Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 28
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The procession of waves behind aboat, he began, is known to be steady with respect to the boat. Therefore, the phasevelocity of this procession must be equal to the velocity of the boat. According to theKelland-Airy formula (2.52), the former velocity cannot be larger than the velocity y'gJi ofinfinitely-long waves. Therefore, the procession can only exist if the boat moves slowerthan this critical velocity, in conformity with Russell's 'accurate observations and welldevised experiments.' 135As the boat must have started from rest, the wave procession necessarily has a finitelength. Its end moves with the group velocity, which is smaller than the phase velocity.Therefore, the length and the global energy of the procession increase in time, and anequivalent work must be spent to propel the boat.
If the boat moves faster, the processionlengthens at a slower rate but the waves are much higher, so that the resistance grows.A crisis occurs when the velocity of the boat approaches that of infinitely-long waves.'Once that crisis has been reached,' Thomson asserted, 'away the boat goes merrily.'Thomson then recalled how 'the discovery [had been] made by a horse' and 'had permittedfor a few years a system of fly-boats between Edinburgh and Glasgow on the Forth andClyde Canal, until, in the early 1 840s, the development of railways had rendered thispoetical notion of speed obsolete.
136In the previous year Thomson had published abundant, complex calculations thatjustified this theory. The basic mathematical problem was to determine the disturbance1 32Rayleigh (!883b].133Thomson (1887./] p. 410.135Thomson (1887./] pp. 415-20.136Jbid. pp. 418-19.1 3"Thomson to Stokes, 8 Nov. 1886, ST.WATER WAVES93of a uniform flow caused by a local pressure on the water surface of a canal. Thomson firstsolved the similar problem. of the waves produced by a bump at the bottom of the canalwhen water flows at constant velocity. These two problems resemble Rayleigh's fishingline problem, except that capillarity is now neglected and the depth is finite. Like Rayleigh,Thomson obtained the desired solution by the superposition of sinusoidal solutions.
Hisexecution of this plan seems awkward to a modern reader. Instead of generating thepressure peak by the direct superposition of sine functions, he used the mathematicalintermediate of a periodic succession of Lorentzian peaks, and then let the distancebetween too successive peaks tend to infinity. He encountered enormous difficulties inevaluating the resulting integral for the surface disturbance. His results were, nonetheless,the same as those of the following calculation based on the method of residues.137For the two-dimensional problem of a local pressure disturbing a uniform flow, thedisturbance is given by the vanishing-capillarity limit of eqn (2.1 1 3):Fer = --27rpg+Ioo eikx dk-oo c2jef - 1 - isk '(2.1 14)in which C£ = (gjk) tanh kh.
The lowest possible value of C£ is its infinite-wavelength limitgh. Therefore, when exceeds Vi/i, the integrand only has imaginary poles, and the integralis an exponentially-decreasing function of x. There is no wave production, and the boat can'travel merrily'. In the opposite case, the integrand has two symmetric, quasi-real poles±kg + is in the upper half ofthe complex k-plane that yield a downstream undulation ofthewater surface, with a period equal to the length 27r/kg of free waves traveling at the speedThis explains why waves are produced by a boat at subcritical speed, and why these wavesalwaysthe boat.
When the speed c is slightly below the critical velocity Vifi, the tworesidues are (3/2kgh2)e±ik,. The amplitude of the resulting oscillations diverges togetherwith their period 27r/kg when the pressure point reaches the critical velocity, in conformance with the 'crisis' described in Thomson's popular lecture.
In the limit of infinite depth,the two residues are kge±ik,x, so that the amplitude of the oscillations is inversely proportional to the wavelength. Lastly, Thomson computed the necessary propelling force bybalancing the energy flux of the waves with the work done by this force.cc.follow2.5.5Echelon wavesThomson's greatest achievement in this area was to derive the ship-wave pattern in thethree-dimensional case.138 Like Rayleigh, Thomson superposed the disturbances produced by pressures constantly applied on straight horizontal lines passing through afixed point 0 of the water surface, to be identified with the location of the boat.
