Forced Surface- Waves on Water. 572 
then from (14) we have 
Neaodypor ss 5 6 5 5 « (lb) 
From (10) we find that the amplitude of the waves 
travelling out on either side of the log would be 
2 ine) 
pay en( e-(KotP)0 dar 
g d 
o*abp 
= e 
9(%o +p) 
A large value of p would correspond to a concentration 
of the outward flow round the lower edges of the log ; 
hence this estimate gives, asan upper limit for the amplitude, 
(GUIDO oo oo (ili) 
4, If, in the general problem of §2, the normal velocity 
at «=0 is a function of y as well as of z, the solution of the 
three-dimensional motion can be obtained by an additional 
Fourier synthesis. 
Assume first that $ is proportional to cos «'(y—{), then 
the potential equation is 
Kod 
peeve ge. 8 o(17) 
oe A, Sie Ke’? 6=0, Pee hoey te (19) 
The time entering as a simple harmonic factor, the 
boundary condition at z=0 is given by (4). 
We have now the following elementary solutions, omitting 
the factors in y and ¢: 
haem hoz inl?) for Ke! < key ; 
2 
pe Ko —z(e%— "0"! For Kk! > Ky 5 
a 
pre Were (Kcosxz—Kysin kz). . . (20) 
The theorem (8) may be generalized, with suitable 
limitations on the function f(y, z), to 
L(y, = 
2K, . a 2 
eee “aa” aa “£0, Ble" cose’ (y—8)ae 
= \ “dal ae” dp i f (a, B) x 
(« Cos KZ — Kp Sin Kz) X 
x (« cos Ka—K, Sin Ke) 
x 
Ke + Ko" 
cosx’(y—8)dk'. (21) 
307 
