Encyclopædia Britannica, Ninth Edition/Tides/Chapter 8

VIII. Tidal Deformation of the Solid Earth.

§ 43. Elastic Tides.

Elastic tides.The tide-generating potential varies as the square of the distance from the earth's centre, and the corresponding forces act at every point throughout its mass. No solid matter possesses the property of absolute rigidity, and we must therefore admit the probable existence of tidal elastic deformation of the solid earth. The problem of finding the state of strain of an elastic sphere under given stresses was first solved by Lamé;^[1] he made, however, but few physical deductions from his solution. An independent solution was found by Sir W. Thomson,^[2] who drew some interesting conclusions concerning the earth. His problem, in as far as it is now material, is as follows. Let a sphere, of radius a and density w, be made of elastic material whose bulk and rigidity moduli are k and n, and let it be subjected to forces due to a potential wr*S 2 per unit volume, where S 2 is a surface spherical harmonic of the second order. Then it is required to find the strain of the sphere. We refer the reader to the original sources for the methods of solution applicable to spherical shells and to solid spheres. In order to write Thomson's solution we put r, X, I for radius vector, latitude, and longitude, and p, /*, v for the corresponding displacements. Then the solution is as follows:

(96). For either tidal or rotational stresses

In the case of tides r=|m/c 3, m and c being the moon's mass and distance, and in the case of rotation T = - ^w 2, w being the angular velocity about the polar axis. The equation to the surface is found by putting r=a + p, where in the expression for p we put r=a. Hence from (96) the form of surface is given by

In most solids the bulk modulus is considerably larger than the rigidity modulus, and in this discussion it is sufficient to neglect n compared with k. With this approximation, the ellipticity e of the surface becomes

Now suppose the sphere to be endued with the power of gravitation, and write

g (99)>

where g is gravity at the surface of the globe. Then, if there were no elasticity, the ellipticity would be given by e=r/Q, and without gravitation by e= r/r. And it may be proved in several ways that, gravity and elasticity co-operating,

e = =-. L- ....(100). r + g Q 1+t/g

If n be the rigidity of steel, and if the globe have the size and mean density of the earth, r/g=2, and with the rigidity of glass r/g=$. Hence the ellipticity of an earth of steel under tide-generating force would be ^ of that of a fluid earth, and the similar fraction for glass would be f. If an ocean be superposed on the globe, then, if the globe rises and falls with the tide as though it were fluid, there will obviously be no tide visible to an observer carried up and down with the solid; and with any degree of rigidity the visible tide will be the excess of the fluid tide above the solid tide. Hence on an earth with rigidity of steel the oceanic tides would be reduced to f, and with rigidity of glass to of the tides on a rigid earth.

§ 44. Rigidity of the Earth.

Rigidity of the earth.Although the computation of oceanic tides is as yet impossible, it cannot be admitted that perfect rigidity in the earth would augment the tides in the proportion of 5 to 2, although they might perhaps be augmented in the proportion 4 to 3. Thus Thomson concludes that the earth's mass must have an effective rigidity at least as great as that of steel. If it were true, as was held until recently, that the earth is a fluid ball coated with a crust, that crust must be of fabulous rigidity to resist the tidal surgings of subjacent fluid. Hence we are led to the conclusion that far the larger portion of the earth's mass, if not all of it, is a solid of great rigidity. Up to the present time the argument by which the tides of long period were proved to have approximately their equilibrium height has generally been accepted without much doubt, but we have (§ 17) shown good cause for rejecting Laplace's argument, at Attempt least for a fortnightly tide. It appeared formerly that, from numerito evalue cal data as to the heights of the tides of long period, we should rigidity be able to compute the actual effective rigidity of the earth's mass. by tides But from 18 we see that, although these tides remain incalculable, of long yet with such oceans as ours the tides of long period must conform period, much more nearly to the equilibrium laws than do the tides of short period. Thus a comparison of the observed heights of the tides of long period with the equilibrium law still remains of interest, although the evaluation of the earth's rigidity appears with present data unattainable. Acting on the old belief, Mr G. H. Darwin has compared the lunar fortnightly and monthly tides, as observed for thirty-three years at various Indian and European ports, with the equilibrium theory, and has found that the tideheights were about two-thirds of the equilibrium height.^[3] From this the conclusion was drawn that the effective rigidity of the earth was as great as that of steel. Whilst, then, this precise comparison with the rigidity of steel falls to the ground, the investigation remains as an important confirmation of Thomson's conclusion as to the great effective rigidity of the earth. When extensive and accurate knowledge of the tides has been attained, the attempted evaluation of the rigidity may conceivably be possible, because there is a minute tide with a period of 18·6 years (§ 23, schedule [A, iii.]) of which Laplace's argument must hold good. The 19-yearly tide.Great accuracy will, however, be necessary, because the height of the tide at the equator only amounts to one-third of an inch, and a preliminary inquiry seems to show that there are other relatively considerable variations of sea-level arising from unexplained causes.^[4]

