J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 60
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This followsfrom the original dynamical equations (10.1.8) and (10.1.9). In fact,when u and c are assumed to be functions of x alone these equations reduce todudc+2c2u~+axcuaxdudcfor the case inwhich h==axand0,=axconst, (and soHx=+0).These equations are=22c 2const, and uc 2const,immediately integrable to yield uand these two relations are simultaneously satisfied only for constantvalues of u and c.
On the other hand, any constant values whatevercould be taken for u andc.Thecaseswediscuss in this section aremotions which result by piecing together two such steady motions(each with a different constant value for the depth and velocity)through a shock which moves with constant velocity. In this case themotion as a whole would be steady if observed from a coordinatesystem attached to the moving shock front. In view of our above discussion it is clear that such a motion with a single shock discontinuityis the most general progressing wave which propagates unchanged inform that could be obtained from our theory.*Let us consider now the problem referred to at the beginning of thepreceding section: a vertical plate or piston, as we have called it at* This result should not be taken to mean that the so-called "solitary wave'*does not exist. (By a solitary wave is meant a continuous wave in the form of asingle elevation which propagates unaltered in form.) It means only that ourapproximate theory is not accurate enough to furnish such a solitary wave.
Thisis a point which will be discussed more fully in section 10.9.WATER WAVES328the left end of a tank full of water at rest is suddenly pushed into thewater at constant velocity w. As we could infer from the discussionat the beginning of the preceding section the motion must be dis-continuous from the very beginning or, as we could also put it, the"first" point on the envelope of the characteristics would occur att0. Since the piston moves with constant velocity we might expectthe resulting motion to be a shock wave advancing into the still waterand leaving a constant state behind such that the water particles move=with the piston velocity w.
The circumstances for such an assumedmotion are indicated in Fig. 10.7.1, which shows the x, J-plane togetherwith the water surface at a certain time tQ We know that the constantstates on each side of the shock satisfy our differential equations. Inaddition, we show that they can always be "connected" through ashock discontinuity which satisfies the shock relations derived in thepreceding section. In fact, the relations (10.6.18) and (10.6.19) yield.through elimination of^==p/^ the relation_fl w _f)(10.7.1)w andthe depth h Q in the still water, when we setQh Equation (10.7.1) is the same as (10.6.21) except that ggoreplaces g lf and the discussion of its roots | follows exactly the samefor | in terms of=.lines as for (10.6.21): foreach A>and anyw ^the cubic equa-tion (10.7.1) has three roots for f one negative, another positive, anda third which has a value between these two.
In the present case the:positive root for | must be taken in order to satisfy our energy condition (cf. the discussion based on (10.6.27) of the preceding section)since the sideisthe front side of the shock. Once | + has been calcuwe can determine the depth of the water h^ behindlated from (10.7.1)the shock from the(10.7.2)firstshock conditionh,(w-+)=- AO| +.a motion of the sort indicated in Fig. 10.7.1can be determined in a way which is compatible with all of our conIt is therefore clear thatditions.*Afew further remarks about the above motion are ofinterest.In* It should bepointed out that our discussion yields a discontinuous solutionof the differential equations, but does not prove that it is the only one whichmight exist. However, it has been shown by Goldner [G.6] that our solution wouldbe unique under rather general assumptions regarding the type of functionsadmitted as possible solutions.LONG WAVES IN SHALLOW WATER829wtFig.
10.7.1.Abore with constant speed and heightI(2)sxsft"2=0i^_t=t,Reflected.t=tShockCDIncldenShock*tltoFig. 10.7.2. Reflection of a borefrom arigid wallWATER WAVES330= cQt, c = Vgh Q> whichFig. 10.7.1 we have indicated the line xwould be the initial characteristic terminating the state of rest if themotion were continuous, i.e. if the disturbance proceeded into stillwater with the wave speed c for water of the depth A We know,however, from our discussion of the preceding section that the shockspeed | is greater than c which accounts for the position of the shockline x = |J below the line x = c Q t in Fig. 10.7.1. On the other handof the water particles behind thewe know that the velocity w.,shock relative to the shock is less than the wave speed c = VgA x inthe water on that side. It follows, therefore, that a new disturbancecreated in the water behind the shock should catch up with it sincethe front of such a disturbance would always move relative to thewater particles with a velocity at least equal to c r For example, if thepiston were to be decelerated at a certain moment a continuous depression wave would be created at the piston which would finallycatch up with the shock front, and a complicated interaction processwould then occur.The case we have treated above corresponds to the propagation ofa bore into still water.
