Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 20
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If the canal was terminated by a wall at oneend, standing waves occurred. If the canal had the finite length L, the period of theoscillations could only be an integral multiple of a fundamental period L I c, as in closed59Kelvin's later tide-predicting machine was based on the same principle.6"¥oung [1823] p. 307.WATER WAVES63organ pipes. Further assuming that the sea was equivalent to a canal along its greatestlength, Young replaced it with a set of pendulums that had the same periods. He was thusleft with the elementary problem of determining the response of a damped harmonicoscillator to a sinusoidal excitation.6 1As is now known to any physics undergraduate, the general solution to this problem isthe sum of a free oscillation that exponentially decreases in time owing to the dampingforce, and a forced oscillation whose amplitude varies as (w5 - w2) - 1 if the eigenfrequencywo is not too close to the excitation frequency w.
When the former frequency exceeds thelatter, the forced oscillations are in phase with the exciting ones. In the opposite case, thetwo oscillations are in opposition. This result has an immediate, fruitful application: forthe known order of magnitude of the depth and size of the oceans, their fundamentalperiod of oscillation is much larger than half a day, so that the phase of tidal oscillations isopposed to the phase of the inducing luminary (as in Fig. 2.1 8(b)).
Through equallyelementary reasoning, Young explained several other well-known properties of the tides.Russell was apparently unaware of Young's insights when he proposed his waveconception of tides, but he knew about Whewell's successful program of tide observationand prediction.
As befits the author of The history of inductive sciences, Whewell'sapproach was inductive:I believe the instances are comparatively few in the history of philosophy, in whichthe general laws of the phenomena have been pointed out by the theory before theyhad been gathered by observation. The law of the tides, thus empirically obtained,may be used either as tests of the extant theories, or as suggestions for the improvement of those portions of mathematical hydraulics on which the true theory mustdepend. ·Like a Ptolemean astronomer, Whewell tried to fit the results of measurements into simpleharmonic formulas. Such was the basis for his reduction of tides in a given port.62In order to connect tides observed in different locations, Whewell followed Young'ssuggestion to draw 'cotidal maps' that represented lines of high water at successive hourson a day of full Moon (see Fig.
2.19). According to Young, 'these lines would indicate . . .the directions of the great waves, to which that of the progress of the tides in successionmust be perpendicular.' Although Whewell did not refer to Young and doubted thepossibility of theoretically deriving these lines, he allowed himself to identify the cotidalline at a given time with 'the summit or ridge of the tide-wave at that time.' He describedthe global. forced wave that followed the motion of the Moon and the Sun, as well as thefreely-propagating waves in smaller open seas, basins, channels, and rivers.
These wavesprogressed with the depth-dependent velocity that Lagrange had derived and the Weberbrothers had verified.632.3.3Airy's wave theory of tidesThe Astronomer Royal, George Biddell Airy, was also unaware of Young's wave theorywhen he wrote the article 'Tides and waves' for an 1845 volume of the Encyclopaedia62Whewell [1 834] p. 19.6 1 Young [1813], [1823].63Whewell [1833] pp.
148 (cotidal maps), 149 (tidewave), 212 (Lagrangian velocity); Young [1823] p. 293.64WORLDS OF FLOWFig. 2.19.A portion of Whewell's first cotidal map ([1833] plate).metropolitana. However, he was familiar with Whewell's and Russell's tide studies. He didnot cite Russell as a stimulus for his own theory, presumably because he had a pooropinion of Russell's theories in general. After noting the 'great value' of Russell's experiments, he warned the reader 'against attaching any importance to the theoretical expressions which are mingled with them in the original account.' 6464Airy[1 845] p. 350.WATER WAVES65As an eminent representative of the new generation of British natural philosophers whohad thoroughly assimilated the methods of French mathematical physics, Airy was notonly able to condemn Russell's loose theorizing but also to precisely assess the merits ofLaplace's formidable calculations. While he found obscurities and even mistakes in thistheory, his overall judgment was admiring: 65We must allow [Laplace's theory] to be one ofthe most splendid works of the greatestmathematician of the past age.
To appreciate this, the reader must consider, first, theboldness of the writer who, having a clear understanding of the gross imperfection ofthe methods of his predecessors, had also the courage deliberately to take up theproblem on grounds fundamentally correct . . . ; secondly, the general difficulty oftreating the motions of fluids; thirdly, the peculiar difficulty of treating the motionswhen the fluid covers an area which is not plane but convex; and, fourthly, thesagacity of perceiving that it was necessary to consider the Earth as a revolvingbody, and the skill of correctly introducing this consideration.
