J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 17
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In section 5.4. we shall give an account ofthis method of attack. Roseau [R.9] has solved the problem of wavesan ocean having different constant depths at the two differentinfinities in the .r-direction which are connected by a bottom ofinvariable depth.Before outlining the actual contents of the present chapter, itmay be well to summarize the conclusions which have been obtainedfrom studying numerical solutions of the problems being consideredhere, which have been carried out (cf. [S.18]) for two-dimensional=90, 45, and 6, and for three90. The results for the case of an-135 have already been discussed earlier.overhanging cliff with a)In the other three cases the most striking and important result isthe following: The wave lengths and amplitudes change very littlefrom their values at oo until points about a wave length from shorehave been reached.
Closer inshore the amplitude becomes large, asit must in accord with our theory. It is a curious fact (already mentioned earlier) that the amplitude of a progressing wave becomesless (for r/j - 6 about 10less) at a point near shore than its valueat oo, although it becomes infinite as the shore is approached. Thiswavesfor slope angles co185,fordimensional wavesthe case o>=%been observed experimentally. This statement holdsfor the three-dimensional waves against a vertical cliff (with anamplitude decrease of about 2 %), as well as for the two-dimensionaleffect has oftencases.solution for the case of a beach sloping atuseful for the purpose of a comparison with the results obtainedThe exact numerical6isfrom the linear shallow water theory (treated in Chapter (10.13) ofPart III) and from the asymptotic approximation to the exact theoryobtained by Friedrichs [F.I 4], The linear shallow water theory, asits name indicates, can in principle not furnish a good approximationto the waves on sloping beaches in the deep water portion since ityields waves whose amplitude tends to zero at oo.
For a beach slopingat 6, for example, it is found that the shallow water theory furnishesa good approximation to the exact solution for a distance of two orthree wave lengths outward from the shore line if the wave lengthis, say, about eight times the maximum depth of the water in thisrange; but the amplitudes furnished by the shallow water theoryWATER WAVES76would be 50 to 60 percent too small at about 15 wave lengths awayfrom the shore line. One of the asymptotic approximations to theexact theory given by Friedrichs yields a good approximation overpractically the whole range from the shore line to infinity (it is inaccurate only very close to shore); this approximation, which evenyields the decrease in amplitude under the value at oo mentioned above,almost identical with one obtained by Rankine (cf. Miche [M.8,p.
287]) which is based upon an argument using energy flux considerations in connection with the assumption that the speed of theenergy flux can be computed at each point in water of slowly varyingdepth by using the formula (cf. (3.3.9)) which is appropriate in waterhaving everywhere the depth at the point in question. Friedrichs thusgives a mathematical justification for such a procedure on beachesof small slope.It has already been made clear that the discussion in this chaptercannot yield information about the breaking of waves, which is anisessentially nonlinearphenomenon. However, it is possible to analyzethe breaking phenomena in certain cases and within certain limitationsby making use of the nonlinear shallow water theory, as we shall seein Part III. For this purpose, one needs to know in advance themotion at some point in shallow water, and this presumably couldbe done by using the methods of the present chapter, combinedpossibly with the methods provided by the linear shallow watertheory.The material in the subsequent sections of this chapter is orderedas follows.
In section 5.2. the problem of two-dimensional progressingwaves over beaches sloping at the angles 7t/2n, n an integer, is discussedfollowing the method of Lewy [L.8] and the author [S.18]. In section5.3 the problem of three-dimensional waves against a vertical cliffis treated, also using the author's method. The reasons for includingthese treatments in spite of the fact that they yield results that areincluded in the more general treatments of Peters [P.6] and Roseauthat they are interesting in themselves as an example ofandalso they can be applied to other problems, such as themethod,problem of plane barriers inclined at the angles n/2n (cf.
F. John[J.4]), which have not been treated by other methods. In section 5.4,the general problem of three-dimensional waves on beaches sloping[R.9]atistreated following essentially the ideas of Peters.In section 5.5 the problem of diffraction of waves around a rigidany angleverticaliswedgeistreated; in case thewedge reducesto a plane theWAVES ON SLOPING BEACHES AND PAST OBSTACLESproblem becomes theclassical diffraction77problem of Sommerfeld[S.I 2] for the case of diffraction of plane waves in two dimensionsnew uniqueness theorem and a new andaround a half-plane barrier.Aelementary solution for the problem are given. Methods of analyzingthe solution are also discussed; photographs of the waves in such casesand comparisons of theory and experiment are made.Finally, in section 5.6 a brief survey of a variety of solved andunsolved problems which might have been included in this chapter,with references to the literature, is given.
