J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 59
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that the water is at rest on oneside of the shock. Also, we write the second of the shock conditions,(10.6.12) in theform^PQ =?- "-?(10.6.17)1,61<?owhich follows from mv l = QO V O V I and mv= ulFrom U Q = we have v^ =f and v(10.6.17) takes the form(10.6.18)- fK -upon making use of pnow==:2gg / 2 ? ( cf (10.6.9)).-giK _This conclusion wasThe first shock condition{)= -f'=firststatedso thatei)so that (10.6.18) can be written(10.6.20,(10.6.12).AQtakes the form(10.6.19)and(cf. (10.6.5))= ~ (go +f)pi^t'oby Rayleigh[R.3J.WATER WAVES322ifueliminated, orisitbe written in the formmayg is eliminated.
Thus (10.6.19) together with either (10.6.20) or0.(10.6.21) are ways of expressing the shock conditions when uare now in a position to discuss some important special cases.0, at most two of the remainingHaving fixed the value of w i.e. U Qif=We=,quantitiesdiscussionandand|,g,Q V and w x can be prescribedarbitrarily.For ourlateruseful to single out the following two cases: Case 1.
JDJ.g are given, i.e. the depth of the water on both sides of the shockthe velocity on one side are given. Case 2.and u are given, i.e.it is^the velocity of the water on both sides of the shock and the depth ofthe water on one side are given. We proceed to discuss these cases indetail.CaseFromwe see that | 2 is determinedfor any arbitraryof the water depths. Henceis determined by (10.6.20) only within sign. Suppose now thatis, as we have seen above, the front61 -^ (?<) ^ n ^is case ^e sideside of the shock, and since uthe shock front must move in thedirection from the side 1 toward the sidein order that the mass flux1.(10.6.20)and g l9values (positive, of course) ofi.e.=should pass through the shock from front to back.Hence if it is once decided whether the side is to the left or to theis uniquely fixed.
If, as in Fig. 10.6.4,right of the side 1 the sign of*-1-u.<0XFig. 10.0.4.Bore advancing intostillwateris chosen to the left of the side 1, and the ^-direction is positive to the right, it follows that f is negative, as indicated. It is usefulto introduce the depths A and A x of the water on the two sides of thethe sideshock:(10.6.22)LONG WAVES IN SHALLOW WATERandinto express (10.6.20) in terms of these quantities.our caseTheresult for fis- - I/" h(10.6.28)K<since h(A x+Ai*= gh>A)/2 <as one readily sees from g tportant conclusion: Since h^Vgh328t,/&x.From(10.6.23)the shock speed.|we drawfAlso, in the caseis|wthe im-greater than=we havefrom (10.6.19)u,(10.6.24)=|(lJsJ,so that the velocity of the water behind the shock has the same sign as1 ) but is less than f numerically.(since h Q /h lFinally, it is very important to consider the speed v l of the shock<front relative to the water particles behindvl(10.6.25)and thisformin turn<it:from (10.6.24) we have=u ^S=-'T^HIlcan be expressed through use of (10.6.23)in the^In other words, the speed of the shock relative tothe water particles behind the shock is less than the ivave propagationspeed Vg/?! in the water behind the shock.
Hence a small disturbancecreated behind a shock will eventually catch up with it. Although theso that v lVgh v=it holds quitedrawn for the special case uof the waterfortheshockvelocitiestothemotionrelativegenerallyon both sides of a shock, in view of earlier remarks on the dependenceconclusion wasof the shock relationsThe caseon theserelative velocities.by Fig. 10.6.4 is that of a shock advancing intostill water.
The fact thatf is in this case of necessity negative is aconsequence of the assumption of an energy loss across the shock. Itisillustratedworth while to restatethis conclusion in the negative sense, asfollows: a depression shock can not exist, i.e. a shock wave whichleaves still water at reduced depth behind it should not be observedin nature.*The observations bear out this conclusion. Bores advancing* Ingas dynamics the analogous situation occurs: only compression shocksand not rarefaction shocks can exist.WATER WAVES324Fig. 1Q.6.5. Reflectionfrom aFig.
10.6.6. Hydraulicrigid walljumpinto still water are well known, but depression waves are alwayssmooth.Instead of assuming that g x > g (or that A t > h ) as in the case ofFig. 10.6.4 we may assume ^ <(or /& x < h Q ), so that the side 1 isthe front side. In other words the water is at rest on the baek side ofthe shock in this case. If the front side is taken on the right, the situation is as indicated in Fig. 10.6.5.