Had hefollowed Rayleigh even further, he could have obtained the wave pattern geometrically, by137Thomson [1886].138Thomson [1887/], [1906]. In 1887, Thomson only gave the formulas for the configuration of the wave crests,which he claimed to have obtained by Stokes's principle of group velocity ([1887f] p. 423). In the following it isassumed that the relevant calculations were similar to those of Thomson [1906]. One could speculate thatThomson reasoned in the more elementary manner given at the end of this chapter. That manner, however,does not seem to yield the height of the waves, which Thomson claimed to have computed in 1887.94WORLDS OF FLOWtracing the envelopes of the component wave crests.
139 However, he preferred a moreanalytical method that also yielded the intensity of the waves.Denote by r and () the polar coordinates of the point P of the wake with respect to theorigin 0 and to the trajectory of the boat, and by if! the angle that the rearward normal ofone of the pressure lines makes with the axis () = 0. The distance 8 of the point P from thisline is r cos (if! 0).
The wave number k of the resulting wave component must be such thatthe corresponding wave velocity Vi[k is equal to the projection V cos if! of the velocity Vof the boat on the wave normal. Hence the phase 4> of this component at the point P is</> = kB =gr cos(f/1 - 0).V2 cos2 f/!(2.1 15)Its amplitude is proportional to k = gj V2 cos2 if!. The angle if! is uniformly distributedbetween 0 - 'IT/2 and 0 + 'IT/2, since P must belong to the wake of the if! line. The resultantdisturbance has the form140df/1.cos2 "'I -e-,./2B+rr/2u r:x.cos </>(2. 1 1 6)In order to evaluate this integral, Thomson appealed to 'the principle of interference, as setforth by Prof. Stokes and Lord Rayleigh in their theory of group-velocity and wavevelocity.' 141 At a distance from the boat much larger than the characteristic wavelengthA = gj V2 , the phase 4> is very large and therefore cos </> oscillates very quickly betweenpositive and negative values when if! varies.
This oscillation implies destrUctive interference,unless there are particular values of if! for which the phase is stationary, that is, d<f>/df/1 = 0.If x and y denote the Cartesian coordinates of the point P, and T is the tangent of theangle if!, we have(2.1 1 7)The condition of stationary phase then givesXT + y(1 + 2�)= 0.(2.1 1 8)This quadratic equation has real roots only if (yjx)2 < 1/8. Hence the .llisturbance isconfined between the two half-lines that originate in the (point-like) boat and make anangle oftan-1 JI78 � 19°28' with the mid-wake of the boat. The curves of constant phaseobey the parametric equations 142(2. 1 1 9)1 39Lamb did so in the 1895 edition of his treatise. The equation of the envelope is easily seen to be identical tothe condition of stationary phase that Thomson presumably used in 1 887.14"Thomson [1906] p.
409.141Thomson [1887g] p. 303.142The formulas of Thomson [1887f] have a different parameter,same curves, despite Larmor's contrary statement ([1907] p. 413n).w=(I-2i')/(l+ 2i'),but represent theWATER WAVES--95S H I P WAVES ._Wave- Pafterrv.Pl.wv af &rves·---- - - ----'!(!!_<:helnn Wave.s .of Echelon WaYes.Fig.
2.26.Kelvin's ship-waves (Thomson [ 1 887f1 plate; perspective view borrowed by Kelvin from R.E.Froude: Thomson must have considered that the impulse of the prow of the long barge approximatelydetermined the wave pattern).96WORLDS OF FLOWwhere a = (A.j27rr)<f>. They have the 'beautiful' shape represented in Fig. 2.26. Thomsonfurther determined the amplitude of the waves by summing the contributions of the tworoots of eqn (2. 1 1 8) to the integral (2.1 16). He even had a clay model made to represent thewave pattern.1432.5.6 The stationary-phase methodThomson's success in completing the theoretical analysis of ship waves crucially dependedon the stationary-phase method.
An anticipation of this method is found in a mathematical paper of 1850 by Stokes, in the context of Airy's spurious rainbows. 144 Stokes did notexplain why the procedure worked, and did not provide a clear criterion for judging whichintegral was amenable to it.
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