J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 32
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We observefirst that the oscillatory factors in the four approximate formulas(6.5.12)(6.5.15) do not differ essentially, but the slowly varyingnonoscillatory factors are different in the various cases: (a) at a fixedpoint on the water surface the disturbance increases in amplitudelinearly in t in the case of an initial elevation concentrated at a point(cf. (6.5.12)), while for a fixed time the amplitude becomes large forsmall x like #- 3/2 (b) in the case of an initial impulse concentratedat a point the amplitude increases quadratically in t at a fixed point,while for a fixed time the amplitude increases like ^~ 5/2 for smallx.
(In these limit cases the approximate formulas are valid for x ^ 0,since the only other requirement is that the quantity gfifax should beis not very surlarge.) The behavior of these solutions near xThethere.sinceisabehaviorat any fixedtheresingularityprising->oo is, however, somewhat startling: the amplitude ispoint x as tseen to grow large without limit as the time increases in both of thesecases.
This rather unrealistic result is a consequence of the fact that;the singularity at the origin is very strong. If the initial disturbancewere finite and spread over an area, the amplitude of the resultingmotion would always remain bounded with increasing time, as onecould show by an appeal to the general behavior of Fourier transforms.* This fact is well shown in the special case of a distributedimpulse, as we see from (6.5.14), whichthe amplitude remains bounded as/:istvalid for allx^=andlarge-> oo.The general character of the waves generated by a point disturbanceis indicated schematically in the accompanying figures which showthe variation in surface elevation at a fixed point x when the time increases, and at a fixed time for all x.
These figures are based on theformula (6.5.12) for the case of an initial elevation; the results for thecase of initial impulse would be of the same general nature.* It is also a curious fact that the motion given by (6.5.14) for the case ofan impulse over a segment requires infinite energy input, since the amplitudeat any fixed point does not tend to y.ero. For the case of an initial elevation confinedto a segment, however, the wave amplitude would die out with increasing time.WATER WAVES168worth while to discuss the character of the motion furnishedby (6.5.12) in still more detail.
It has already been remarked that anyparticular phase such as a zero, or a maximum or minimum of r\ isof necessity propagated with an acceleration since each such phase isassociated with a particular constant value of the quantity g* 2 /4#: ifthe phase is fixed by setting g* 2 /4# = c, then this phase moves inaccordance with the relation x = g* 2 /4c. The formula (6.5.12) holds2only where the quantity gt /4>x is large, and hence the individual phaItisses are accelerated slowly in the region of validity of this formula; or,Fig.
6.6.1a,b Propagation ofwaves due to aninitial elevationUNSTEADY MOTIONSin other words, thephasesmove169such regions at nearly constantx or t the waves behaveinvelocity. Also, for not too great changes inharmonic waves of a certain fixed period andwave length. This can be seen as follows. Suppose that we vary talone from / == t Q to t = t Q + At. We may write for the phase <p:very nearlylike simpleas one readily verifies.ThusifAt/t Qissmall,i.e. ifthe change At in thesmall compared with the total lapse of time since the motionwas initiated, we have for the change in phase:timeisConsequently the period T = At of the motion corresponding to the2n in the phase is given approximately by the formulachange Aq>T(6.6.1)this formula is good, as we know, if T/t Q c^Lthisis the case since g/M#o s always assumed to be large.andsmall,Thus the period at any fixed point varies slowly in the time.
In thesame way one finds for the local wave length A the approximateThe accuracy ofis*formulaA(6.6.2)goby varying with respect to <r alone, and this is also easily seen to beaccurate if gt%/4x Q is large. Thus for a fixed position x the period andwave length both vary slowly, and they decrease as the time increases,while for a fixed time the same quantities increase with x, as is borneout by the figures shown above.of considerable interest next to compute the local phase velo2ccitythe velocity of a zero of 77, for example from gt /4xIt is=when x andtvary independently; the resultdx(6 6 3>--for the velocity of=any phase; thusis2xTfor fixedx the phases move moreWATER WAVES170slowly as the time increases, but for fixed t more rapidly as x increasesthat is, the waves farther away from the source of the disturbancemove more rapidly, and they are also longer, as we know from (6.6.2).The wave pattern is thus drawn out continually, and the waves asthey travel outward become longer and move faster.
