Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 27
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Thomson was the first, however, to solve the hydrodynamicequations in this case.127In a similar manner to Poisson, Thomson sought solutions of the form cos (kx - wt) forthe linearized equations of motion. The only difference with the Lagrange-Poisson conditions for the velocity potential is the substitution of ga- - Ta'' for ga- in the pressureequation at the free surface, where T denotes the superficial tension per unit density.Consequently, g must be replaced by g + � T in the dispersion formula w2 = gk for waveson deep water. The corresponding celerity is(2.105)Hence, for a given value of the celerity there are two possible values of the wavelength27Tjk, as observed at the front and rear of the fishing line. When Cl is larg�compared to@, the smaller waves approximately obey c = VTk as capillarity waves would exactlydo, and the larger waves approximately obey c = ViJk as gravity waves would exactly do.The formula (2.
105) further indicates the existence of a minimum velocity, @, belowwhich the waves can no longer be formed. Thomson verified this last point on his yachtwith the help of an eminent guest, Hermann Helmholtz.1281260n a possible anticipation in William Rowan Hamilton's optics, cf. Lamb (1932] p. 381n.127Thomson to Stokes, 3 March 1871, ST; Thomson (1871b]; Ponce1et [1831].
In the same papers, Thomsontreated the wave-generating instability of a water surface under wind, see Chapter 5, pp. 1 88-90.1 28Thomson [1871a], (1871c] p. 88 (He1mho1tz).89WATER WAVES2.5.3 Rayleigh's solutionThomson only reasoned on free waves and did not try to analyze the process throughwhich the fishing line caused the waves. Rayleigh accomplished this much more difficulttask in1883.Using a favorite stratagem, he first turned the problem of progressive wavesinto a steady-wave problem by selecting the reference system bound to the perturbingcause (the fishing line). Then he computed the distribution of surface pressure thatcorresponds to a sine wave in the restricted two-dimensional problem. Although he onlytreated the case of infinite depth, the fmite-depth formulas are given here to allow a9parallel discussion of later related works.
1 2The assumed expressions of the potential cp and the stream function:!'. = x + a coshckyeikx,'f. =cy+ia sinhkyif! areikxe ,(2.106)where a is a small constant (the extraction of the real part of complex expressions isunderstood). The unperturbed motion (adirection of increasing X. The stream linewater. The free surface fits the stream line= O) is a uniform flow at the velocity c in theif! = 0 corresponds to the bottom y = 0 of theif! =eh.khThe corresponding surface deformation isu = -ia sinhikxe .(2.107)The pressure P applied on the free surface differs from the fluid pressure by the capillaryforcepTa".
As the latter pressure obeys Bernoulli's law, we haveP; = -gu + Tu" - 2I (u2 + v-2 - c--_2).(2.108)To first order in the small quantity a, this gives!!_ = ia[(g + T�) sinh kh - !?k cosh kh)]eikx.p+Joo(2.109)Consequently, the surface deformation that corresponds to the pressure pointFP = F8(x) = -21Tof intensity"kxe'dx(2.1 10)F is(2.111)withc� =1 29Rayleigh [I883b].(�+ Tk)tanhkh.(2.112)90WORLDS OF FLOWWhen the wave number k is such that the velocity of the corresponding free wave is equalto the velocity c of the stream, this integral is ill-defmed.
In order to circumvent thisdifficulty, Rayleigh introduced a small, fictitious frictional force J.L(c - v) that damped anyfree oscillation of the uniform stream. As he had already shown in his Theory ofsound,Lagrange's theorem for the existence of the potential remains true in the presence of thisforce.
Its only effect on the previous calculation is an additional term JL(cx - q;) in thepressure equation. 1 30From a formal point of view, Rayleigh thus anticipated the adiabatic turning on of theperturbing force that is commonly used in modem scattering theory. Indeed, a slowvariation of the coefficient a implies an additional term -fJq;jfJt = (ixja)(cx q;) in thepressure equation (2. 1 08).
