Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 19
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. . identical with the great primary wave of translation;its velocity diminishes and increases with the depth of the fluid, and appears toapproximate closely to the velocity due to half the depth of the fluid . . . -The tideappears to be a compound wave, one elementary wave bringing the first part of theflood tide, another the high water, and so on: these move with different velocitiesaccording to the depth. On approaching shallow shores the anterior tide waves movemore slowly in the shallow water, while the posterior waves, moving more rapidly,diminish the distance between two successive waves. The tide wave becomes thusdislocated, its anterior surface rising more rapidly, and its posterior surface descending more slowly than in deep water.-A tidal bore is formed when the water is soshallow at low water that the first waves of flood tide move with a velocity so muchless than that due to the succeeding part of the tidal wave, as to be overtaken by thesubsequent waves, or wherever the tide rises so rapidly, and the water on the shore orin the river is so shallow that the height of the first wave of the tide is greater than thedepth of the fluid at that place.Russell thus explained a few basic facts: that for river tides, the time of ebb is larger thanthe time of flood, with the difference increasing with the distance from the mouth of theriver; that tides can be very different in nearby locations, that they depend on the bottomof the sea or the form of channels and rivers, and that strong river tides are oftenaccompanied by a breaking surge or tidal bore (mascaret in French).
In his later watertank experiments, Russell verified that 'compound solitary waves' evolved during theirpropagation so that the front became steeper than the rear (see Fig. 2. 16, p. 58). 562.3.2From Newton to WhewellAlthough the idea that tides were a wave phenomenon was not as new as Russellsuggested, it departed from the then current approaches· to tide theory and prediction.The historical background of these approaches must first be recalled.
57In his Principia Newton gave the correct expression for the force that is responsible fortides, namely, the combined action of the Moon's and the Sun's attractions. His derivationof the resulting deformation of the surface of the oceans was only tentative and retrospectively erroneous. He seems to have adopted an equilibrium theory, with retardation due tofriction. According to the pure equilibrium theory that Colin MacLaurin, Leonhard Euler,and Daniel Bemoulli developed in their competition for the 1 740 prize oMhe FrenchAcademy, under every instantaneous configuration of the Moon and the Sun, the watersurface takes the form it would have if the corresponding forces were acting permanently.Retaining only the lunar action in a first approximation, the net force exerted by the Moonon oceanic water is the Newtonian gravitational force, which is proportional to the inverse"Russell[1837c] p.
426.Although Russell's identification of the tidal wave with a compound solitary wavemakes little sense from a modern point of view, his theory appears to be similar to Partiot's more correct theory,discussed later on p.56Russell [1 837c],82.[1838], [1 845].57The following account is based on Cartwright[1999].WATER WAVES,, 8Fig.
2.18.610Moon0MoonTides on an ocean of uniform depth as given the equilibrium theory (a); as inferred from observedtides (b).squared distance of the water from the Moon, minus the inertial force due to the acceleration of the Earth toward the Moon, which is proportional to the inverse squared distanceof the center of the Earth from the Moon.
Therefore, this net force is a maximum at thepoints closest to and furthest from the Moon. For a uniform ocean covering the wholeEarth, the resulting equilibrium surface (obtained by making the total potential of theterrestrial and lunar forces a constant) has the form indicated in Fig. 2.1 8(a). Unfortunately, observed tides more closely correspond to the form indicated in Fig. 2.1 8(b).For this reason, in 1776 Laplace proposed a dynamic theory of tides.
Assuming that thehorizontal velocity of the water was the same on a vertical line, and neglecting secondorder quantities, he obtained the equations of motion (in modern notation)1 aau- - 2fiv cos e = - - - (g? - V - o U)atR aeav1a- (g? - V - o V)- + 2fiu cos e = - -.R sm & aq,at'(2.46)where u and v are the velocity components along the meridians and the parallels, respectively, 0 and 4> are the colatitude and the longitude, respectively, n is the angular velocity ofthe Earth, R is the radius of the Earth, ? is the elevation of the water surface, V is thecombined gravitational potential from the Moon and the Sun, and o V is the gravitationalself-potential of the water.
The first terms on the left-hand side of these equationscorrespond to the acceleration of the water particles, and the second to the Coriolisforce (not yet named so, of course). The right-hand side corresponds to the sum of pressureforces (depending on the elevation of the sll1face) and gravitational forces. These equations are to be solved in combination with the continuity equation:0 (uh sin 0) + � (vh) + R sin & �� = 0,where h is the original depth of the water.
5858Laplace [1775/76].(2.47)62WORLDS OF FLOWLaplace decomposed the potential U from the Moon and Sun into three terms that hadmonthly, diurnal, and semi-diurnal variations, and then solved his equations throughperturbative methods in the analytically simple case for which the depth h varies as thesine-squared of the latitude. As he himself realized, this assumption could not pass for arealistic representation of the oceans. In the last section of his memoir, he switched to asemi-empirical method in which the elevation of the water in one harbor was representedas a sum of sine functions with the frequencies of the perturbing forces.
In modern terms,we would say that he understood that the forced oscillations of the water surface necessarily had the same spectrum as the perturbing forces, owing to the linearity of the basicequations. 59Laplace's memoir looked and still looks forbiddingly complex, not only because of theidiosyncratic notation and the elliptic style, but also because most of the developmentswere purely algebraic. Physical discussion was confmed to the first assumptions and to thefmal results, whereas a modern tide-theorist would anticipate and comment on theintermediate algebraic steps by appealing to general notions of forced oscillations andwave propagation. That Laplace's equations in fact describe a wave motion modified bythe Coriolis force is easily seen by combining them to get, for n = 0 and constant h,{j2� - gM� = -M(U + 8U),ot2(2.48)where 11 is the two-dimensional Laplacian.
Although Laplace must have recognizedd'Alembert's equation of vibrating strings, he did not exploit this analogy in his theoryof tides. Instead, he appended to this theory the water-wave calculations with which ourstory began.Laplace's theory was alien to contemporary British physics, which remained dependenton older Newtonian methods and tended to ignore the newer French mathematicalphysics. In 1 8 13, the founder of the wave theory of light, Thomas Young, judged thatthe theory of tides was too practically important to be treated with Laplace's abstrusemethods.
Instead of the learned calculus of partial differentials, he offered a simpleanalogy between the ocean and a pendulum:The oscillation ofthe sea and oflakes, constituting the tides, are subject to laws exactlysimilar to those of pendulums capable of performing vibrations in the same time, andsuspended from points which are subjected to compound regular vibrations, of whichthe constituent periods are completed in half a lunar and half a solar day.uIn modern words, he assimilated tides with the forced oscillations of harmonic oscillatorssubjected to the superposition of two periodic forces.60In order to justify this analogy (perhaps suggested by Laplace's equations), Young firstshowed that, in a canal of constant depth h, long waves of small amplitude were propagated with the Lagrangian velocity c = Vifi.
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