316 TIDAL WAVES. [CHAP. VIII 



It was pointed out in Art. 87 that S n will not be finite over 

 the whole sphere unless n be integral. Hence, for an ocean 

 covering the whole globe, the form of the free surface at any 

 instant is, in any fundamental mode, that of a ' harmonic spheroid ' 



r=a + h + S n cos((rt + ) (4), 



and the speed of the oscillation is given by 



the value of n being integral. 



The characters of the various normal modes are best gathered 

 from a study of the nodal lines (S n = 0) of the free surface. Thus, 

 it is shewn in treatises on Spherical Harmonics* that the zonal 

 harmonic P n (//,) vanishes for n real and distinct values of p lying 

 between + 1, so that in this case we have n nodal circles of 

 latitude. When n is odd one of these coincides with the equator. 

 In the case of the tesseral harmonic 



n _ 2 ^ d s P n (A*-) cosj 

 d/ji s sinj 



the second factor vanishes for n s values of //,, and the trigono- 

 metrical factor for 2s equidistant values of o>. The nodal lines 

 therefore consist of n s parallels of latitude and 2s meridians. 

 Similarly the sectorial harmonic 



has as nodal lines 2n meridians. 



These are, however, merely special cases, for since there are 

 2n + 1 independent surface-harmonics of any integral order n, 

 and since the frequency, determined by (5), is the same for each 

 of these, there is a corresponding degree of indeterminateness in 

 the normal modes, and in the configuration of the nodal lines -f. 



We can also, by superposition, build up various types of 

 progressive waves; e.g. taking a sectorial harmonic we get a 

 solution in which 



?oc(l - IJL*)*" cos (nt - <rt + e) (6); 



this gives a series of meridional ridges and furrows travelling 



* For references, see p. 117. 



f Some interesting varieties are figured in the plates to Maxwell's Electricity and 

 Magnetism, t. i. 



