228-229] CASE OF FINITE DEPTH. 405 



As in Art. 226, the pressure is greatest over the troughs, and 

 least over the crests, of the waves, or vice versd, according as the 

 wave-length is greater or less than that corresponding to the velo- 

 city c, in accordance with general theory. 



The generalization of (5) by Fourier's method gives, with the help and in 

 the notation of Art. 227 (15) and (17), 



. J.L ;. 



dk... ...(i\ 



as the representation of the effect of a pressure of integral amount pQ applied 

 to a narrow band of the surface at the origin. This may be written 



cos 



ucothu-gh/c 2 

 Now consider the complex integral 



/i 



where =u + iv. The function under the integral sign has a singular point at 

 f = + i oo , according as x is positive or negative, and the remaining singular 

 points are given by the roots of 



(tanh )/=<%/* ................................. (iv). 



Since (i) is an even function of x, it will be sufficient to take the case of x 

 positive. 



Let us first suppose that c 2 > gh. The roots of (iv) are then all pure 

 imaginaries; viz. they are of the form ip, where /3 is a root of 



The smallest positive root of this lies between and ^77- , and the higher roots 

 approximate with increasing closeness to the values (s + %)7r, where s is integral. 

 We will denote these roots in order by /3 , (3 lt j3 2 >---- Let us now take the 

 integral (iii) round the contour made up of the axis of u, an infinite semicircle 

 on the positive side of this axis, and a series of small circles surrounding the 

 singular points f=fc/3 , ifa ^3 2 v The part due to the infinite semicircle 

 obviously vanishes. Again, it is known that if a be a simple root of /()=0 

 the value of the integral 



taken in the positive direction round a small circle enclosing the point =a is 

 equal to 



* Forsyth, Theory of Functions, Art. 24. 



