Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 22
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To sustain this opinionhe did not use Airy's objection, which only excluded solitary waves of arbitrarily longlength. Rather, he referred to recent calculations by Samuel Earnshaw. The Reverendmathematician had integrated the equations of motion for a wave of permanent shape thatmet a condition experimentally verified by Russell, namely, that fluid particles originallyin the same vertical plane remained so during the passage of the wave.
In Earnshaw'sopinion, this result confirmed the existence of solitary waves. Stokes drew the oppositeconclusion from the same calculation, for he noted that Earnshaw waves could not beconnected to the surrounding fluid at rest without an absurd discontinuity of the velocity.As Stokes did not question the experimental truth of parallel-plane motion, he concludedthat there was a necessary non-frictional decay of solitary waves.80Not only did Stokes deny the properties of solitary waves that Russell judged mostessential, but he also condemned-without naming Russell-applications of solitarywaves to tides and to sound:81With respect to the importance of this peculiar wave .
. . it must be remarked that theterm solitary wave, as so defmed (as a phenomenon sui generis] must not be extendedto the tide wave, which is nothing more . . . than a very long wave, of which the formmay be arbitrary. It is hardly necessary to remark that the mechanical theories of thesolitary wave and the aerial sound wave are altogether different.2.4.2 Stokes on finite oscillatory wavesIn 1 846, Stokes believed permanent, solitary waves of finite height to be impossible.
Butthe existence of permanent, oscillatory waves of finite height remained plausible. Also,Russell had found that the (phase) velocity of oscillatory waves obeyed the Kelland-Airyformula (2.52) (for infinitely-small waves) even when the waves were no long!!r small withrespect to the depth. Stimulated by this result and its apparent contradiction with Airy'svelocity formula (2.59) for finite waves, Stokes sought a perturbative solution of Euler's78Stokes [1846a] pp. 1 6 1-4 Oong waves), 1 71-5 (tides); Lagrange [1781]; Green [1838]; Kelland [1840]; Airy[1845]. On early British wave theory, cf. Craik [2004] 8-24.79Stokes [1846a] p. 1 64; Green [1839]; Kelland [1840]. Kelland believed the motion to have a form independentof the height of the waves, for he used erroneous boundary conditions.80Stokes (1846a]: pp.
168--70; Earnshaw (1849] (read in Dec. 1 845). Cf. Craik [2004] pp. 1 7-18.81 Stokes (1846a] p. 170.71WATER WAVESequations that made the fluid velocity the gradient o f a potential and a function o f xand y only.82As in Lagrange's theory of waves, the potential must satisfy eqn[)2cp8x2+82cpBy2=(2.14),- etnamely0.The equation of the surface isa('V)2_! + _'P_ + g(y - h)Bt2=(2.61)0.The boundary condition at the bottom of the channel is8cpf8y = 0when y =0.Thecondition that a particle on the surface should remain on the surface is(2.62)at any point of the surface.
The general integral of eqn (2. 14) that meets the first boundarycondition iscp = Cx + L cosh ky(Akkcos kx + Bk sinkx).The first term may be dropped as it represents a constant velocity. To first order in(2.63)cp, thesecond boundary condition (2.62) and the condition that the velocity is a function of x and y only imply eqn(2.52), namely2=�tanhetkh,for every term of the sum over k. Since there is only one value of k that meets thiscondition, the sum is reduced to a sine wave.
83As a corollary, the propagation of a solitary wave without change of form is impossibleat frrst order. In modem terms, we would say that the dispersion (dependency of celerityon wavelength) of infinitely-small monochromatic water waves implies the spreading ofwave packets.
Stokes concluded:84Thus the degradation in the height of such waves, which Mr. Russell observed, is notto be attributed wholly, (nor I believe chiefly,) to the imperfect fluidity of the fluid . . .but is an essential characteristic of a solitary wave. It is true that this conclusiondepends on an investigation which applies strictly to indefinitely small motions only:but if it were true in general that a solitary wave could be propagated uniformly,without degradation, it would be true in the limiting case of indefinitely smallmotions; and to disprove a general proposition it is sufficient to disprove a particularcase.83Ibid. [1847a].
pp. 199-204.84Ibid. 204. This objection is invalid, because it assumes that the length of the waves is kept constant in the82Stokes [1 847a]. Cf. Craik [2005].zero-amplitude limit, whereas for a solitary wave the length grows indefinitely when the amplitude tends to zero.WORLDS OF FLOW72After this new blow to Russell's interpretation of the solitary wave, Stokes proceeded togive a theoretical justification of Russell's experimental results on oscillatory waves. 85 Tosecond order in the amplitude a of the wave, the celerity of the waves still obeys theKelland-Airy formula (2.52), in conformity with Russell's measurements. This result doesnot contradict Airy's formula (2.59), Stokes explained, because the latter assumes wavesmuch longer than the depth, whereas the smallness of Stokes's perturbations is easily seento contradict this condition.
