From the PBILosoPHICAL MaGazinF, vol. viii. October 1929. 
Forced Surface- Waves on Water. 
By T. H. Havetocr, F.R.S. 
Ih, ee following notes deal with some problems of forced 
waves on the surface of water, the waves being 
forced in that the normal fluid velocity has an assigned 
value at every point of a given vertical surface ; the problems 
treated here are the elementary cases when the given 
surface is an infinite plane or a circular cylinder. The 
motion of the water surface consists in general of travelling 
waves together with a local disturbance, and the type of 
solution is one which may have possible application to the 
waves produced in water by the small oscillations of a solid 
body. 
2. Consider first deep water, and take the origin in the 
free surface with Ow horizontally and Oz vertically down- 
wards. The velocity potential satisfies 
OH) ID gooenes @) 
on? 02 
Neglecting the square of the fluid velocity at the free 
surface, and omitting the effect of capillarity, the condition 
at the free surface is 
op dh _ 
aay hs 8 8 5 2 0 
and the surface elevation € is given by 
= ee 3 
BAe) Rate tenant) 
For simple harmonic motion we assume a time-factor 
et, and (2) gives 
Kop + OY =0, at z=0, ies eA) 
with «9=07/g. 
Suppose now that we are also given 
= 22 26) oe, ch oO; ee aC) 
where f(z) is given for all positive values of z; and we 
require a solution of (1), (4) and (8) suitable for positive 
values of w. 
