236-237] VARIABLE DEPTH. 427 



The asymmetrical oscillations are given by 



(f)-\-i\^ = iA {cosh k (y + iz}~ cos k(y+iz)} ............... (vi), 



or = - A (sinh ky sin kz + sin ky sinh kz\ j 



ty A (cosh ky cos kz - cos ky cosh kz) f 



The stream-line ^=0 consists, as before, of the lines y=z\ and the surface- 

 condition (iii) gives 



cr 2 (sinh ky sin kh + sin ky sinh M) = gr (sinh ky cos M + sin ky cosh M). 

 This requires o- 2 sin kh=gkcoskh, \ ..... 



<r 2 sinh khgk cosh M / " ' 



whence tanh JrA=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 



coswcoshw = l .............................. (x)*, 



where m = %kh. 



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 =S, and making k infinitesimal, the 

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



, = J -.o-6 ( '' 



whilst from (viii) 



1 

 J 



The corresponding form of the free surface is 



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 



* Cf. Lord Kayleigh, Theory of Sound, t. i., Art. 170, where the numerical 

 solution of the equation is fully discussed. 



