J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 84
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For the details of the calculations and aquantitative analysis in terms of the parameters, the paper of Dressiershould be consulted, but a few of the results might be mentioned here.Once the values of the slope S and the roughness coefficient r 2 areprescribed by the physical situation, and the wave propagation speedUis arbitrarily given, there exists a one-parameter family of possibleroll-waves. As parameter the wave length A, i.e. the distance betweentwo successivethe roll wavebores, can be chosen; if this parameter is also fixed,solution is uniquely determined.specific solutionAMATHEMATICAL HYDRAULICS469wouldalso be fixed if the time period of the oscillation were to befixed together with one other parameter theaverage discharge rate,say. Perhaps it is in this fashion that the roll waves are definitelyfixed in some cases for example, the roll waves down theof aspill-damare perhaps fixed by the period of surface waves in theat the crest of the spill-way.
Schonfeld [S.4a] discusses thewaydamproblemfrom the point of view of stability and arrives at the conclusion thatonly one of the solutions obtained by Dressier would be stable, andhence it would be the one likely to be observed.11.4.Unsteady flows in open channels.
The method of characteristicsIn treating unsteady flows it becomes necessary to integrate thenonlinear partial differential equations (11.1.1) and (11.1.6) for prescribed initial and boundary conditions. It has already been mentionedthat such problems fall into the same category as the problems treatedin the preceding chapter, since they are of hyperbolic type in twoindependent variables and thus amenable to solution by the methodis true that theequations (11.1.1) and (11.1.6)more complicated than those of Chapter 10 because of the occurrence of the variable coefficient A and of the resistance term, so thatsolutions of the type called simple waves (cf.
Ch. 10.3) do not exist forof characteristics. Itarethese equations. Nevertheless the theory of characteristics is stillavailable and leads to a variety of valuable and pertinent observations regarding the integration theory of equations (11.1.1) and (11.1.6) which are very important. The essential facts have already beenstated in Chapter 10.2, but we repeat them briefly here for the sakeof preserving the continuity of the discussion. Our emphasis in thischapter is on numerical solutions, which can be obtained by operatingwith the characteristic form of the differential equations, but since weshall not actually use the characteristic form for such purposes weshall base the discussion immediately following on a special case, al-and observations are applicable in the most generalcasein question is that of a river of constant rectanspecialuniformandsectionslope, with no flow into the river from thegularin (11.1.2) and (11.1.6)).
In this case the differentialbanks (i.e. qthough thecase.resultsThe=equations can be written as follows:v xy(11.4.1)(11.4.2)vt= 0,+ E = 0.+vy x ++ vv x + gy xytWATER WAVES470Wehave introduced the symbolEfor the external forces per unitmass:E= -(11.4.3)E differsThe termof y orgS+ gSSf9from the othersin that=itconst.contains no derivativesv.The theoryof characteristics for these equations can be approached* in thevery directlypresent special case by introducing a newctoreplace y, as follows:quantityc(11.4.4)-^gy.This quantity has great physical significance, since it represents aswe have seen in Chapter 10 the propagation speed of small disturbances in the river. From (11.4.4) we obtain at once the relations2cc x(11-4.5)and the= gyX92cc tdifferential equations (11.4.1)2cc x(11.4.6)and+ v + vv x + E =2w? a + 2c = 0.t'= gyt,(11.4.2) take theform0,tThese equations are next added, then subtracted, to obtain the following equivalent pair of equations:a>Idxdtj(11.4.7)We observe that the derivatives in these equations now have the formof directional derivativesthe transformationindeed, to achieve that was the purpose ofand v in the first equation, for example,so that c++are both subject to the operator (cv)d/dxd/dt, which meansthat these functions are differentiated along curves in the a% -planewhich satisfy the differential equation dx/dtcv.
In similar= +and vin the second equation are both subjectto differentiation along curves satisfying the differential equationfashion, the functions c=+cfldx/dtit is entirely feasible to develop the integration theory of equations(11.4.7) quite generally on the basis of these observations (as is done,for example, in Courant-Friedrichs [C.9, Ch. 2]), but it is simpler, andleads to the same general results, to describe it for the special case in* For a treatment which showsquite generally how to arrive at the formulation of the characteristic equations, see Courant-Friedrichs [C.9, Ch. 2].MATHEMATICAL HYDRAULICS471Ff is neglected so that the quantity E inaconstant(11.4.7)(see (11.4.3)).
