J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 54
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We can approximate theu and c for small values of t as follows: consider a scries ofpoints on the #-axis (cf. Fig. 10.2.1) a small distance 6x apart. At allof these points the values of u and c are known from (10.2.8). Consequently the slopes of the characteristics C l and C 2 at these points areinvalues 01tFig. 10.2.1. Integrationbyfinite differencesknown from (10.2.4). From the points 1, 2, 3, 4 straight line segmentswith these slopes are drawn until they intersect at points 5, 6, and 7,and if dx is chosen sufficiently small it is reasonable to expect thatLONG WAVES IN SHALLOW WATER297the positions of these points will be good approximations to the intersections of the characteristics issuing from the points 1, 2, 3, 4 sinceweare simply replacing short segments of these curves by theirThe values of both x and t at points 5, 6, and 7 are nowtangents.knownthey can be determined graphically for example andthrough the use of (10.2.5) and the initial conditions we can alsodetermine the approximate values of u and c at these points.
For thispurpose we observe that along any particular segment issuing fromrntthe points 1, 2, 3 or 4 the values of u2c2cmt and uare known constants since the values of u and c are fixed by (10.2.8)for t = 0; hence we have+Ialong\alongCx uC2 u:+ 2cmt:2cmt==uu-f-2c,and2c.At the points 5, 6, and 7 we know the values of t and hence (10.2.9)furnishes two independent linear equations for the determination ofthe values of u and c at each of these points.
Once u and c are known5, 6, and 7 the slopes of the characteristics issuing from thesebe determined once more from (10.2.4) and the entire procanpointscess can be carried out again to yield the additional points 8 and 9and the approximate values of u and c at these points. In this waywe can approximate the values of u and c at the points of a net overa certain region of the x, <-plane, and can then obtain approximateat pointsu and c at any points in the same region either by interor by refining the net inside the region.
It is quite plausiblevalues forpolationand could be proved mathematically that the above process wouldconverge as dx ->Q to the unique solution of (10.2.4) and (10.2.5)corresponding to the given initial conditions for sufficiently smallvalues of t (i.e. for a region of the #, /-plane not too far from the #-axis)provided that the prescribed initial values of u and c are sufficientlyregular functions of x for example, if they have piecewise continuousfirst derivatives.It shouldbe clear that once the characteristics are known the valuesu and c for all points of the x t -plane covered by them are alsoknown, since the constants A: x and k 2 in (10.2.5) are known on eachcharacteristic through the initial data and hence the values of u and cof9can be calculated by solving the linear equationthrough that point. This statement ofcourse implies that each one of the two families of characteristicscovers a region of the x, /-plane simply and that no two members offorany point(x, t)(10.2.5) for the characteristicsWATER WAVES298in other words it isdifferent families are tangent to each otherforma regular curvicharacteristicsoftwofamiliesthattheimpliedsystem over the region of the x, J-plane in question.One of the points of major interest in the later discussion centersaround the question of determining where and when the characteristics cease to have this property, and of interpreting the physicalmeaning of such occurrences.The method of finite differences used above to determine the characteristics can be interpreted in such a way as to throw a strong lightlinear coordinatesolution.
Consider the point 10 ofrecall that the approximate values u loon the physical properties of theWeFig. 10.2.1 for example.and c 10 of u and c at point 10 were obtained through making use of theinitial values of u and c at points 1, 2, 3, 4 on the #-axis only, andfurthermore that the values u lo and c 10 required the use of points con-fined solely to the region within the approximate characteristics joining point 10 with points 1 and 4. Since the finite difference schemeto yield the exact characteristicsoutlined above converges as dx ->we are led to make the following important statement: the values of uandc at any point P(x, t) within the region of existence of the solutionare determined solely by the initial values prescribed on the segment ofthe x-axis which is subtended by the two characteristics issuing from P.Domain ofC,EL\determinacyJRangeofinfluence ofFig. 10.2.2.QDomainDomainof dependenceofdependence of Pand range of influenceIn addition, the two characteristics issuing from P are also determinedsolely by the initial values on the segment subtended by them.
