J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 91
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11.A.2. Behavior nearwaveWATER WAVES508using the parameter values introduced above. In this case we find:8.06 e 146 *)- 1 with ^ and f in miles and hours. This re(1.05Clsult has the following physical meaning: The angle a of the profile=+',measured between the wave front and the undisturbed water surfacedies out exponentially: oc~ 1/(1 + ae bx ), with a and b constants depending on the river and the boundary condition at x = 0. Theoretically, a could also increase exponentially downstream so that a borewould eventually develop, but only if the increase in level at x = isextremely fast; in our example no bore will develop unless the waterthe extremely rapid rate of at least 1 ft per minute.Unfortunately, the evaluation of c 2 (), which yields the curvature ofrises at'thours10=50100!9.7tx150,milesFig.
11.A.3.Region of practically undisturbed flowthe profile at the wave front, is already very cumbersome. The curvature is found to decrease for large x like xc~ bx b being a positive con,With the twohighest order terms in the expansion known, it ispossible to estimate the region adjacent to the first characteristicwhere the flow is practically undisturbed.
It is remarkable how farstant.behind the forerunner thefirstmeasurable disturbance travels (seeFig. 11.A.3).In a similarway, an expansion as a power series in T has been carriedout for the problem of the junction of the Ohio and Mississippi, asdescribed in earlier sections. Here even the lowest order term wasobtained only after a complicated computation, since it was necessaryto work simultaneously in three different x, J-planes, with boundaryconditions at the junction.
The differential equations for c l are, in allthree branches, of the same type as for the Ohio, and their solutionfor the junction problem with the parameters of section 11.6MATHEMATICAL HYDRAULICS= .00084= .00084509145 *for the upstream branch of the Mississippi,229 *and c 1<r~for the downstream branch of the Mississippi,c l and f both being given in miles and hours.
This means that theangle a also dies out exponentially in the Mississippi, a little fasterdownstream than upstream, as might have been expected, since theoncoming water in the upstream branch has the affect of making theare c xwavec-front steeper.In the problem of the idealized Ohio River and of the idealizedproblem of its junction with the Mississippi River the expansionswere carried out numerically in full detail and were used to avoidcomputation by finite differences in a region of practically undisturb-ed flow.*This would become more and more importantputed beyond 10 hours.ifthe flow were to be com-PARTIVCHAPTER12Problems in which Free Surface Conditions areExactly.The Breaking of a Dam.SatisfiedLevi-Civita'sTheoryThis concluding chapter constitutes Part IV of the book.
In Part Ithe basic general theory and the two principal approximate theorieswere derived. Part II deals with problems treated by means of thelinearized theory arising from the assumption that the motion is asmall deviation from a state of rest or from a uniform flow. Part IIIis concerned with theapproximate nonlinear theory which ariseswhen the depth of the water is small, but the amplitude of the wavesneed not be small. Finally, in this chapter we deal with a few problemsin which no assumptions other than those involved in the basic generaltheory are made. In particular, the nonlinear free surface conditionsarc satisfied exactly.The first type of problem considered in this chapter belongs in thecategory of problems concerned with motions in their early stagesafter initial impulses have been applied.
A typical example is themotion of the waterdam whendamsuddenly broken. Thisproblem will be treated along lines worked out by Pohle [P.ll],[P.12], Similar problems involving the collapse of a column of liquidin the form of a circular half-cylinder or of a hemisphere resting on arigid bottom have been treated by Penney and Thornhill [P.2] bya method different from that used by Pohle.The second section of the chapter deals with the theory of steadyprogressing waves of finite amplitude. The existence of exact solutionsof this type is proved, following in the main the theory worked outby Levi-Civita12.1.inatheis[L.7].Motion of water due to breaking of a dam, and related problemsof the present section we employ throughoutEulerthis book the so-calledrepresentation in which the velocity andpressure fields are determined as functions of the space variables andWith the exception513WATER WAVES514the time.
In this sectionmonlycalled theit isLagrangemakeuse of what is comwhich the displacementsare determined with respect to theconvenient torepresentation, inof the individual fluid particlestime and to parameters which serve to identify the particles. Usuallythe parameters used to specify individual particles are the initialand weOnly a two-dimensional problempositions of the particles,shallconform here to that practice.willbe treated in detail here; con-sequently we choose the quantities a, 6, and t as independent variables,with a and b representing Cartesian coordinates of the initial positions0.
The displacements of the particlesof the particles at the time tare denoted by X(a, b; t) and Y(a, b; t), and the pressure by p(a, b; t).The equations of motion areXtt=--- Pxe>Y = --ttPY-SQwith Newton's second law. We assume gravity to be theexternalforce. These equations are somewhat peculiar becauseonlyof the fact that derivatives of the pressure p with respect to the dein accordpendent variablesXaandFathe result,X and Y occur.respectively,Toeliminateand add, thenalsothem we multiply byby X b Y b and add;,<isX X +tta(Y tie(12.1.1)X X +ttb(Y tt+ g)YbQand these are the equations of motion in the Lagrangian form.
