J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 56
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Thus u and c would be known asIft=we wisht Q9functions of x for that particular time.j\Of course, the surface elevationwould also be known fromc=Vg(h+fi).10.4. Propagation of disturbances intostillwater of constant depthIn the preceding section we have seen how the method of characthe notion of a simple wave in terms of which we candescribe with surprising ease the propagation of a disturbance initiated at a point into water of constant depth moving with uniform speed.In the present section we consider in more detail the character of theteristics leads tosimple waves which occur in two important special cases.
We assumeand that it then propagatesalways that the pulse is initiated at xin the positive ^-direction into still water. Thus we are considering=cases in which the flowissubcritical at the outset.Oneof the most striking and important features of our whole discussion is that there is an essential difference between the propagationof a pulse which is created by steadily decreasing the surface elevationand of a pulse which results by steadily increasing the eleat xrf0.
If the pulse is created by initiating a change in thevation at x===particle velocity u at xmoving a vertical barrier at(which might be achieved simply bywith the prescribed particle veloxthesurfaceelevation rj the same typicalofinsteadchangingbycity)differences will result if u is in the first case decreased from zerothrough negative values, and in the other case is gradually increased=WATER WAVES306=are given init becomes positive (i.e.
if the particles at xthe first case a negative acceleration and in the second case a positiveacceleration.) The qualitative difference between the two cases fromso thatthe physical point of view is of course that in the first case it is adepression in the water surface and in the second case an elevationsometimes referred to later on as aabove the undisturbed surfacewhich propagates into still water.were to consider waves of very small amplitude so that we mightlinearize our equations (as was done in deriving equation (10.1.13))there would be no essential qualitative distinction between the motionsin the two cases; that there is actually a distinction between the twois a consequence of the nonlinearity of the differential equations.In the preceding section we have seen that the motions in either ofour two cases can be described in the #, /-plane by means of a family ofstraight characteristics which issue from the /-axis.
In Figure 10.4.1we show these characteristics together with a curve indicating a sethumpIf weof prescribed values for cVg(A+^)=<*(0atoc=0,whichinturn result from prescribed values of rj at that point. We assume thatin the zone of quiet /. Hence the slope docjdt of any straightuUQ=from a pointcordance with (10.3.6) bydxcharacteristic issuing~r on the /-axis_=8c(T)-2c c(10.4.1)When r is varied(10.4.1Thethe zone //.tics intisgiven, in ac-.yields the complete set of straight characterisvalues oft/ and c along the same characteristic)are constant (as we have seen in the preceding section), so that thevalue of u along a characteristic is determined, from (10.3.5) byu(r)(10.4.2)sinceUQ-2[c(r)-c],assumed to be zero andc(r) is given.are now in a position to note a crucial differencecases described above. In the first of the two casesisWetwodepressionmovingintostillwaterthe elevationassumed to be a decreasing function ofwith increase of/.i.e.that of aat xisso that c(t) also decreasesIt follows that the slopes dxjdt of the straight line/characteristics as given by (10.4.1) decrease asfamily of straight characteristics diverge on*rj(t)between the/increases* so that themoving out from theOne should observe that decreasing values of dx/dt mean that the characmake increasing angles with the #-axis, i.e.
that they become steeper withteristicsrespect to the horizontal.LONG WAVES IN SHALLOW WATER307(This is the case indicated in Fig. 10.4.1). In the second case,however, the value of 77 and thus of c is assumed to be an increasingtf-axis.function oftatx=so that the straight characteristicsmust cven-c(0,t)Fig. 10.4.1.
Propagation of pulses intostillwaterhave an envelope in generalandriot be expected to have at beyond those for whichsuch intersections exist. In the first case the motion is continuousthroughout. What happens in the second case beyond the pointwhere the solution is continuous can not be discussed mathematicallyuntil we have widened our basic theory, but in terms of the physicalbehavior of the water we might expect the wave to break, or to develop what is called a bore,* some time after the solution ceases to betually intersectin fact, they willmeans that our problem cancontinuous solution for values of ^r andthis in turnwe propose to discuss the question of thebreakersandbores in some detail.ofdevelopmentaboveare the exact analogues of two casesdiscussedThe two caseswell known in gas dynamics: Consider a long tube filled with gas atcontinuous. In later sectionsand closed by a piston at one section.
If the piston is moved awayfrom the gas with increasing speed in such a way as to cause ararefaction wave to move into the quiet gas, then a continuous motionresults. However, if the piston is pushed with increasing speed intorestthe gas so as to create a compression wave, then such awave always* In certain estuaries in variousparts of the world the incoming tides from theocean are sometimes observed to result in the formation of a nearly vertical wallof water, called a bore, which advances more or less unaltered in form overquite large distances.
What is called a hydraulic jump is another phenomenonof the same sort. Such phenomena will be discussed in detail later on.308WATER WAVESdevelops eventually into a shock wave. That is, the development ofa shock in gas dynamics is analogous to the development of a bore(and also of a hydraulic jump) in water.10.5. Propagation of depression waves into still water of constant depthIn this section we give a detailed treatment of the first type ofmotion in which a depression of the water surface propagates intowater. However, it is interesting and instructive to prescribe thedisturbance in terms of the velocity of the water rather than in termsof the surface elevation.
We assume, in addition, that the velocity isx(t) of the water particlesprescribed by giving the displacement xstilloriginally in the vertical plane atx== 0,* and this, as we have remark-ed before, could be achieved experimentally simply by moving a vertical plate at the end of a tank in such a way that its displacementisx(t).** Figure 10.5.1 indicates the straight characteristicswhicht= const.Fig. 10.5.1.Adepressionwaveon the "piston curve" x = x(t). The piston is assumed tofrom rest and move in the negative direction with increasingspeed until it reaches a certain speed w < 0, after which the speedremains constant. That is, x decreases monotonically from zero atuntil it attains the value w, after which it stays constant at thatt =initiatestarttvalue.
In Fig. 10.5.1 this point*ismarked B;clearly the piston curveisIn our theory, it should be recalled, the particles originally in a verticalplane remain always in a vertical plane.**Moving such a plate at the end of a tank of course corresponds in gas dynamics to moving a piston in a gas-filled tube.LONG WAVES IN SHALLOW WATER309a straight line from there one.
At any point A on the piston curve wehave U A = x t (t), corresponding to the physical assumption that thewater particles in contact with the piston remain in contact with itand thus have the same velocity. If we consider the curved characteristic drawn from A back to the initial characteristicCj which terminates the zone / of rest we obtain from (10.3.5) the relationsince in our caseis= Ki + *o.CA(10.5.1)uthus given by0.(cf.Theslope of the straight characteristic at(10.4.1)):dx(10.5.2)3= -U Awe have assumed that U A~ w it follows fromSinceAxt(t)always decreases astincreasesxt(10.5.2) that dx/dt also decreases as t increases in this range of values of t so that the characteristics divergeas they go outward from the piston curve. Beyond the pointtheuntilB=wstraight characteristics are parallel straight lines, since U Aconst, on that part of the piston curve, and the state of the watertherefore constant in the zoneismarked ///in Fig.
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