Included are brief referencesto researches in oceanography, seismology, and to a selection of papersdealing with simple harmonic waves by using mathematical methodsdifferent from those employed otherwise in this chapter. In particular, a number of papers employing integral equations as a basicmathematical tool are mentioned and occasionthe Wiener-Hopf techniqueof solvingiscertaintaken to explainsingularintegralequations.5.2.Two-dimensional waves over beaches sloping at angles a)=n/2nWe consider first the problem of two-dimensional progressingwaves over a beach sloping at the angle co = nf2n with n an integerFig.
5.2.1. Sloping(cf.asFigure5.2.1), in spite ofwas mentionedbeach problemthe fact that the problem can be solved,by a method which is notin the preceding section,WATER WAVES78restricted to special anglesPeters [P.6], and Roseau [R.9]).(cf.solved here by a method which makes essential use ofthe fact that the slope angle has the special values indicated becauseThe problemisthe method has some interest in itself, and it yields representationswhich have been evaluated numerically in certain cases. In addition,the relevant uniqueness theorems are obtained in a very natural way.We assume that the velocity potential 0(x y; t) is taken in theiatforme <p(x, y).
Hence <p(x* y) is a harmonic function in the9=sector of anglecon/2n.Theboundary condition thenfree surfacetakes the forma2(5.2.1)y<p y=for0,y=x0,>0,Saswe have often seen(cf. (3.1.7)),??(5.2.2)Itiswhile the condition at the bottomis= 0.onuseful to introduce thesame dimensionlcss independent variablesas were used in the preceding chapter:(5.2.3)xl=mx,=ymmy,a*/g.The function q>(x, y) obviously remains harmonicand conditions (5.2.1) and (5.2.2) become(5.2.1)'<p y-<p=0,cp n(5.2.2)'y-=0,x0,>in these variables,0,after dropping subscripts.The simple harmonic standing waveseverywhere are given byincos (xsin (xSH{oowater ofinfinitedepth+ a)+ a)'we write these down because we expect that they will represent thebehavior of the standing waves in our case at large distances fromthe origin, that is, far away from the shore line.Thesolution of the problempotential f(z) defined by(5.2.5)The function/(*)isobtained in terms of the complex= f(x + iy) = <p(x, y) +f(z) should,likeq> 9iX (x,y).be regular and analytic in theWAVES ON SLOPING BEACHES AND PAST OBSTACLESentire(including thesectorboundaries,* except for the origin).(5.2.1)' and (5.2.2)' are given in terms ofThe boundary conditions(5.2.6)<p y-<pVn(5.2.7)=&31 e (if=3le~We(5.2.8)(5.2.9)-on/)=(f(z))=-for zThe second condition3flIj-==**79(/(*))rresultsfor z real0fcexpand(- iexp (w/2n}{9rpositive,in/2n)>/')0.from- ~- #*(- r dO= #* {-'>*/'(*)}.(/(*)))1)introduce the two following linear differential operators:Li(D)= -iL 2n (D)exp=inftn) D,{-i/)-1with Z> meaning d/dz.
The basic idea of the method invented byH. Lcwy is to find additional linear operators, L 2 L 3.,2 n-isuch that the operation L^ L 2L 2n applied on /(s) yields a,..,...function F(z) whose real part vanishes on both boundaries of our sector.Once this has been done, the function F(z) can be continued analytically over the boundaries of the sector by successive reflections toyield a single-valued function defined in the entirecomplex planebeshowntheItcanthen(see [S.18]), essenexcept possiblyorigin.thefunctionthatLiouville'stheorem,F(z) is uniquelytially by usingdetermined within a constant multiplying factor by boundednessconditions on the complex potential f(z) at oo together with the orderof the singularity admitted at the origin. After F(z) has been thusdetermined, the complex potential f(z)the ordinary differential equation L^L 2.obtainedis..L 2nf(z)=as a solution ofF(z).Of course,necessary in the end to determine the arbitrary constants in thegeneral solution of this differential equation in such a way as toit issatisfy all conditions of the problem, and this can in fact be doneexplicitly.
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