIC OSCILLATIONS49in a two-dimensional motion would be the same (per unit width ofthe plane) for all positions of the plane provided that no singularitiesare passed over.
This fact makes it possible, if one wishes, to con-sider the energy in the fluid as though the energy itselfincompressible fluid, and to speak of its rate of flow.were anIn the literature dealing with waves in all sorts of media, butparticularly in dispersive media, it is indeed commonly the customto introduce the notion of the velocity of the flow of energy ac-companying a progressing wave, and to bringthis velocity in relationvelocity (to be discussed into the kinematic notion of the groupthe next section). The author has foundand feels thatself to these discussions,ititdifficult to reconcilehim-would be better to discardthe difficult concept of the velocity of transmission of energy, sincethis notion is not of primary importance, and nothing can be ac-complished by its use which cannot be done just as well by usingthe well-founded and clear-cut concept of the flux of energy througha given surface.
On the other hand, the notion is used in the literature(and probably will continue to be used) and consequently a disincluded here, following pretty much the derivationgiven by Rayleigh in an appendix to the first volume of his Sound[R.4], In the next section, where the notion of group velocity iscussion ofit isintroduced, some further comments about the concept of the velocityof transmission of energy will be made.Weconsider the energy flux per unit breadth across a verticalplane in the case of a simple harmonic progressing wave in waterof uniform depth (or, in view of the above remarks, across any surfaceof unit breadth extending from the bottom to the free surface).velocity potentialgiven by(3.2.1))(cf.= A cosh m(y + h) cos(3.3.5)andisThe(mx+ at + a)(3.3.2) yields(3.3.6)F=+2 * /<JA*Q<jm J*ft*C osh 2m(y+ h)dy\ sin2+ at) dtT = 2jt/a(mxfor the flux across a strip of unit breadth in the time9the period of the oscillation. Hence the average flux per unit timeisgiven by(* q 7^(3.3.7)tw avF=-since the average of sin 2 6 over a periodis1/2.Wehave also takenWATER WAVES50=in the upper limit of the integral in (3.3.6) and thus neglecteda term of higher order in the amplitude.
It is useful to rewrite theformula (3.8.7) in the following form through use of the relationsor*a/m:gm tanh mh and cr)==F av =(8.8.8)si- cosh*mhU,Owith U a quantity having the dimensions of a velocity and givenby the relationU(3.3.9)Next we calculate the average energy stored in the water as a resultof the wave motion with respect to the length in the direction ofpropagation of the wave.
This is obtained from (3.3.1) by calculatingfirst the energy JEj over a wave length A = 2n/m at any arbitraryfixed time. In the present caseEKE =m2Qr+ A+(3.3.10)2we haveAJ [$A2+ h) cos (mx + at + a)+ h) sin (mx + at + a)] dxdy2sinh 2 m(ycosh 2 m(y2Ewhich the constant Q refers to the potential energy of the waterof depth h when at rest.
On evaluating the integrals, and ignoringcertain terms of higher order, we obtain for the energy between twoinplanes awave length apartarisingfrom the passage of the wave theexpression(3.3.11)EI- A cosh-E =2nth,2gEas one finds without difficulty. Thus the average energyav in thefluid per unit length in the ^-direction which results from the motionisgiven by(3.3.12)E av =-cosh 2 mh.2gequation (3.3.8) we observe that E av isthecoefficientofU in the formula (3.3.8) for the averageexactlyenergy flux per unit time across a vertical plane.
It therefore follows,assuming that no energy is created or destroyed within the fluidUpon comparison withSIMPLE HARMONIC OSCILLATIONSitself,that the energywave ontheis51transmitted in the direction of propagation ofU. As we see from (3.3.9)the average with the velocityU is not the same as the phase or propagation velocity c;the velocityU isalways less than c: for water of infinite depth it has thevalue c/2 and it increases with decrease in depth, approaching thephase velocity c as the depth approaches zero.in fact,3.4.Groupvelocity.
DispersionIn any body of water the motion of the water in general consistsof a superposition of waves of various amplitudes and wave lengths.For example, the motion of the water due to a disturbance over acan be analyzed in terms of the superposition of infinitely many simple harmonic wave trains of varyingamplitude and wave length; such an analysis will in fact be carriedout in Chapter 6. However, we know from our previous discussion(cf. (3.2.7)) that the propagation speed of a train of waves is anrestricted area of the surfacewave length in other words, the waveweconcerned arc subject to dispersionwithwhicharephenomenaand thus one might expect that the waves would be sorted out astime goes on into various groups of waves such that each groupwould consist of waves having about the same wave length.
We wishto study the properties of such groups of waves having approximatelyincreasing function of thethe samewavelength.Suppose, for example, that the motion can be described by thesuperposition of two progressing waves given by(341)with=Adm andof the(3.4.2)([m+ dm]x[a+ da]t)da considered to be small quantities. The superpositiontwo wave=sin2Atrains yieldscos-2(xdmVtda) sinm+I\L\x2 J\aL-\--\t\2j/= B sin (m'x o't)with m' = m + dm/2, a' = a + da/2. Since dm and da are small itso that <Pfollows that the function B varies slowly in both x andtisan amplitude-modulatedsine curve at each instant of time, asindicated schematically in Figure 3.4.1.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
