J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 55
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To change r] at one point watermight be either poured into the tank or pumped out of it at that point at anappropriate rate.LONG WAVES IN SHALLOW WATER301Our statement is an immediate consequence of the following fundamental fact: if the values of u and c on any characteristic curve, C say(i.e. a solution curve of the first of the two ordinary differential equations(10.2.4)), are constant, then CJ is a straight line and furthermore it isembedded in a family of straight line characteristics along each of whichu and c are constant, at least in a region of the x, f -plane where u(x, t)and c(x, t) are without singularities and which is covered by theThe proof is easily given.
Tothecurveisalineif u and c are constant alongbegin with,straightCJofsincethethecurveisconstant in that case from (10.2.4).it,slopeconsider any twoNext, let C l be another characteristic near to CJ.twodistinct families of characteristics.WeBQ on CJ together with the characteristics of the familyC 2 through A Q and BQ and suppose that the latter characteristicsintersect C at points A and B (cf. Fig. 10.3.1To prove our statementpoints A$ andl):Fig. 10.3.1.Region containing a straight characteristicwe need only show that u(A )~u(B) and c(A )= c(B) since then u and cwould be constant on C l (because of the fact that A and B are anyarbitrary points on CJ and hence the slope of the curve C r would beconstant, just as was argued for Cj.
We have u(A ) = u(BQ ) andc(A Q ) = c(BQ ) and consequently we may write(1081)by making useuAUBUA2cA-2c B=U BQ2cA-2<?Bo,=of the second relation of (10.2.5) (which holds alongsince the originalC 2 ) and observing thatthe characteristicsm=WATER WAVES802depth of the waterisassumed to be constant. Next we make use ofthe first relation of (10.2.5) forUA(10.3.2)But from(10.3.1)+ 2cA =to obtainUB+ 2c BUB-.we haveUA(10.3.3)Cl-2c A=2c B ,UAUBtherefore proved.The problems formulated in the first paragraph of this section areat once seen to have solutions (at least in certain regions of theandand(10.3.2)CA=CBand.(10.3.3) are obviously satisfied only ifOur statementisx /-plane) of the type we have just defined as simple waves sincethere is a region near the #-axis in the x, /-plane throughout which theparticle velocity u and wave speed c arc constant, and in which therefore the characteristics are two sets of parallel straight lines.
The circumstances are illustrated in Fig. 10.3.2 below: There is a zone / along9Fig. 10.3.2.Asimplewavethe 07-axis which might be called the zone of quiet* inside which theconst. (Thesecharacteristics are obviously straight lines xc t=lines arenot drawn in the figure). This regionupper side by an"initial characteristic"xisterminated on theerfwhich divides the* In a "zone ofquiet" we permit the particle velocity u to be a non zeroconstant, but the free surface elevation r\ is taken to be zero in such a region.In case uUQconst.
7^initially, the motion can be thought of as observedfrom a coordinate system moving with that velocity; thus there is no real lossof generality in assuming UQ0, as we frequently do in the following.==LONG WAVES IN SHALLOW WATER303region of quiet from the disturbed region above it. The physical interpretation of this is of course that the disturbance initiated at the=propagates into the region of quiet, and the water at anypoint remains unaffected until sufficient time has elapsed to allowthe disturbance to reach that point. The exact nature of the motionin the disturbed region is determined, of course, by the character oftime/=the disturbance created at the point x0, i.e., by appropriate dataprescribed along the /-axis.* One set of characteristics, i.e., the setcontaining the initial characteristic C?, therefore consists of straight(That the characteristics C 2 in the zone// are necessarily curvedlines and not straight lines can be seen from the fact that they wouldotherwise be the continuations of the straight characteristics from thezone / of quiet and hence the zone // would also be a zone of quiet, asone sees immediately).
Furthermore, the set of straight characteristicsC\ in zone // is completely determined by appropriate conditions prescribed at xfor all /, i.e., along the /-axis. What these conditionsshould be can be inferred from the following discussion. Consider anystraight characteristic issuing from a point / = T on the /-axis. Weknow that the slope dx/dt of this straight line is given in view oflines.(10.2.4),bydx(10.3.4)=u(r)+ c(r).Suppose now that there is a curved characteristic C 2 going backfrom / = T on the /-axis to the initial characteristic C (see the dottedcurve in Fig. 10.3.2).
We have the following relation from (10.2.5):u(r)(10.3.5)-2c(r)-u-2c,knownvalues of u and c in the zone of quiet.Hence the slope of any of the straight characteristics issuing from the/-axis can be given in either of the two forms:inwhich u andC Q are thedx1r-=;-[3u (T )-u Q + cQ](10.3.6)as one sees*,or- . 8c(r)(UVfrom (10.3.4) and(10.3.5).Thusifeither u(r) or c(r) isdiscussion in the preceding section centered about the initial valuein which the initial data are prescribed on the #-axis, but onesees readily that the same discussion would apply with only slight modificationsto the present case, in which what is commonly called a boundary condition (i.e.at the boundary point x0), rather than an initial condition, is prescribed.Ourproblem for the caseWATER WAVES304given, i.e.
if either u or c is prescribed along the J-axis, then the slopesof the straight characteristics C l and with them the characteristics C lthemselves are determined. Since we know, from (10.3.5), the valuesu and c along the /-axis if either one is given, and since u and care clearly constant along the straight characteristics, it follows that weknow the values of u and c throughout the entire disturbed region inof bothother words, the motion is completely determined.So far, we have considered only the case in which the curved characteristics (i.e., those of the type C 2 ) which issue from the boundaryx = c J of the disturbed region actually reach the /-axis. This, however,need not be the case.
Suppose, for example, that UQ is positive andu>c=Vg/T. In this case the slope dxjdt of the curves C 2 is positive,will turn to the left, as in Fig. 10.3.2.and we cannot expect that theyIndeed, in such a case one does not expect that a disturbance will propagate upstream (that is, to the left in our case) since the stream velocity is greater than the propagation speed. In gas dynamics one wouldsay that the flow is supersonic, while in hydraulics the flow is saidto be supercritical.
One could also look at the matter in another way:For not too large values of t the velocity u can be expected to remainsupersonic and hence for such values of t both sets of characteristicsissuing from the f-axis would go into the right half plane (u beingagain taken positive). Thus we would have the situation indicatedin Fig. 10.3.3, in which a segment of the /-axis is subtended by twot*<V.cFig. 10.8.3.The)tsupercritical caseLONG WAVES IN SHALLOW WATEE805drawn backward from P. In this case, as in the case ofthe initial value problem treated in the preceding section, we mustprescribe the values of both u and c along the /-axis.
If we do so, thenthe solution is once more determined through (10.8.4) and the factthat u2c is constant along one set of characteristics and u2c ischaracteristics+constant along the other.In either of our two cases,i.e. of subcritical or supercritical flow,see therefore that the simple wave can be determined. One seesalso how useful the formulation in terms of the characteristics can bewein determining appropriate subsidiary conditions such asboundaryconditions.to know the values of u and c for any particular timeonce the simple wave configuration is determined, we needonly draw the line t = J and observe its intersections with thestraight characteristics since the values of u and c are presumablyknown on each one of the latter.
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