Sucha segment of the #-axis is often called the domain of dependence of theLONG WAVES IN SHALLOW WATER299point P. Correspondingly we may define the range of influence of apoint Q on the #-axis as the region of the x, -plane in which the valuesu andby the initial values assigned to point Q. InFig. 10.2.2 we indicate these two regions. It is also useful on occasionto introduce the notion of domain of determinacy relative to a givendomain of dependence. It is the region in which the motion is determined solely by the data over a certain segment of the #-axis. Theseofc are influencedregions arc outlined10.2.2, inbycharacteristic curves, as indicated in Fig.in view of the discussionan easily understandable fashionabove.Weare now in a position to understand why it is appropriate tothe quantity c the propagation or wave speed.
To this end wesuppose that a certain motion of water exists at a definite time, whichcallwe taketo be0.tthat time, and, asThis means, of course, that u and c are known atjust seen, the motion would be uniquelywe havedetermined for t > 0. However, we raise the question: what differencewould there be in the subsequent motion if we created a disturbancein some part of the fluid, say over a segment QxQ 2 f *he #-axis (cf.Fig. 10.2.3)? This amounts to asking for a comparison of two solutionsof our equations which differ only because of a difference in the initialQ20,Fig. 10.2.3. Propagation of disturbancesconditions over the segmenttwosolutions in questionQ 1 Q 2 Our whole discussion shows, that the.only in the shaded region ofpoints of the x, J-plane influenced bywoulddifferwhich comprises allthe data on the segment QxQ^ an d which is bounded by characteristicsC 2 and C x issuing from the endpoints Q t and Q 2 of the segment.
TheseFig. 10.2.3,curves, however, satisfy the differential equations dx/dtuc. Since u represents the velocity of the water,dx/dt=+=uc,it isthenWATER WAVES300clear that c represents the speed relative to the flowing stream atwhich the disturbance on the segment Q X Q 2 spreads over the water.This implies that the data in our two problems really differ at pointsQ x and Q 2 and that these differences persist along the characteristicsissuing from these points. Actually, only discontinuities in derivativesat Q l and Q 2 (and not of the functions themselves) are permitted in theabove theory, and it could be shown that such discontinuities wouldnever smooth out entirely along the characteristics C l and C 2 We are.therefore justified in referring to the quantity c(local) propagation speed of small disturbances=+h) as theVg(rjthat is, small in thesense that only discontinuities in derivatives occur at the front of adisturbance.10.3.The Notion of a Simple WaveThereisan importantclass ofproblems in which the theory ofcharacteristics as presented in the preceding section becomes particularly simple.
These are the problems in which (1) the initial un-disturbed depth h of the waterisconstant so that the quantitymin(10.2.1) (cf. also (10.1.10)) is zero, (2) the water extends from theorigin to infinity at least in one direction, say in the positive directionof the a?-axis, and (3) the water is either at rest or moves with constantvelocityand the elevation ofIn other words, the waterisin asurfaceiszero at the timeuniform state at timet=t=0.such thatand c = C Q = Vgh = const, at that instant. Ourdiscussion from here on is modeled closely on the discussion given byCourant and Friedrichs [C.9], Chapter III.u=u =its freeconst,We now suppose that a disturbance isinitiated at the originso that either the particle velocity u, or the surface elevationwavevelocity c=r/x=(or the+h)) changes with the time in a prescribedVg(7/a disturbance at one point in the water propagatesmanner.* That is,into water of constant depth and uniform velocity.
Under thesecircumstanceswe showthat one of the two families of characteristicsfurnished by (10.2.4) consists entirely of straight lines along each of whichu and c are constant. The corresponding motion we call a simple wave.* Onemight accomplish this experimentally in a tank as follows: To obtaina prescribed velocity u at one point it would only be necesary to place a verticalplate in the water extending from the surface of the water to the bottom of thetank and to move it with the prescribed velocity.
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