Theseequations are not often used because the nonlinearities occur in anawkward way; however, they have the great advantage that a solution is to be found in a fixed domain of the a, 6-plane even thougha free surface exists. For an incompressible fluid the only caseconsidered herethe continuity condition is expressed by requiringthat the Jacobian ofand Y with respect to a and b should remainXunchanged during the flow (since an area element composed alwaysof the same particles has this property); but since Xa and Y =- binitially, it follows(12.1.2)thatXaY - X Y a =bbITHE BREAKING OF A DAM515the condition of continuity. If the pressure p(12.1.1) by differentiation the result isis(X a X bt(12.1.3)+ Y a Y,Integration with respect to(X a X tt(12.1.4)+ Ya Ybl )/-=.t )t(X b X atis+YYbeliminated fromat ) t.leads to(Xb X af+YY6at)-f(a, b)an arbitrary function.
It can easily be shown by a calculationtheEulcrian representation that the left hand side of this equausingtion represents the vorticity; consequently the equation is a verifi-with/cation of the law of conservation of vorticity. If the fluid starts fromrest, or from any other state with vanishing vorticity, the functionwould bej(a, b)zero.The method used by Pohlcand(12.1.1)which(12.1.2)[P.ll], [P.12] to solve the equationsfurnish the necessary three equations forthe three functions X, F, and p consists in assuming that solutionsexist in the form of power series developments in the time, with coefficients which depend on a and b:X(a(12.1.5)Y(a,p(a.+ X(a.
b) + X^(a, b)b + Y(a. b)-t + F< >(0, b)-p+ p<(a. b)b) + p=--b; t)9a2-ft;(l)(0)ft;1*1(fl,t*2)2++...,...,(0, b)t2X+YandIn these expansions we observe that the terms of order zero inare a and b in accordance with the basic assumption that theseIt should also bequantities fix the initial positions of the particles.noted thatandX(2)Xandand(l)FF(1)are thecomponents of theinitial velocity,similarly for the acceleration; in general, we would(1) would be(l) andthatprescribed in advance as(2)Xtherefore expectpart of the initial conditions.FOf course, boundary conditions imposedon .Y, F, and p would lead to boundary conditions for the coefficientfunctions in the series developments. The convergence of the seriesfor the cases discussedbelow has not been studied, butitseems likelythat the scries would converge at least for sufficiently small values ofthe time.
The convergence of developments of this kind in some simplerproblemsinhydrodynamics has been proved by Lichtenstein[L.12].Theequation (12.1.2) and theequated to zero with the followingseries (12.1.5) are inserted first incoefficient of eachresult for the firstpower of / istwo terms:WATER WAVES516F< 2WeX= ->Fandare subject to the above relation andhence cannot both be prescribed arbitrarily; however, if the fluid(l} = F (1)starts from rest so that0, the condition is automatic-observe that(l}(1)XThe equationally satisfied.ties,Xbut nonlinear inXgeneral:jqw+y<)y9and(2)FandwouldF (n)= F(X<and(n)Xfor(1)(1).F(2)is linear in these quantiThis would be the situation insatisfyan equation of the form*< 2) F< 2 >,(1),,...,F<"-),JC<-i>,(i1. In\ Y (l \ i = 1, 2,., nmotionsthe following we shall consider onlystarting from rest. Con(l) = F (1) =0, and equation (12.1.4) holdssequently, we havewith /0; a substitution of the series in powers of t in equationFwithXa nonlinear function in..X=(12.1.4) yields (for the lowest order term):The higher orderX(n)_y(n)function ofF- Ff -X?>(12.1.7)=X(icoefficientsG(JC<, F<\F(t),i2=>,...,2, 3,.0.an equation of the formXi*-u F< w ~ 1 >), with G a nonlinearsatisfy9..,n-1.Thus we observe thatXsatisfy the Cauchy-Riemann equations and arc thereforeconjugate harmonic functions of a and b.
The higher order coefficientsand(2)satisfy Poisson's equation with a right hand side a knownfunction fixed by the coefficient functions of lower order. Thus theand F can be determined step-wise bycoefficients in the series forwouldXX(t)andsolving a sequence of Poisson equations. Once the functions(i)have been determined, the coefficients in the series for theYpressure p can also be determined successively by solving a sequenceof Poisson equations.
To this end we of course make use of equations(o)(12.1.1); the result for p(a,b) isp(*(12.1.8)fromhandandFor p (n) (a,(12.1.6)function.+ pfl = - 2Q(XP + Ff = 0,= F^> = 0. Thus p (a, b)JC)is a harmonicone would find a Poisson equation with a right(1)b)side determinedby(0}X(i)andF(t)fori=2, 3,...,n+2.would be possible to consider boundary conditions in a generalway, but such a procedure would not be very useful because of itsItcomplexity.
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