J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 57
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10.5.1.isThe zone //thus a region of non-constant state connecting two regions of differ-=ent constant states. Since C A^/g(h-\-r] A ), where r\ A refers to theelevation of the water surface at the piston, it follows from (10.5.1)thatt]A decreases in thezone // astincreases,i.e.the water surface atthe "piston" moves downward as the piston moves to the left, since weassume* that U A decreases as A moves out along the piston curve.
Sinceu and c are constant along any straight characteristic it is not difficultto describe the character of the motion corresponding to the disturbedconst. Its interzone // at any time t: Consider any straight line t=section with a characteristic yields the values of u and c at that pointthey are the values of u and c which are attached to that characteristic.Since the characteristics diverge from the piston curve one seesincreases upon moving from the piston torj steadilythat the elevationthe right and the particle velocity decreases in magnitude, until theinitial characteristicCj is reached after which the water is undisturbed.if attention is fixed on a definite point x >in thewater and the motion is observed as the time increases it is clear oncemore because the characteristics diverge that the water remains unafterdisturbed until the time reaches the value determined by x = cwhich the water surface falls steadily while the water particles passingOn the other hand,,WATER WAVES310that point move more and more rapidly in the negative ^-direction.In the foregoing discussion of a depression we have made an assumption without saying so explicitly,=+Csuch that CA\UAturn requires thati.e.that the speed UA of the piston isis not negative, and this in(10.5.1))Q (cf.- UA ^ 2c(10.5.3).UA increases monotonically to the terminal valuew itSincefollows thatw must be assumed in the above discussion to have atFig.
10.5.2.Alimit caselimit case in whichw just equals 2c isSincethehavethe slope dxfdtcharacteristicsinteresting.straightuc and since C Afrom (10.5.1) when UA2c it follows inthis case that dx/dtUA on the straight part of the piston curve.most the value 2c Q The.===+=,Butthis means that the straight characteristics have all coalescedinto the piston curve itself in this region, or in other words that thezone /// has disappeared in this limit case. The circumstances areindicated in Fig. 10.5.2.ofthe elevationleftBAt thefront of theof the waterr\ Aiswavefor values ofequal tox to theh fromCA=+ V!A ) == 0> which means that the water surface just touches thebottom at the advancing front of the wave.It is now clear what would happen if the terminal speedw of theThe zone // would terminate on thepiston were greater than 2ctothecurvedrawnfrom the point where the pistontangentpiston2cxThet just equalsspeedregion between this terminal characteristic and the remainder of the piston curve beyond it might becalled the zone of cavitation, since no water would exist for (x, t):.LONG WAVES IN SHALLOW WATER311values in such a region.
In other words, the piston eventually pullsitself completely free from the water in this case. Quite generally wesee that the piston will lose contact with the water (under the cir-cumstances postulated in this section, of course) if, and only if, itOnce this happens it is clear that thefinally exceeds the speed 2cnofurtheronhaseffectthemotion of the water. These circumpiston.stances are indicated in Fig. 10.5.3.CovitotionZoneFig.
10.5.3. Case of cavitationIf the acceleration of the pistonisassumed to beinfinite so thatspeed changes instantly from zero to the constant terminal valuemotion which results can be described very simply by exThe general situation in the x, 2-plane is indicated informulas.plicititsw, theFig. 10.5.4. This case might be considered a limit case of the oneindicated in Fig. 10.5.1 which results when the portion of the pistonshrinks to a point. Thecurve extending from the origin to pointBu=wFig. 10.5.4. Centered simplewaveWATER WAVES812consequence is that the straight characteristics in zone // all passthrough the origin. The zone /// is again one of constant state. Inthe zone // we have obviously for the slopes of the characteristicsdx(10.5.4)Ttx=7At the same time we have from?-(10.5.5)It follows that thezone //zonefu(10.5.2) dxjdt^ u+ c+cso that.terminated on the upper side by the line*(10.5.6)Fromis=.(10.5.5)and(10.5.1)we can obtainthe values ofu andcwithinHiu(10.5.7)C(10.5.8)Since c^=2 (x(-1=_tt+Co==we must havex/t\cA1andIx_^_^2cw mustso thatbe^2c2cconformity with a similar result above.
If wthe terminal characteristic of zone // is given, from (10.5.6), byfromx=c=(10.5.6) in,=2cQtwt and this line falls on the piston curve since the slopeof the piston curve is w. In this limit case, therefore, the zone ///collapses into the piston curve. If the piston is moved at still higherspeed, then cavitation occurs as in the cases discussed above sinceat the front of the wave, or in other words, the water surfacetouches the bottom.Fromsince c(10.5.9)(10.5.8)=Vg(hwe can+77)calculate the elevationrjof the water surface;>?+h=+ 2c=In the case of incipient cavitation, i.e.hw 2c we have r\ =at the front of the wave.
