Images in Some Problems of Surface Waves 276 
If, therefore, we wish to obtain the complete expression for this part of the 
surface elevation at a point in the range — a2/4f <a <0, we should have to 
integrate with respect to p between appropriate variable limits. We shall 
consider only points to the rear of this range, so that the limits for p are Qand a. 
This being understood, the distribution of doublets on the semicircle CC, 
contributes to the regular waves a part given by 
e) 202 Kof az 
= 8rK-ate— “of | e *+4f cos { (x =F are 
n 0 5 0 rep 
Putting p = 2f tan 40, this becomes 
n = 2rKpPrat ft e~ “4S +7/4F (A cos «oz — B sin Koo), (23) 
where 
ee [et* cos (6+ h sin 6 —& tan 46) d0, 
0 
B = | tm? sin (0+ h sin 6 —k tan 46) d0, 
0 
with h = xoa/4f and k = 2xof. 
5. In the applications to be made, h and & are positive, h is less than unity 
and is usually a small fraction. In these circumstances, the integrals may be 
evaluated by expansion in power series of h. It can be shown, after a little 
reduction, that we have 
ye ao hr A sy t) h” ; 
A= 22 Tyas B= 227 Mas; (24) 
where 
i [- cos (2ré — k tan ¢) dd 
0 
M, = f WO in (Op — [bein VTE. (25) 
0 
The quantities L and M may be evaluated in terms of known functions by a 
reduction formula. It can readily be shown that 
(7+ 1) Lys = KL,” — kL,’ + rL,, (26) 
the accents denoting differentiation with respect to k; or denoting this opera- 
tion by D, we have 
7! L, = (kD? — kD +r —1)(KD?— kD +7 —2) ... (KD?— AD) Lp. (27) 
The quantity M satisfies similar relations. 
273 
