Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 26
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As modernsoliton theorists know, the possibility of solitary waves rests on the exact compensationbetween a linear dispersive term and a nonlinear term in the equation of motion. For agiven height of the wave, this compensation only occurs for a definite shape and length.1182.5 The principle of interference2.5.1Group velocityIn his report on waves of 1 844, Russell wrote:One observation which I have made is curious. It is that in the case of oscillatingwaves of the second order, I have found that the motion of propagation of the wholegroup is different from the apparent motion of wave translation along the surface.The remark went largely unnoticed, until William Froude privately communicated asimilar observation to Stokes and to Rayleigh in the early 1 870s.119o f the wave.
Strictly speaking, they did not write Boussinesq's equation(2. I 02),which i s now called the KdVequation. On the precise connection between their equation and the KdV equation, cf. Miles117p.[1981].Miles [1981]137, who notes that Korteweg and de Vries's expression for the relation between cnoidal waves and Stokes'sKorteweg and de Vrieswaves is not quite correct.1 18Korteweg and de Vries"9Russell[1845] p.
67.[1 895] p. 424; Boussinesq [1877] pp. 390-6. Cf. Lamb [1932] p. 426-7,[1895] pp. 438-43;Levi-Civita [1925]; Stokes [1880c],[1891].86WORLDS OF FLOWAt that time, Stokes was working on the measurement of sea waves for the Meteorological Council. In particular, he was asked to determine the origin of the strong swellssometimes observed in fine weather. Stokes immediately explained these swells by wavepropagation from distant storms, and commented: 'It is curious to see that captains seemto have so little idea of the propagation of waves excited in a stormy region into a regionwhere as regards the wind, it is comparatively calm.' According to the formulac = w/k = Vi/k of small deep-water waves, Stokes explained, the velocity c of a periodicwave is related to its time period r = 27T/w through c = gr/27T.
A measurement of r wouldthus provide information on the location of the storm. 1 20In 1 873, William Froude read the relevant section of Stokes's memorandum. Hecommented to the author:Primdfacie, the speed of such waves would determine the duration of their passageover a given distance. But this is not really so: because the foremost waves areperpetually dying out, as they invade the undisturbed water, and are undergoingmetempsychosis in the ranks behind them.For example, FJ;"oude went on, if the wheels of a paddle ship are stopped while its speed iskept constant by other means, the waves remain stationary with respect to the ship buttheir front moves away from the ship.
From the perspective of an observer at rest, thismeans that the undulations within the train of waves advance faster than the front of thetrain. Froude had seen a lot of that in his towing tanks.1 21In January 1 876, Stokes reported to Airy:I have lately perceived a result of theory which I believe is new-that the velocityof propagation of roughness on water is, if the water be deep, only half of thevelocity of propagation of the individual waves. This is of importance in connectingrecords of long swells which may be found in ships' logs with records like those ofAscension or St. Helena.The following month he proposed the following problem for the Smith prize examinationpapers at Cambridge University: 1 22Find the expression for the velocity of propagation of a series of simple periodicwaves in water of uniform depth, the motion being small and in two dimensions.-Iftwo such series, of equal amplitude and nearly equal wavelength, travel in the samedirection, so as to form alternate lulls and roughness, prove that in deep water theseare propagated with half the velocity of the waves; and that as the ratio of the depthto the wavelength decreases from oo to 0, the ratio of the two velocities increas6's from� to 1 .Denoting by k and k + dk the wave numbers o f the two superposed waves, and w andw + dw the corresponding pulsations, the amplitude of the superposition varies ascos ! (xdk - tdw).
The resulting modulation travels with the velocity dw/dk. For small120Stokes to Captain Toynbee, 5 Sept. 1878, in Stokes [ 1907] vol. 2, p. 141; Stokes to Colonel Sabine, 22 Sept.1 870, ibid. p. 136.121 Froude to Stokes, 17 Jan. 1873, ibid. pp. 1 56-7.122Stokes to Airy, 5 Jan. 1 876, ibid. p. 1 77; Stokes (1876].87WATER WAVESh,waves in water of depth according to Kelland and Airy,sponding ratio between the group and phase velocities,dwldk 1= z (lwlk+ k tanh- 1hw2 = gk tanhkh. The correkh - khtanh kh),(2.104)varies from � to 1 when kh varies from oo to 0, as Stokes asked the Smith prize competitorsto demonstrate.123The following year, the Manchester engineering professor Osbome Reynolds reportedhis own observations of wave groups produced by throwing a stone into a pond, by theinterference ofsea waves, or by the motion ofa ship.