The last fact isnot too surprising since the waves in the vicinity of a particular pointhave essentially the simple character of the sine or cosine waves offixed period that we have studied earlier, and such waves, as we haveseen in Chapter 3, propagate with speeds that increase with the wavelength.
All of theabove phenomena can be observed as the result ofthrowing a stone into a pond; though the motion in this case is threedimensional it is qualitatively the same, as one can see by comparing(6.5.15) with (6.5.12).another way of looking at the whole matter which isprompted by the last observations.
Apparently, the disturbance atthe origin acts like a source which emits waves of all wave lengths andfrequencies. But since our medium is a dispersive medium in whichthe propagation speed of a particular phase increases with its wavelength, it follows that the disturbance as a whole tends with increasingtime to break up into separate trains of waves each of which has ap-Thereisproximately the same wave length, since waves whose lengths differmove with different velocities. However, it would be a mistake tothink that such wave trains or groups of waves themselves move withthe phase speed corresponding to the wave length associated with thegroup. If one fixes attention on the group as a whole rather than onan individual wave of the group, the velocity of the group will be seento differ from that of its component waves.
The phase velocity for thepresent case can be obtained in terms of the local wave length readily(6.6.3) by expressing its right handwave length through use of (6.6.2); thefrom the equationside in termsof the localresultdx(6.6.4)'*V=2x=*iisIn ;l/^L.V 271Onthe other hand, the position x of a group of waves of fixed wavelength A at time t is given closely by the formula=(6.6.5)aswesee directlyfrom(6.6.2), sothat the velocity of the groupisUNSTEADY MOTIONSwhich171its component waves. In other words, the component waves in a particulargroup move forward through the group with a speed twice that of theis,evidently, just half the phase speed ofgroup.Finally, we observe that these results are in perfect accord with thediscussion in Chapter 3 concerning the notions of phase and groupvelocity.The phase speed c for a simple harmonic waveof wave length=A in water of infinite depth is given (cf. (3.2.3)!), by cVgA/2jr,and this is also the phase speed of the waves whose wave length isA as we sec from (6.6.4).
We have also defined in section 3.4 thenotion of group velocity for simple harmonic waves in water of infinitedepth, and found it to be just half the phase velocity. The kinematicdefinition of the group velocity given in section 3.4 was obtained bythe superposition of trains of simple harmonic waves of slightly differ-ent wave length and amplitude, while in the present case the wavesarc the result of a superposition of waves of all wave lengths andHowever, the principle of stationary phase, which furnishesthe approximate solution studied here, in effect says that the mainmotion in certain regions is the result of the superposition of waveswhose wave lengths and amplitudes differ arbitrarily little from aperiods.certain given value.
The results of the analysis in the present case arethus entirely consistent with the analysis of section 3.4.At any time, therefore, the surface of the water is covered by groupsof waves arranged so that the groups having waves of greater lengthare farther away from the source. These groups, therefore, tend toseparate, as one sees from (6.6.4). The waves in a given group do notmaintain their amplitude, however, as the group proceeds: one seesreadily from (6.5.12) in combination with (6.6.2) that their amplitudeis proportional to l/^/x for waves of fixed lengthLThe aboveborne out byinterpretations of the results of the basic theory are allexperience.
Figure 6.6.2 shows a time sequence of photo-graphs of waves (given to the author by Prof. J. W. Johnson of theUniversity of California at Berkeley) created by a disturbanceconcentrated in a small area: the decrease in wave length at a fixedpoint with increasing time, the increase in the wave lengths near thefront of the outgoing disturbance as the time increases, the generaldrawing out of the wave pattern with time, the occurrence of welldefined groups,Aninterestingetc.are well depicted.developmentinoceanography has been based on thetheory developed in the present section. Deacon [D.6, 7]andhisWATER WAVES172Fig. 0.6.2.Waves dueto a concentrated disturbanceUNSTEADY MOTIONSFig.
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