294 Dr. T. H. Havelock. [Apr. 1, 
The method of evaluating these integrals approximately so as to give the 
regular wave-trains has been discussed in a previous paper and it is followed 
now in the case of finite depth.* We take, under certain limitations, the 
value of an integral such as 
y = | 6) sin {9} a 
to be the value of its principal group, viz., 
2a 3 7 
=4—=—5 —4 22a 
Yo { ares b (uo) cos {9 (uo)—47}, (224) 
where wp is such that g’ (w%) = 0. 
Now in the integrals in (22) we have to find successively two principal 
groups, first with regard to « and then in the variable ¢; and thus we may 
evaluate the amplitude factor in the resulting regular wave-trains. 
For water of depth 2 we may write 
JS(k) = v-V =v y/ (Zianh Kh). 
The group with respect to « gives a term proportional to 
, cos {tx2f’ (x) +4}, 
where « has the value given by 
S(e)+4f («) = —F. (23) 
From (224), this introduces into the amplitude a factor 
T/V/[t {27 («) +f" («)} I. (24) 
Further, the group with respect to ¢ occurs for 
S {bef (sO & f=O. 
Also we have in these circumstances 
CP en caren pea oe con a Cae Ae 
Bier} = 4 Ler +t} =F (-xf(o) 
= _& ifar Kf _ 1 (f+«f')? = (xf)? (25) 
2 yeaa fe t OF +f” ACH aa) 
Hence from (224), (24), and (25) the selection of the two groups adds to 
the amplitude a factor 1/«/(«), where 
AO=0S0= a/ (Ztanh Kh). 
* Havelock, ‘Roy. Soc. Proc.,’ A, vol. 81, p. 411, 1908. 
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