86 The equation of the surface isy=h + a cos kx -[cosh)]kh(2 cosh2kh + 1ka2 cos 2kx.4 sinh3kh(2.64)For infinite depth and to third order, it isy=13h + a cos kx - :z ka2 cos 2kx + 8 �a3 cos 3kx,(2.65)fair agreement with the trochoids that Russell had inferred from observations of highsea waves (see Fig. 2.2 1 ). To the same order, the deep-water celerity becomesinc=�V!i."k (l + 2 �aZ)(2.66)Lastly, Stokes found that, for high waves, the propagation of the waves was accompaniedby a net flux of water.
He even recommended taking into account this flux in the deadreckoning of the position of ships. 872.4.3 Gerstner 's waves and ship rollingStokes returned to water waves in the 1 870s, when he had to write a memorandum on themeasurement of waves for the Meteorological Council. 88 A good knowledge of the height(a)(b)Fig. 2.2 1 .Wave of finite height according to Stokes's theory ([1847a] p. 212) (a); according to Russell'scycloidal interpretation of ocean waves (b).too pale?85Stokes [1847a] pp. 205-8.
To second order, Stokes also gave finite-depth results.86Ibid. p. 209. Moreover, Airy dealt with a different problem, namely, the deformation of a wave that has a sineshape near the origin.87Stokes (1 847a] pp. 198-9, 208-9.88Cf. Froude to Stokes, 17 Jan. 1 873, in Stokes (1907].WATER WAVES73and length of sea waves, he argued, was necessary for a proper control of ship rolling.
Thispreoccupation and discussions with William Thomson-who was involved in similarquestions-probably led him to improve his theory of high waves and to reflect on thehighest possible wave. In 1 8 80, he used the publication of the first volume of his collectedpapers as an opportunity to update his views on this topic.
89In the first place, Stokes expressed his opinion on an old theory of finite, oscillatorywaves on infmitely-deep water that had become popular among naval engineers. Thistheory, published in 1 802 by the Prague mathematics professor, engineer, and knight,Franz Joseph von Gerstner, assumed a circular motion of the fluid particles with a radiusdiminishing with the distance from the surface:90x= X + k-1ekY cos k(X - et), y = Y - k-1ekY sink(X - et),(2.67)where x and y are the coordinates at time t of the particle that has the mean coordinates Xand Y, 27T/k is the wavelength, and e is the celerity of the wave. This motion is easily seento satisfy the continuity condition and the equations of motion.
The pressure for a givenfluid particle is independent oftime if and only if e = Vi/k. It is then a function of l' only,so that the wave surface can be any of the lines for which Yis a negative constant. Fig. 2.22illustrates the resulting waves for different values of this constant. The highest waves, forwhich the constant vanishes, have an infinitely-sharp edge. Their surface is a cycloidgenerated by a circle of radius k-1 rolling on the underside of the line y = k-1 • Theother waves are trochoids with an eccentricity decreasing with their amplitude.
Gerstnerbelieved his waves to be the only ones compatible with the general principles of mechanics.In fact, as the Leipzig mathematician Ferdinand Moebius noted some twenty years later,Gerstner's derivation relies on the specific assumption that the pressure around anyparticle of the fluid remains the same in the course of time (whereas general principlesrequire this to be true only for the particles at the free surface of the fluid).91The Webers' Wellenlehre included a detailed analysis of Gerstner's waves.
They foundreasonable agreement with the observed motion of suspended particles, although theradius of the circular motions did not quite vary as Gerstner predicted. Their overalljudgment was laudatory: 'Even if these conditions [for Gerstner's calculation to apply] arenot completely met in reality, Gerstner's investigation remains not only interesting butalso useful.' Russell, who became acquainted with Gerstner's waves through the Webers'book, found even better agreement with observation than the Webers had. His judgmentwas enthusiastic: 'Gerstner's theory is characterized by simplicity of hypothesis, precisionof application, its conformity with the phaenomena, and the elegance of its results.m89Both Stokes and Thomson implicitly assumed a simple relation between observed ocean waves and thetheory of finite waves of permanent shape.
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