294 TIDAL WAVES. [CHAP. VIII 



182. In the case of simple harmonic motion, assuming that 

 ?; x cos (at + e), the equation (4) of the preceding Art. becomes 



(1). 



Some particular cases of considerable interest can be solved 

 with ease. 



1. For example, let us take the case of a canal whose breadth varies as 

 the distance from the end #=0, the depth being uniform ; and let us suppose 

 that at its mouth (x = a) the canal communicates with an open sea in which 

 a tidal oscillation 



77 = acos(&amp;lt;r + e) ................................. (i), 



is maintained. Putting h = const., 6oc#, in (1), we find 



provided k 2 = o- 2 /gh .................................... (iii). 



The solution of (ii) which is finite when #=0 is 



or, in the notation of Bessel s Functions, 



t=AJ Q (kx) .................................... (v). 



Hence the solution of our problem is evidently 



The curve yJ Q (x} is figured on p. 306; it indicates how the amplitude of 

 the forced oscillation increases, whilst the wave length is practically constant, 

 as we proceed up the canal from the mouth. 



2. Let us suppose that the variation is in the depth only, and that this 

 increases uniformly from the end #=0 of the canal, to the mouth, the remain 

 ing circumstances being as before. If, in (1), we put h=h ( p/a t K = &amp;lt;r 2 a/gh , 

 we obtain 



This solution of this which is finite for #=0 is 



KX K% 2 



or i=*AJ Q (2 K a) .................................... (ix), 



whence finally, restoring the time-factor and determining the constant, 