The curve of the water surface at any time tis a parabola from the front of the wave to the point x = c Q t (cor,responding to the characteristic which delimits the zone of quiet),after which it is horizontal. In Fig. 10.5.5 the total depth rjh of+LONG WAVES IN SHALLOW WATERthe waterwaterisplotted against x for a fixed timetangent to the bottom at the front xist.The313surface of the2c Qt of themovingwhich the water is in motion extends from thispoint back to the point x = c Qt. From (10.5.7) we can draw the following somewhat unexpected conclusion in this case: Since t may bewater.The regioningiven arbitrarily large values it follows that the velocity u of the waterat any fixed point x tends to the valuescas t grows large.The case of cavitation may have a certain interest in practice: themotion of the water might be considered as an approximation to theflow which would result from the sudden destruction of a dam builtin a valley with very steep sides and not too great bottom slope (cf.Waterx=-2cxsctolFig.
10.5.5. Breaking of athe paper ofRe[R-dam5 ])- If the water behind thedamwere 200 feethigh, for example, our results indicate that the front of the wavewould move down the valley at a speed of about 110 miles per hour.=in (10.5.9) we observe that the depth of the watersetting xat the site of the dam is always constant and has the value \h,Byi.e.Thefour-ninths of the original depth of the water behind the dam.velocity of the water at this point is also constant and has the==value uf Vgh, as we see from (10.5.7).
The volumef CQrate of discharge of water at the original location of the dam is thusconstant.So far we have not considered the motion of the individual waterparticles.u(x,t) isHowever, that is readily done in all cases once the velocityknown: We have only to integrate the ordinary differentialequationdx(10.5.10)In zone // in our present case we haveWATER WAVES314dxBytial2Ix+ 2c one finds readily that== Atequation d$/dt = 2/3t, from whichsettingconstant.xtHence we have(10.5.12)for the position x(t)x=t{At~11 *2csatisfies the differen-Aan arbitraryof any particle in zone //2t3with}.In the case of cavitation this formula holds for arbitrarily largewe have for large t the asymptotic expression for x:tsothatx~ - 2c(10.5.13)Q t.~fc(This is not in contradiction with our above result that ufor large t and fixed x since in that case different particles pass thepoint in question at different times, while (10.5.13) refers always tothe same particle).In the first section of Chapter 12 this same problem of the breakingof a dam will be treated by using the exact nonlinear theory in sucha manner as to determine the motion during its early stages after thedamhas been broken in other words, at the times when the shallowwater theory is most likely to be inaccurate.10.6.
Discontinuity, or shock, conditionsThewhich propagates intostillhump has already beeninthethemotioniscontinuousfirstout:casethroughout, butpointedin the second case the motion can not be continuous after a certaintime. The general situation is indicated in Fig. 10.6.1, which showsthe characteristics in the x, -plane for the motion which results whena "piston" at the end of a tank is pushed into the water with steadilyincreased speed.
As before, the slope dxjdt of a straight characteristic= x(t) is given (cf. (10.5.2)) byissuing from the "piston curve" xdx/dt = ^UA + CD i n which U A = x (t) is the velocity of the piston.Since UA is assumed to increase with t it is clear that the characteristics will cut each other. In general, they have an envelope as indidifference in behavior of a depressionwater as compared with the behavior of atcated by the heavy line in the figure. The continuous solutionsfurnished by our theory, which have been the only ones under consideration so far, are thus valid in the region of the x, 2-plane between-the initial characteristiccharacteristic(indicatedand the piston curve up to the curvedby the curve segment ED) through theLONG WAVES IN SHALLOW WATER315"first" point E on the envelope of the straight characteristics, butnot beyond ED.What happens "beyond the envelope" can in principle thereforeFig.
10.6.1. Initial point of breakingnot be studied by the theory presented up to now. However, it seemsvery likely that discontinuous solutions may develop as the time increases beyond the value corresponding to the point E which arethen to be interpreted physically as motions involving the gradual9development of bores and breakersx*<Ut)O(t)inthe water.xa,(t)IFig. 10.6.2. Discontinuity conditionsa particularly simple limit case of the situation indicatedin Fig. 10.6.2 for which a discontinuous solution can be found oncewe have obtained the discontinuity conditions that result from theThereisWATER WAVES816fundamental laws of mechanics. That is the case in which the "piston"accelerated instantaneously from rest to a constant forward velocityso that the piston curve is a straight line issuing from the origin inisthe x, J-plane.
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