Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 29
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In contrast, Thomson imagined the destructive interference ofthe rapidly-oscillating integrand, and showed that the method applied to the integral ofany rapidly-oscillating function. In 1 887, he published a striking application of thismethod to the Poisson-Cauchy integral given in eqn (2.30), namelyu� I dk cos kxcos wkt,+oo=0that represents, in two dimensions, the water-surface disturbance caused by a localdeformation around x = 0. 1 45The progressive part of this disturbance is the real part of the integral(2.120)The phase is stationary when x dk - t dwk = 0, that is, when the group velocity dwk/dk isequal to xjt.
For gravity waves on infinitely-deep water, Wk = ..fik. The phase is stationary for k = K = gt2j4x2. Its value around this stationary point isgt2(2.121)+ (3(k K)2,4xwhere (3 = x3 jgt2 is the value of fd2 <f>/dk2 for k = K. The resulting approximation of theintegral for large values of gt2 j4x is4> """ -+u= � e-igfl(4x27r+ooI dke-oo-ifl(k-•)' = � e-igt2(4x f!leirr/4 = At27rV f3fg e-i(gt2(4x":..rr/4) ' (2.
122)2 V -;;;;;;in conformance with Poisson's equation (2.36).In the same spirit, Horace Lamb later noted that, in the large-phase approximation andfor relatively small variations of the distance x from the origin, the disturbance at a given143Thomson [1906] p. 413; [1887/] p. 424.144Stokes [1 850a]; Airy [1838]. Cf. Darrigo1 [2003] pp. 85-6.145Thomson [1887g].97WATER WAVEStime differs very little from a sine wave with the wave number k = gfl /4x2. Therefore, thedisturbance created by a non-local deformation with the profile f (a) results from theinterference of a system of sine waves with an amplitude proportional to f(a) and a phaseshifted by -ka.
As in diffraction theory, this picture leads to replacing the coefficient A bythe F ourier transform of the profile in the expression (2.122) of the disturbance created bya point-like perturbation. Lamb thus short-circuited Poisson's delicate and lengthy derivation of eqn (2.38). 146In summary, Thomson and Lamb substituted a physico-mathematical analysis for thepurely formal developments of Poisson, Cauchy, and Stokes. The gain was enormous:whereas the anterior treatment rested on formal tricks that required much ingenuity andworked only in particular cases, the stationary-phase method offered an intuitive strategythat automatically gave the asymptotic behavior of large-phase integrals.
Under Thornson's magic wand, much of the enormous memoirs of Cauchy and Poisson collapsed into afew lines of physico-mathematical common sense. Moreover, the formidable problem ofship waves received a strikingly simple solution.2.5.7After-mathThis solution was not quite definitive. Although Thomson gave the result and the methodof stationary phase in 1887, he only published the calculations in 1906, at age 83. As heknew, the pressure obtained by isotropic superposition of pressures localized on straightlines passing through a fixed point varies as the inverse of the distance from this point.
Inmodem notation, this results from the identity Jg" o(rcos e)de = 2/r. Such a slowlydecreasing function cannot realistically represent the pressure exerted by a boat on thewater surface. In the year of his death, Thomson was still working on an improved versionof his theory in which the perturbation was more sharply localized. 147In his conference of 1 887, Thomson suggested a more direct approach. The disturbanceproduced by the ship, he noted, may be regarded as the superposition of the disturbances produced by a succession of impulses along its path.
A Newcastle lecturer in appliedmathematics, Thomas Havelock, managed to do the corresponding calculations in 1908,thanks to a repeated application of the method of stationary phase. At a point P in the wakeof the ship A (see Fig. 2.27), the wave created by an individual impulse is the superpositionof monochromatic, circular waves with the phase wr - kd (up to a constant), where k is thewave number, d is the distance EP between the impulse and the point P, w is the deep-waterpulsation yg!{, and t is the time that has elapsed since the ship was at E.