If we were to superimpose a constant velocity| on the water in the motion illustrated by Fig. 10.7.1 the resultwould be the motion called a hydraulic jump in which the shock frontis stationary. We need not consider this case further.We treat next the problem of the reflection of a shock wave froma rigid vertical wall by following essentially the same procedure asabove. The circumstances are shown in Fig. 10.7.2. We have anincoming shock moving toward the rigid wall from the left into stillwater of depth h Q The shock is reflected from the wall leaving stillwater of depth h 2 behind it. Since the water in contact with the wallshould be at rest, such an assumed motion is at least a plausible one.We proceed to show that the motion is compatible with our shockconditions and we calculate the height h 2 of the reflected wave...We assume that h^ and u v = 10, the depth and the velocity of thewater behind the shock, are known.
The shock speed f + is then determined by taking the largest of the three roots of the cubic equation(10.6.21), which we write down again in the formOnce |+ has been determined, the depth h in front of the shock isfixed from the first shock condition, which is in the present caseLONG WAVES IN SHALLOW WATER(10.7.4)(To determineW - S+fa= - |A-we may once morethe reflected shockIL4III26evidentlymakeIrOl3811.4I8I10I12II141618(a)ho100806040201I8I10I12II14I16(b)Fig. l().7.3a, b.
Reflection of a bore=from arigid wallw remain the same on one side of theuse of (10.7.4), since A t and u^nowthe smallest of the three roots ofmustchoosewebutshock,WATER WAVES382shock speed f_ since the side (1) is now obviously theThe depth h 2 of the water behind the shockafter the reflection that is, of the water in contact with the wallafter reflectionis then obtained in the same way as hby usinghhininofandofwith(10.7.4)2placeplaceQ+|_(10.7.3) as thefront side of the shock.:(w(10.7.4)!and A 2 /A as functions of/?_/* 2 .w wetaking a series of values forBy/&2/&J=_)&!1 /Ahave determined the ratiosThat is, the height h 2 of the.wave has been determined as a function of the ratio of thedepth AJ of the incoming wave to the initial depth h at the wall.
Theresults of such a calculation are shown in Figs. 10.7.3a and 10.7. 3b:In Fig. 10.7.4 we give a curve showing (h 2A )/^o as a function ofAO )/AO> that is, we give a curve showing the increase in depth(Ajreflectedafter reflection as a function of the relative height (A x/*)/^o*t'ieincoming wave.*\>100806040201io426For A!/A O near to unity,show thati.e.fin(1(\ 7 K\^AvF.i.tjy10fin- r*u_*-small,this relationi.e.14we mayh Q )/h small,for (/^fl-t916hr ho'it isnot difficultlle\.tf&QFrom12Height of the reflected boreFig.
10.7.4.to1I8.f?/Qwrite h 2h~ 2(7^h)if (AjA)the increase in the depth of the water after reflectionisisLONG WAVES IN SHALLOW WATER333wave when the latter is small. Thiswhat one might expect in analogy with the reflection of acousticwaves of small amplitude. However, if h^h^ is not small, the waterincreases in depth after reflection by a factor larger than 2. Fortwice the height of the incomingis~A ); while if h^h^ is 10,hA X /A is 2, then h 23(/& xseesfromthe graph of Fig.asoneAAthen A 2),35(A t10.7.4. In other words, the reflection of a shock or bore from a rigidwall results in a considerable increase in height and hence also in pressure against the wall if the incoming wave is high.
In fact, for veryhigh waves the total pressure p per unit width of the wall could beshown to vary as the cube of the depth ratio hjh^.In the upper curve of Fig. 10.7.3a we have drawn a curve for theanalogous problem in gas dynamics, i.e. the reflection of a shock fromthe stopped end of a tube. In the case of air with an adiabatic expo1.4 the density ratio Q 2 /6i as a function of Q^QQ (in annent yobvious notation) is plotted as a dotted curve in the figure.
As wesee, the density in air on reflection is higher than the correspondinginstance,if^quantity, the depth, in the analogous case in water. However, thecurve for air ends at Q l /Q Q6, since it is not possible to have a shockhas a higher density ratio. In water1.4whichwithinawavegasy==The explanation for this difference liesis assumed to hold across a shock in gaslawenergyournotinbuttheory for water waves.dynamics,thereisno suchrestriction.in the fact that the10.8.The breaking of a damAt the end of section 10.5 we gave the solution to an idealized version of the problem of determining the flow which results from the sudden destruction of a dam if it is assumed that the downstream sideDamFig.
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