Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 21
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The verticalcomponent of this action amounts to a negligible modification of gravity. However, theequation of motion now includes the horizontal component F of this action:JS JJ""".tU.z/, I'Omr. r:r·tUU.,;A¥1!... a. .rJ..Jk,., rirr.t.o tirinl. npp�n-. widr. .rm4Jl, t;i..U f2.lJl!u....... � .c::::::-... � �� �� �� -<""Fig. 2.20.The evolution of a sine wave along a canal to second and third order in its amplitude (Airy [1845]plate).69Airy [1845] pp. 297, 300.believed the solution to be still valid far from the mouth.tion requires that x < < h/ka.70AiryInreality, the consistency of the approxima71 Airy [1845] pp.
300-l . As Stokes, Saint-Venant, and Boussinesq later made clear, Airy's formula applies to thecrest of long, non-permanent waves, whereas Russelfs formula applies to permanent waves whose length iscomparable to the depth ofwater. Simple derivations ofAiry's formula are found in Lamb [1932] pp. 261-2, 278-80.68WORLDS OF FLOW(2.60)Here Fis the superposition of harmonic components with a latitude-dependent phase andamplitude. Airy determined the resulting forced oscillations for circular canals runningalong a parallel and along a meridian, and for canals closed at both ends.
In each case,there are free oscillations at frequencies that are integral multiples of a fundamentalfrequency. The amplitude of the forced oscillations depends on how close these eigenfrequencies are to the frequencies of the tidal force F. Airy also introduced friction propor72tional to the velocity, and discussed the consecutive damping of free oscillations.In conclusion to this analysis, Airy admitted that his assumption of tidal canals ofuniform depth and breadth was no more realistic than Laplace's assumption of an Earthcovering ocean with a special law of depth. The main advantage he saw in his method wasthat it permitted a more detailed consideration of the interplay of the lunar, solar, andfrictional forces, since all the equations could be solved in finite terms through elementaryanalysis.
In brief, his theory failed as much as Laplace in quantitative tide prediction, but it73offered more qualitative insights.2.3.5 The inverse method, for and against RussellAiry did not confine his study of waves to aspects relevant to tide theory. He alsoexplained commonly-known properties of water waves, and some of Russell's moresurprising results. As he was generally unable to integrate his hydrodynamic equationsfor the actual forces that produced the wave motion, he ingeniously inverted this procedure: he sought to compute, for a hypothetical form of fluid motion, the forces that wouldmaintain this motion.
This is much easier to do, since differentiations are involved insteadof integrations. From the knowledge of these forces, he then inferred what the actualmotion would be in their absence, or what additional action on the water could producethe hypothetical motion.As a first example, consider the breaking of waves on a sloping shore.
Airy computedthe forces necessary to maintain a constant shape of the waves when they approach theshore. The result is forces that pull the tip of each wave in the direction opposite to that oftheir progression. Since in reality these forces do not act, the tips of the waves must bendforward, as should happen at the beginning of the breaking process. Another example isthe swelling of waves under wind. Airy injected a swelling motion in the equations ofmotion.
The resulting forcesturn out to be pressures applied to the rear of tl;te waves, aswould naturally be expected for waves before the wind.74A third example is the 'great primary wave', or forced wave that accompanies a canalboat in its motion. In this case, the horizontal disturbance g and the surface disturbancetuare functions of x - v only, where v is the velocity of the boat. In the small-long-wave(2.60) gives F = (if - gh)g", while the continuityu = -hg' .
Therefore, the force that is necessary to maintain thisapproximation, the equation of motionequation(2.53)givesmotion has the same sign as the slope of the surface when the velocity of the boat is inferior72Airy [1 845] pp. 310-39. For a concise account of Airy's theory of oceanic tides, cf. Lamb [1932] pp.
267-73.73Jbid. p. 363.14Ibid. p. 314.WATER WAVES69to that of free waves; it has the opposite sign in the reverse case, and it vanishes when thetwo velocities are equal. This conclusion agrees with the relative position of a canal boat7and its forced wave, and with the drop in resistance in the critical case.
5Airy thus explained Russell's observations, but implicitly rejected his intuitive theory ofsolitary-wave riding. Through the same kind of argument, he dismissed Russell's solitarywave. For waves of finite height, the equation of motion is the nonlinear equation(2.56).Without additional force and for a disturbance propagating without any change of shape,it can only hold if the slope I;' of the disturbance is a constant. As this slope must vanish atinfmity, there is no such disturbance. Airy concluded that the solitary wave was mathematically impossible. What Russell had observed was a wave small enough forLagrange's theory to apply approximately:76We are not disposed to recognize this wave [Russell's] as deserving the epithets 'great'or 'primary' . .
. and we conceive that, ever since it was known that the theory ofshallow waves of great length was contained in the equation 82t;/8t2 = gh82t;j8x2. . . the theory of the solitary wave has been perfectly well known.As we have already seen, this authoritative judgment failed to disturb Russell's belief in thenovelty of his solitary waves.2.4 Finite waves2.4.1InStokes's BA report1 846 a new leader of British hydrodynamics, the Cambridge professor George GabrielStokes, reviewed the state ofthis field for the British Association. Since the previous report byChallis, there had been much British work on waves, in a good part stimulated by Russell'sexperiments. Stokes played down the importance ofPoisson's and Cauchy's memoir: 'Themathematical treatment of such cases [waves produced by emersion] is extremely difficult;and, after all, motions of this kind are not those which it is most interesting to investigate.
' Inthe wake of Russell's and Airy's works on waves, tides, and navigation, what had becomemost important was the study of 'simpler cases of wave motion, and those which are morenearly connected with the phenomena which it is most desirable to explain. mAmong the simpler cases of motion, Stokes retained waves with a length much longerthan the depth. As Lagrange, George Green, Philip Kelland, and Airy had shown, thesewaves propagated without deformation in a canal of constant section as long as theirheight was much smaller than the depth. Their velocity obeyed a simple formula. Greenand Airy had computed their deformation for a slowly-varying canal depth or breadth.75Ibid. pp.
349-50.76Ibid. p. 346. As Stokes later noted, Airy overlooked the fact that his equation of motion applied to waveslonger than those observed by Russell.77Stokes [1846a] p. 161. This opinion echoed an earlier remark by Kelland ([1840] p. 497): 'I doubt much . . .whether such men as Laplace and Lagrange would have been induced, with the expectation ofjoining experimenton her lower and more trodden fields, to reconsider and remodel their investigations; nor have I any reason tohope, that such men as Poisson and Cauchy will quit the delectable atmosphere in which they are involved, ofabstruse analysis, for the more humble, but not less important task ofendeavouring to treat the simpler problemsin a manner not made general arbitrarily to lead to the most elegant formulae, but general to that extent, and inthat mode, in which the problem in nature is so.
•70WORLDS OF FLOWAiry had shown how finite height affected their propagation. Stokes also dwelled on thefruitful application that Airy had given of this sort of wave to the theory of tides.78Another case of special interest was given by 'waves which are propagated with aconstant velocity and without change of form, in a fluid of uniform depth, the motionbeing in two dimensions and periodical.' By implicit analogy with monochromatic planewaves in optics, he regarded these waves as 'the type of oscillatorywaves in general'.
Greenhad given the expression Vi/k for the velocity of such waves in the case of infinite depth,and Kelland had anticipated Airy's results in the case of finite depth.79Stokes then turned to the controversial issue of solitary waves. Stokes admitted thatRussell's experiments made the sui generis character of solitary waves probable, but hedenied that friction was the only cause of the decay of such waves.
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