236-237] VARIABLE DEPTH. 427 



The asymmetrical oscillations are given by 



or (f) = - A (sinh Jcy sin kz + sin ky sinh kz\ } 



\js = A (cosh Icy cos kz - cos ky cosh /#) J 

 The stream-line ^ = consists, as before, of the lines y = z; and the surface- 

 condition (iii) gives 



o- 2 (sinh ky sin kh + sin ky sinh kh} =gk (sinh % cos kh 4- sin y cosh kh}. 

 This requires a 2 sin M=^ cos kh, } , .... 



&amp;lt;r 2 smhA=^coshM f&quot; 

 whence tanhA = tanM ................................. (ix). 



The equations (v) and (ix) present themselves in the theory of the lateral 

 vibrations of a bar free at both ends ; viz. they are both included in the 

 equation 



cosmcoshm = l .............................. ( x )*&amp;gt; 



where m=2M. 



The root M = 0, of (ix), which is extraneous in the theory referred to, is 

 now important ; it corresponds in fact to the slowest mode of oscillation in 

 the present problem. Putting Ak 2 = B, and making k infinitesimal, the 

 formulae (vii) become, on restoring the time-factor, and taking the real parts, 



whilst from (viii) 



The corresponding form of the free surface is 



i r/7/^n 



(xiii). 



The surface in this mode is therefore always plane. 



The annexed figure shews the lines of motion (^ = const.) for a series 

 of equidistant values of \//-. 



* Of. Lord Bayleigh, Theory of Sound, t. i., Art. 170, where the numerical 

 solution of the equation is fully discussed. 