Weight of continents.Sir W. Thomson's solution of the strain of an elastic sphere has been also used to determine what degree of strength the materials of the earth must have in order that the great continental plateaus and mountains may not sink in.^[5] In another investigation it has been shown that local elastic yielding on the coast-lines of conti nents may produce an augmentation of apparent tide in certain places on account of the flexure of the upper strata, when a great weight of water is added and subtracted from the adjacent oceanic area at high and low tide.^[6] There is reason to believe that such flexure has actually been observed by a delicate form of level on the coast of the Bay of Biscay.^[7]

§ 45. Viscous and Elastico-Viscous Tides.

It might be supposed that the earth is composed of a viscous Viscous fluid of great stiffness, or that it possesses an elasticity which and breaks down under continued stress. Both these hypotheses have elasticobeen considered, and the results are confirmatory of the conclusion viscous that the earth is made of very stiff material.^[8] These problems tides. appear to have been worthy of attack, although the existence of measurable oceanic tides of long period negatives the adoption of the hypothesis of true viscosity, at least under stresses comparable with tide-generating forces.

If a sphere of radius a, density w, viscosity modulus v, be under the action of forces due to a potential per unit volume ivr-S 2 cos nt, so that n is the speed of the tide, .the solution of the problem shows that the tide of the sphere is expressed by

Cf - cos e cos (nt - e) (101), r 19i;,3 a bwa 1 a T a where

Thus the tides of the viscous globe are to the equilibrium tides of a fluid globe as cos e to unity, and there is a retardation e/n of the time of high tide after the passage of the tide-generator over the meridian. Further, by arguments similar to that applied in the case of elastic tides, it is found that oceanic tides are reduced by the yielding in the proportion of sin e to unity, and that there is an apparent acceleration of the time of high water by ( ir - e)/,. It appears by numerical calculation that, in order that the oceanic semi-diurnal tide may have a value equal to two-thirds of the full amount on a rigid globe, the stiffness of the globe must be about twenty thousand times as great as that of pitcli at freezing temperature, when it is hard and brittle. We must here pass by the results of the hypothesis of an elasticity degrading under the influence of continued stress.

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1. Théorie Math. de l'Élasticitë, 1866, p. 213.
2. Thomson and Tait, Nat. Phil., §§732-737 and 833-842, or Phil. Trans., pt. ii., 1863, p. 583.
3. Thomson and Tait, Nat. Phil., vol. i. pt. ii., 1883, § 847 sq.
4. Darwin, "On 19-yearly Tide at Karachi," in Brit. Assoc. Report, 1886.
5. G. H. Darwin, Phil. Trans., pt. i., 1882, p. 187, with correction, Proc. Roy. Soe., 1885.
6. Id., Brit. Assoc. Rep., 1882, or Phil. Mag., 1882.
7. 5 D Abbadie, Annales Soc. Sc. de Bruxelles, 1881, or quotation by Darwin, loc. cit.
8. 6 G. H. Darwin, Phil. Trans., pt. i., 1879, p. 1; see also Lamb, "On the Oscillations of a Viscous Spheroid," Proc. Lond. Math. Soc., Nov. 1881, p. 51.