217-218] PROGRESSIVE WAVES. 375 



in the negative or positive direction of x, respectively, with 

 the velocity c given by 



y ..................... (4), 



k \k J 



where the value of a has been substituted from Art. 217 (5). In 

 terms of the wave-length (X) we have 



When the wave-length is anything less than double the depth, 

 we have tanh kh=l, sensibly, and therefore 



=() ..................... &amp;lt;&amp;gt; 



On the other hand when X is moderately large compared with h 

 we have tanh kh = kh, nearly, so that the velocity is independent 

 of the wave-length, being given by 



(7), 



as in Art. 167. This formula is here obtained on the assumption 

 that the wave-profile is a curve of sines, but Fourier s theorem 

 shews that this restriction is now unnecessary. 



It appears, on tracing the curve y = (tanh x)jx } or from the 

 numerical table to be given presently, that for a given depth h 

 the wave-velocity increases constantly with the wave-length, 

 from zero to the asymptotic value (7). 



Let us now fix our attention, for definiteness, on a train of 

 simple-harmonic waves travelling in the positive direction, i.e. we 

 take the lower sign in (1) and (3). It appears, on comparison 

 with Art. 217 (7), that the value of % is deduced by putting 

 e = ^TT, and subtracting TT from the value of &#f, and that of rj. 2 

 by putting e = 0, simply. This proves a statement made above as 

 to the relation between the component systems of standing waves, 



* Green, &quot;Note on the Motion of Waves in Canals,&quot; Camb. Trans., t. vii. (1839); 

 Mathematical Papers, p. 279. 



t This is of course merely equivalent to a change of the origin from which x is 

 measured, 