This last point alone,in our opinion, gives the greater claim for reputation than the boasted explanation ofthe long inequality of Jupiter and Saturn.Airy's main reason for abandoning Laplace's theory was not its mathematical difficultynor any fundamental incorrectness in its assumptions, but the practical impossibility ofsolving the tidal equations for the actual form of the bottom of the sea. His own approachwas based on the properties of canal waves. These directly concerned the behavior of rivertides. They also shed light on oceanic tides, as far as an ocean could be replaced by a seriesof adjacent canals.
Accordingly, Airy began with a thorough analysis of wave propagationin a canal. Lagrange's theory was too restrictive since it only applied to small, long waves.Cauchy's and Poisson's theories were even less relevant, since they supposed a mode ofproduction of the waves that was never encountered in tide theory.66Airy's analysis was based on the Lagrangian picture of fluid motion, as was Laplace'stheory of 1776. Denote by X and Y the coordinates of the fluid particles when the fluid is atrest, and X + g and Y + 7J their coordinates when the fluid is in motion.
As before, the Xaxis lies along the bottom of the canal, and the Y-axis is vertical. Like Laplace, thoughwith more elementary methods, Airy proved that the harmonic expressionsg = 8 Cosh k Y coskX coswt,7J = 8 sinh k Y sin kX coswt,(2.49)7J = 8 Sinh k Y cos kXsinwt(2.50)7J = 8 sinh k Y sin (kX - wt),(2.51)with w2 = gk tanh kh, satisfied the continuity equation, the equations of motion, and theboundary conditions as long as the motion was small. Unlike Laplace, he combined thissolution with the other solutiong = -8 cosh k Y sinkX sinwt,to get the solutiong = 8 cosh k Y cos (kX - wt),65Airy [1 845] p. 279.66Jbid. pp.
280-1 .66WORLDS OF FLOWwhich propagates with the velocity�c = wjk such thatc2 =tanhkh.(2.52)In this state of motion, the fluid particles perform elliptical oscillations that tend tocircular ones for infinite depth. As Airy noted, this result agrees with the earlier observations of suspended solid particles made by the Webers and by Russell.2.3.467From river tides to ocean tidesIn the case of tides, the wavelength is much larger than the depth. Then the previousequations imply that the horizontal motion is sensibly the same from the surface to thebottom, and the vertical motion is comparatively very small.68 Airy assumed this propertyto hold even in the case of river tides, for which the elevation of the water was no longernegligible compared to the depth. This enabled him to reach more exact, nonlinearequations of motion.
He reasoned as follows.The volume of the vertical slice of fluid lying between the planesX and X + 8X is h8X in(h + o")[X + 8X + 4'(X + 8X) - X - 4'(X)] in the disturbedcondition (CTdenotes the elevation of the surface above its original height h). Therefore, thethe undisturbed condition, andcontinuity of the fluid implies(2.53)The pressure on each side of the slice varies hydrostatically, since the vertical accelerationis neglected. Therefore, its longitudinal gradient only depends on the slope of the surface:fJPfJu(2.54)ax = pg 8X "Newton's second law applied to the fluid slice then givesph 8XEliminating&gfJt2f)p00"= - f) 8X(h + CT) = -pg 8X(h + u) ax .X(2.55)CT through the continuity equation, Airy fmally obtained(2.56)4'0 = e cos (wt - kX) in the4'1 by integrating the equationAiry solved this equation perturbatively.
The motion beinglowest approximation, he obtained the next approximation(2.57)67Airy [1845] pp. 290 (solution), 344 (Weber), 347 (Rnssell).68Ibid. p. 294.67WATER WAVESwith the condition that for X = 0 the oscillation should still be s cos wt. This gives, for thecorresponding elevation of the surface,O"t. (. (wt - kx) + 3 a2 kxsm= -a sm2 wt - kx) ,4h(2.58)with a = khs and x = X + g1 • 69This solution represents the evolution of a tidal wave as it propagates from the mouthx = 0 along a flat, prismatic river without intrinsic current.70 As is seen from Fig. 2.20, thefront of the waves becomes steeper than the rear.
This explains why the rise of the watertakes more time than its descent at a station far from the mouth. Airy further derived thevelocity of the wave crests (for which dut /dx = 0) at the same approximation:( H)c = hk l +(2.59).He found this formula to be compatible with the velocity measurement of high waves bythe Webers and Russell, despite Russell's claim that the velocity of a solitary wave ofheight u obeyed the formula c = )g(h + u) .71In the case of oceanic tides, the height of the waves is negligible compared to the depth,so that the continuity equation (2.53) and the equation of motion (2.55) can be linearized.However, the direct action of the Moon and the Sun is no longer negligible.
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