In this case the equations (11.4.7)can be written in the formwhich the resistance forceis(11.4.7 ) tas one can readily verify. But the interpretation of the operations defined in (11.4.7)! has just been mentioned: the relations state that thefunctions (v2cEt) are constant for a pbint moving through the+with the velocity (vc), or, as we may also put it, for a pointwhose motion in the x /-plane is characterized by the ordinary differential equations dxjdt = vc. That is, we have thefollowingsituation in the tr, /-plane: There are two sets of curves, C l and C 2called characteristics, which are the solution curves of the ordinaryfluid9,differential equationsdx_=t;-4-c,anddt(11.4.8)"cL/2and we have the12c+Et =kv-v"'trelations'v-\-2c-}-Et=k l(11.4.9)dXOf course the constants= const,along a curve=^ const, along a curve2k^and k 2willC x andC2.be different on different curvesshould also be observed that the two families of characteristics determined by (11.4.8) arc really distinct because of the factsince we suppose that ythat c0, i.e.
that the water\/gyin general. It>^surface never touches the bottom.reversing the above procedure it can be seen rather easily that thesystem of relations (11.4.8) and (11.4.9) is completely equivalent toBythe system of equations (11.4.6) for the case of constant bottom slopeand zero resistance, so that a solution of either system yields a solutionv2cEt and observe thatof the other. In fact, if we set /(#, /)f(x, t)=+citv(11.4.10)= +=const, along any curve xkfollows that along such curvesdt+x(t) forwhich dxjdt=WATER WAVES472In the samethe function g(x,way(11.4.11)ftt)=2cv- c)gx =dx/dt = v+along the curves for whichC l and C 2 cover the+ Et satisfies relation(vc.Thus wherever the curveJ-plane in such a way as to furnish acurvilinear coordinate system the relations (11.4.10) and (11.4.11)hold. If now equations (11.4.10) and (11.4.11) are added and thefamiliesas,definitions of f(x 1) and g(x, t) are recalled it is readily seen that theof equations (11.4.6) results.
By subtracting (11.4.11) from9first(11.4.10) the second of equations (11.4.6) is obtained. In other words,any functions v and c which satisfy the relations (11.4.8) and (11.4.9)will also satisfy (11.4.6) and the two systems of equations are there-now seen to be completely equivalent.As we would expect on physical grounds, aforesolution of the originaldynamical equations (11.4.6) could be shown to be uniquely determined when appropriate initial conditions (for t = 0, say) and boundary conditions are prescribed;itfollows thatany solutions of(11.4.(11.4.9) are also uniquely determined when such conditionsare prescribed since we know that the two systems of equations are8)andequivalent.Atfirst sight one might be inclined to regard the relations (11.4.8)(11.4.9) as more complicated than the original pair of partialdifferential equations, particularly since the right hand sides of (11.4.8)andare notknown and hence the characteristiccurves are also not known.Nevertheless, the formulation in terms of the characteristics is quiteuseful in studying properties of the solutions and also in studyingquestions referring to the appropriateness of various boundary andinitial conditions.lines is given;weIn Chapter 10.2 a detailed discussion along thesenot repeat it here, but will summarize the con-shallThe description of the properties of the solution is given inthe x J-plane, as indicated in Fig.
11.4.1. In the first place, the valuesof v and c at any point P(x t) within the region of existence of the solution are determined solely by the initial values prescribed on the segmentclusions.99which is subtended by the two characteristics issuing from P.In addition, the two characteristics issuing from P are themselves alsodetermined solely by the initial values on the segment subtended byof the x-axisthem.
Such a segment of the ff-axis is often called the domain of dependence of the point P. Correspondingly we may define the range ofinfluence of a point Q on the #-axis as the region of the x, -plane inwhich the values of v and c are influenced by the initial values assignedMATHEMATICAL HYDRAULICSto point Q,i.e., it isfrom Q. In Fig.473the region between the two characteristics issuingwe indicate these two regions.11.4.1tDomain ofdetermmacyJRangeofinfluence ofQDomainFig.
11.4.1.WenowTheofdependenceofProle of the characteristicsa position to understand the role of the characteristics as curves along which discontinuities in the first and higherderivatives of the initial data are propagated, since it is reasonable toareinexpect (and could be proved) that those points P whose domains ofdependence do not contain such discontinuities are points at whichthe solutions v and c also have continuous derivatives. On the otherhand, it could be shown that a discontinuity in the initial data at acertain point docs not in general die out along the characteristicdisturbance in theissuing from that point.
Such a discontinuity (orof thewater) therefore spreads in both directions over the surfaceinotherthecinvanddirectiononeincvwater with the speedthecharacteristicstheview of the interpretation given tothroughrelations (11 .4.7 ) r Since v is the velocity of the water particles we seethat c represents quite generally the speed at which a discontinuityin a derivative of the initial data propagates relative to the moving+water.as theWe are therefore justified in referring to the quantity c =wave speed or propagation^/gyspeed.Weconsidered above a problem in which only initial conditions,and no boundary conditions, were prescribed.
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