Like Russell and Froude, he noted thatgroups of waves in deep water traveled slower than the individual waves ofwhich they weremade. To explain this result, he first noted that the velocity of a wave group obviouslyrepresented the velocity of propagation of energy. He then showed that the latter velocitydiffered from the phase velocity. For instance, the waves produced by wind in a corn fieldobviously do not propagate any energy, since the motions of the individual corn stert;J.s areindependent. In the more complex case of a sine wave on deep water, the particles of watermove on circles with constant velocity, so that no kinetic energy is transmitted by the wave.In contrast, the potential energy is transmitted at the phase velocity.
Since the potentialenergy of such waves is half their total energy, the speed of energy propagation is half thephase velocity. Therefore, the group velocity is half the phase velocity.124In his influentialof 1877, Rayleigh included Stokes's derivation of thegroup velocity, which he had independently obtained under Froude's stimulus. In acontemporary article, he proved Reynolds's equality between energy and group velocityin a precise mathematical manner.
In the case of small waves on water of finite depth, hedid this by computing the ratio between the work of pressure forces on a transverse sectionof the water and the energy density of the waves. In the general case of waves in anarbitrary dispersive medium, he astutely introduced a fictitious friction proportional to theabsolute velocity of the parts of the medium. Assuming vibrational energy to be created atx = 0 and to propagate in the direction of increasing x, he computed the damping effect ofthe frictional force by noting that it turned the operator EP1 at2 into 82 Iot2 + J.LO1 ot,wherein f.L is the friction coefficient divided by the fluid density p.
This is nearly equivalentto changing the pulsation w into w - ! iJ.L. The corresponding change of k is - � iJ.Ldkldw.Consequently, the oscillating factor ei(.,t-kx) of a forced oscillation at the pulsation w isturned into e-!JLXdk/d"' ei(wt-kx) . The dissipated energy in the region x > 0 is the integral ofJ.Lr:nl-. It is therefore equal to 2J.L times the kinetic energy, or else J.L times the total energy inthis region (according to a well-known theorem for harmonic oscillations). Denoting by Ethe energy per unit length near the sou;rce, this remark leads to the expressionJ.LE fo"" e-!'-Xdkfdwdx = Edwldk for the dissipated energy.
By energy conservation, thisdissipation must be compensated for by the energy flux EcE through the section x = 0 ofthe water. Therefore, the velocity cE of energy propagation must be identical to the groupvelocity dwldk.125Theory ofsound1 23A more general argument with a continuous distribution of k is found in Rayleigh [1881].124Reynolds [1 877b].1 25Rayleigh [1877], [1877-78].88WORLDS OF FLOWThe concept of group velocity could plausibly have emerged in the fields of physicswhere dispersion was first known, namely optics and acoustics.126 In reality it did not.
Aswe have just seen, observations made on deep-water waves played a crucial role. Theymotivated Stokes's and Rayleigh's theoretical considerations, although these assumedfamiliarity with interference and beat phenomena in optics and acoustics. As for Froude'sunderstanding of group velocity, it derived from his engineering concern with the energycarried by the waves.2.5.2Thomson 'sfishing lineIn early 1 87 1 , the catastrophic sinking of the HMSCaptain prompted the British Admiralty to name a 'Committee on designs for Ships of War'. On behalf of this committee,William Thomson asked his friend Stokes a few questions about waves: 'The longest wavesthat have been observed?-by whom?-their length from crest to crest?-and height fromhollow to crest?' The following summer, while sailing on his personal yacht the LallaRookh, he observed a gentler but no less interesting phenomenon: a fishing line hangingfrom the slowly-cruising yacht caused very short waves or 'ripples' directly in front of theline, and much longer waves in its wake.
The whole pattern was steady with respect to theline, so that the celerity of both kinds of waves was equal to the velocity of the line'sprogression through the water. Unknown to Thomson, the French military engineermathematician Jean Victor Poncelet had already described this phenomenon with hiscolleague Joseph Aime Lesbros, and Scott Russell had already identified capillarity asthe cause of the ripples.
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