This term has exactly the same form as Rayleigh's frictional term.Taking into account the frictional term, Rayleigh replaced eqn (2. 1 1 1) byerF= 27Tpg+Joo-ooeikx(I + Tk2 jg)(c2 /q - 1 - iek) dk,(2.1 1 3)where Bk is a small quantity that has the same sign as k. In the case of infinite depth,Rayleigh expressed this integral in terms of elementary or already tabulated functions (thesine integral 'Si') . It is more convenient, however, to retain a large but finite depth (for theintegrand to be meromorphic) and to make use of Cauchy's theorem of residues. 1 3 1 Forpositive x, the integration path can be closed in the complex k-plane by the upper half ofan infinite circle centered on the origin, as shown in Fig. 2.
24. Hence the integral is givenby the sum of the residues in the upper half of the complex k-plane. Symmetrically, for•Fig. 2.24.Integration curve and poles in the complex k-plane for evaluating a certain integral.130Rayleigh [1877n8] par. 239.131Cf. Lamb [1895] pp. 396-7; [ 1932] pp. 406-10.WATER WAVES91negative x the integral is given by the sum of the residues in the lower half of the complexk-plane. The poles of the integrand are represented in the figure. The four poles close to thereal axis correspond to the two wavelengths for which the celerity of free waves is equal tothe velocity of the stream.
Two of the poles marked on the imaginary axis correspondto the wavelength for which the free waves have minimum celerity in deep water. Theremaining poles on the imaginary axis correspond to the infmite sequence of imaginarywavelengths for which the celerity of free waves is equal to the velocity of the stream. Theirdistance lkl from the origin is approximately given by the successive zeros of the functiontan lklh - (Cl jgh)lklh.The contribution of the imaginary poles is a series of terms that decrease exponentiallywith x. The physically important terms are the oscillatory terms given by the quasi-realpoles. For positive x, the two symmetric poles of larger wavelength contribute an oscillation at this wavelength; for negative x, the contributing poles are those of smallerwavelength.
Concretely, the pressure point induces shorter capillary waves upstream,and longer gravity waves downstream, in conformance with Thomsen's observations.A fuller analysis of the wave pattern created by a fishing line requires a three-dimensional analysis. For this purpose, Rayleigh superposed the disturbances produced bypressures constantly applied on straight horizontal lines passing through a fixed point ofthe water surface, the direction of the line being uniformly distributed. The individualwave patterns are those of the two-dimensional problem. Their wavelengths 21rjk are suchthat the corresponding celerity Ck is equal to the projection c cos !fr of the velocity of thestream on their wave normal.
The crests of the various component waves thus formcontinuous families of straight lines whose distance from the origin is a given functionof their orientation. Presumably inspired by an analogy with caustic surfaces in optics,Rayleigh obtained the crests of the combined disturbance as the envelopes of the successive families of straight lines (see Fig. 2.25).
132Fig. 2.25.Rayleigh's construction ofthe waves created by a drifting fishing-line ([1883b] p.267). The plane ofthe figure represents the water surface, the point 0 the upwards drifting trace of the line. The two familiesofstraight lines represent the first crest of the capillarity (in front ofO) and gravity (behind 0) waves caused bystraight lines ofpressure passing through 0. Their curved envelope represents the first waves created by the line.92WORLDS OF FLOW2.5.4Houston's paradox solvedThree years later, Thomson studied the similar problem of the waves produced by auniformly-moving boat.
As he eloquently argued: 133Of all the beautiful forms of water waves that of Ship Waves is perhaps mostbeautiful, if you can compare the beauty of such beautiful things. The subject ofship waves is certainly one of the most interesting in mathematical science. Itpossesses a special and intense interest, partly from the difficulty of the problem,and partly from the peculiar complexity of the circumstances concerned in theconfiguration of the waves.In the two-dimensional canal case, Thomson pushed the analysis far enough to explainHouston's old towing paradox. He enthusiastically reported to StokesY4I have been getting out some very curious things about waves (water), amongthem complete confirmation of Scott Russell's doctrine of sudden diminution offorce, in towing a boat in a canal, when the velocity is got to exceed ,fili. I find(which is now quite obvious) that if water were inviscid, zero force would suffice tokeep a boat moving at any constant speed > ,fili, whether in a canal or in openwater.As Thomson explained in an evening lecture for a popular audience, his theory relied onthe group-velocity concept, and on balancing the work produced by the towing force andthe energy emitted by the boat in the form of waves.
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