314 TIDAL WAVES. [CHAP. VIII 



Thus the gravest symmetrical mode 0'=2) has a nodal circle of radius '707a ; 

 and its frequency is determined by o- 2 = 8ffk Q /a 2 . 



Of the unsymmetrical modes, the slowest, for any given value of s, is that 

 for which w = s-f 2, in which case we have 



^ 

 = A 8 cos s6 cos (art + e), 



the value of a- being given by 



The slowest mode of all is that for which * = 1, n=3 ; the free surface is 

 then always plane. It is found on comparison with Art. 187 (16) that the 

 frequency is "768 of that of the corresponding mode in a circular basin of 

 uniform depth A , and of the same radius. 



As in Art. 188 we could at once write down the formula for the tidal 

 motion produced by a uniform horizontal periodic force ; or, more generally, 

 for the case when the disturbing potential is of the type 



Q oc r* cos s6 cos (<rt + e). 



190. We proceed to consider the case of a spherical sheet, or 

 ocean, of water, covering a solid globe. We will suppose for the 

 present that the globe does not rotate, and we will also in the first 

 instance neglect the mutual attraction of the particles of the water. 

 The mathematical conditions of the question are then exactly the 

 same as in the acoustical problem of the vibrations of spherical 

 layers of air*. 



Let a be the radius of the globe, h the depth of the fluid ; we 

 assume that h is small compared with a, but not (as yet) that it is 

 uniform. The position of any point on the sheet being specified 

 by the angular coordinates 0, co, let u be the component velocity of 

 the fluid at this point along the meridian, in the direction of 6 

 increasing, and v the component along the parallel of latitude, in 

 the direction of w increasing. Also let f denote the elevation of 

 the free surface above the undisturbed level. The horizontal 

 motion being assumed, for the reasons explained in Art. 169, to be 

 the same at all points in a vertical line, the condition of con- 

 tinuity is 



-j-x (uha sin 08o>) 80 + -7- (vhaSd) So> = a sin 08o> . a80 . -77 , 

 where the left-hand side measures the flux out of the columnar 



* Discussed in Lord Eayleigh's Theory of Sound, c. xviu. 



