1908. | Groups of Waves in Dispersive Media, ete. 426 
not a constant ratio in this case; we had in §11 the expressions for the two 
velocities as functions of «. It can be shown that in certain cases the 
equations for cw lead to four possible roots, giving four wave-branches through 
the point. 
§ 15. Point Impulse moving on Water of Finite Depth. 
With the same problem we have now, if the water is of depth h, 
1 
a 
5) 
WV = (4 tanh Kh) 
K 
sinh 2ah (We) 
p— (2 tanh eh) (1+ : isl ): 
K 
If we write 
U=4(~41)V, 
nm varies between 0 and 1, being dependent upon the value of «. We use the 
notation 
_ gh _ tanh ch _  2kh 110 
rcs ae eet akG ~ sinh 2xh’ oD) 
Then m and n are monotonic functions of « with the following limiting 
values : 
i—HOn ih = is Te = 1. 
Ki— cons i= 0? n= 
The two equations for cu and « become 
Gyan, 27)2)8 Fi 
(oie ta cose teey = 4(pm)? (14+), (111) 
MSGS @ = (pm). (112) 
(w?—2cusm cos a+ cu?) 
From these we obtain 
2, — {1—tom(+n)}?. 
00s" * = 1 Tom (1+) (3—2)’ ur) 
cu = F0=m) [(3—n) cos «+ {(8—n)? cos? a—8 (1—n) }}]. (114) 
Combining the last two we have the values of cw as 
me = (1—4pm (142) (8—2)}3, (115) 
or cu = w/{1—jpm(1+2) (8—n)}. (116) 
We have two cases to consider according as p > or <1. 
(a) c<a/(gh); p>1—From (114) we see that the equal values of cw, 
29 