1 48E�.."'dFig. 2.27.146Lamb [1932] pp. 392-4.p--''" - - -()---- ->ArDiagram for Havelock's calculation of Kelvin's ship-wave pattern.147Thomson [1907].148Thomson [1887fl; Havelock [1908]. See also Lamb [1916] and Lamb [ 1932] pp. 433-7.WORLDS OF FLOW98In order to avoid destructive interference between the waves created by successiveimpulses, this phase must be stationary with respect to a variation of the time t. Hencethe phase velocity wjk must be equal to i1 = V cos a, where V is the velocity of the boatand a is the angle that EP makes with the direction of motion of the ship.
Furthermore, thephase must be stationary with respect to a variation of k. This implies that the groupvelocity dwjdk must be equal to the ratio djt.The first condition of stationarity and some trigonometry lead to the expression<P=gr cos (aV2-cos2 a0)(2.123)for the phase, where r is the distance from P to the ship's present location A, and e is theangle that AP makes with the direction of motion of the ship.
This expression is the sameas Thomson's equation (2.1 1 5), although the angles a and 1/J have different interpretations.As the variation with respect to k is equivalent to a variation with respect to a, Havelock'scalculation yields the same lines of constant phase as Kelvin's. Only the height of the wavecrests differs, because the amplitude of Havelock's spherical component waves differsfrom the amplitude of Kelvin's straight-line component waves.149 This correction does notreally improve the comparison with the experimental pattern, because the latter dependson the form of the ship, and because at the cusps of the curves of constant phase thetheory leads to a divergent amplitude that is incompatible with the original small-wave1assumption. 50In Kelvin's and Havelock's derivations of the echelon shape of ship waves, there seemsto be a disproportion between the simplicity of the results and the complexity of thecalculations.
In his popular lecture of 1 887, Thomson hinted at a more elementaryderivation. After noting that the disturbance produced by the ship could be regarded asthe superposition of the disturbances produced by a succession of impulses along its path,he declared that the point E in Fig. 2.26(top) (such that EC = CA) represented the positionof the ship at the time when it caused the impulse responsible for the disturbance aroundC. His justification holds in one sentence: 'Calculate out the result from the law thatthe group-velocity is half the wave-velocity-the velocity of a group of waves at sea ishalf the velocity of the individual waves.' Indeed, if the disturbance travels from E to C atthe group velocity, and if the phase velocity along the x-axis is equal to the velocity of theship, this law implies that the ship must move twice as fast as the disturbance.
Thomsonseems to have grasped these two conditions intuitively, through the picture of a train ofwaves made of individual waves that are steady with respect to the ship. As we have just1seen, they can be justified through the method of stationary phase. 1 5Thomson's consideration may now be extended to the disturbance around a point P thatis no longer on the x-axis. This disturbance has traveled from E with the group velocity inthe direction EP. For steadiness with respect to the ship, the phase velocity in this directionmust be equal to the projection of the ship velocity on this direction. Hence the angle EPC149This agreement should be expected, because the disturbance created by a diffuse pressure is the superposition of geometrically-similar echelon patterns created by symmetrically-distributed pressure points."00n more realistic theories of ship waves, cf.
Lamb [1932] pp. 437-9."'Thomson [1887/] p. 426.WATER WAVESFig. 2.28.99Diagram for elementary calculation of Kelvin's ship-wave pattern.is a right angle, and the point P must be located on the circle of diameter EC (see Fig. 2.28).The significantly disturbed part of the water surface is therefore confmed between thetangents AH and AH', which make the angle sin- 1 (OH/OA) = sin- 1 1 /3 = tan-1 .JI78with the axis.152Denoting by Xthe diameter EC and a the angle that the emission line EP makes with theaxis, the Cartesian coordinates of the point P are (see Fig.
2.28)x= X(2 - cos2 a), y = X sin a cos a(2. 124)On a given curve of constant phase, y is a function of x; or, equivalently, X is a function ofa. This function can be determined through the condition dy/dx = cot a, which meansthat the curve of constant phase that goes through P is normal to the direction EP ofpropagation.
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