In this case | must be positiveandnegative in order that the mass flux should take place from theside 1 to the side 0. The value of t^ is given by (10.6.24) in this casealso. The case of Fig. 10.6.5 might be realized in practice as the resultof reflection of a stream of water from a rigid wall so that the water incontact with the wall is brought to rest. We shall return to this case%later.In the above two cases we considered u to be zero. However, weknow that we may add any constant velocity to the whole systemwithout invalidating the shock conditions.
It is of interest to considerthe motion which arises when the velocityis added to u Q9 u r and| in the case shown in Fig. 10.6.4. The result is the motion indicatedby Fig. 10.6.6 in which the shock front is stationary. This case oneof frequent occurrence in nature- is commonly referred to as theFrom ourpreceding discussion we see that the wateralways moves from the side of lower elevation to the side of higherhydraulic jump.elevation.ThevelocitiesU Q and uare both positive, andUQ>uvLONG WAVES IN SHALLOW WATERAlso the velocity u on the incoming sideis325greater than thewavepropagation speed Vgh on that side while the velocity u is less thanVgh lt This follows at once from the known facts concerning the reare the velocitieslative shock velocities and the fact that u andrelative to the shock front in this case. The hydraulic engineers refer%to this as a transition from supercritical to subcritical speed.recall that in this case uCase 2.0, u^ and p L (or h^) areassumed given and f and h are to be determined.
The value of f isto be determined from (10.6.21). To study this relation it is conve-=Wenient to set x=fand y=u^so that (10.6.21) can be replacedbyy(10.6.27)y= k x/(x= HI + x.22In Fig. 10.6.7 we have indicated these two curves, whose intersectionsx of (10.6.21). The first equation is reyield the solutions f=presented by a curve with three branches having two asymptotesk. As one sees readily, there are always three different realno matter what values arc chosen for the positive quanroots for%u v Furthermore, one root + =x + is negative, while thex_ is always positive, another _ =third | ~x lies between the other two. However, the third root= x must be rejected because it is not compatible with (10.6.19):|=Since Q I and Q O are both positive it follows that x =f and ytityk2=g/4/2andfor the velocity-conditionsFig. 10.6.7.
Graphical solution of shockWATER WAVES326y correspondingf must have the same sign. But the sign ofyis always the negative of x as one sees from Fig. 10.6.7.0, but there is no shock discontinuity in,then x(If U-Lythis case.) The other two roots, however, are such that the signs ofM!tox=x==and u x=f are the same. In the case 2, therefore, equationtwo different values of f which have oppositefurnishes(10.6.21)signs and these values when inserted in (10.6.19) furnish two valuesof the depth10.6.4and-/2The two.10.6.5.Ancases are again those illustrated in Figs.appropriate choice of one of the two roots mustmadein accordance with the given physical situation, as will beillustrated in one of the problems to be discussed in the next section.beBefore proceeding to the detailed discussion of special problemsinvolving shocks it is perhaps worth while to sum up briefly the mainfacts derived in this section concerning them: the five essential quanM O u^\ g g x (or, what is the same,tities defining a shock wavef,h and h^) must satisfy the shock conditions (10.6.12).
If it is as,,sumed in addition that the water particles mayenergy on crossingfound that the shock wave travelslosethe shock but not gain it, then it isalways in such a direction that the water particles crossing it pass fromh l9 so that thethe side of lower depth to the side of higher depth. If h Q<sideis the front side of the shock, thespeeds\v\and\v\of the waterrelative to the shock front satisfy the inequalities(10.6.28)In hydraulics it is customary to say that the velocity relative to theshock is supercritical on the front side (i.e. greater than the wavepropagation speed corresponding to the water depth on that side)andsubcritical on the back side of the10.7. Constantshock*shocks: bore, hydraulic jump, reflection fromarigid wallIn the preceding section shock discontinuities were studied for thepurpose of obtaining the relations which must hold on the two sidesand nothing was specified about the motion otherwisethattheshock under discussion should be the only discontiexceptof the shock,* Ingas dynamics the analogous inequalities lead to the statement that theflow velocity relative to the shock front is supersonic with respect to the gason the side of lower density and subsonic with respect to the gas on the other side.LONG WAVES IN SHALLOW WATER327nuity in a small portion of the fluid on both sides of it.
In the presentand following sections we wish to consider motions which are continuous except for the occurrence of a single shock. Furthermore weshall limit our investigations in this section to cases in which the motion on each of the two sides of the shock has constant velocity anddepth. These motions, or flows, are evidently of a very special character, but they are easy to describe and also of frequent occurrencein nature.It isperhaps not without interest in this connection to observe thatwave motions (i.e., motions in whichthe only steady and continuous=the velocity u and wave propagation speed cVg(& +77) are intimefurnishedourtheforofthecase of constant)bytheorydependentdepth h are the constant states u = const., c = const.
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