436 SURFACE WAVES. [CHAP. IX 



sides. The speed, <r = (#&), is exactly what we should expect from the general 

 theory of waves on relatively deep water. 



If in this case we transfer the origin to one edge of the water-surface, 

 writing z + h for z, and y ^Zh for y, and then make h infinite, we get the case 

 of a system of waves travelling parallel to a shore which slopes downwards at 

 an angle of 30 to the horizon. The result is 



<t> = H{e kz +e-* k( ' JSv+z) -3e-* k( ' J3y - z) }coskx.cos(<rt+') ....... (xxx), 



where c= (#/)*. This admits of immediate verification. At a distance of a 

 wave-length or so from the shore, the value of <, near the surface, reduces to 



<^ = H^ Z GQ&kx. cos(<r + f) ........................ (xxxi), 



practically, as in Art. 217*. Near the edge the elevation changes sign, there 

 being a longitudinal node for which 



(xxxii), 



The second of the two roots (xxix) gives a system of edge- waves, the results 

 being equivalent to those obtained by making =30 in Stokes' formula. 



Oscillations of a Spherical Mass of Liquid. 



241. The theory of the gravitational oscillations of a mass of 

 liquid about the spherical form has been given by Lord Kelvin f. 



Taking the origin at the centre, and denoting the radius vector 

 at any point of the surface by a + f, where a is the radius in the 

 undisturbed state, we assume 



where f n is a surface-harmonic of integral order n. The equation 

 of continuity V 2 < = is satisfied by 



where S n is a surface-harmonic, and the kinematical condition 



* The result contained in (xxx) does not appear to have been hitherto noticed. 



t Sir W. Thomson, "Dynamical Problems regarding Elastic Spheroidal Shells 

 and Spheroids of Incompressible Liquid," Phil. Trans., 1863; Math, and Phys. 

 Papers, t. iii., p. 384. 



