Forced Surface- Waves on Water. 574 
The corresponding elementary solutions are 
p=elst—Ko coshx(z—h),. . . . (26) 
where Xp is the real positive root of 
Ghayealn OCB 5 56 6 8 o (0) 
and 
OH OX-“=] O95 2 (C=) 6 > 6 ce (4) 
where « is any real positive root of 
getankhto?=0. . ... . (29) 
This equation has an infinite sequence of real roots, 
together with an imaginary root ix. We assume then the 
possibility of expanding a function f(z) in the range 
O<ezch, in the form 
f (2)=A cosh e(z—h)+=Bcosx(z—h), . (30) 
where the summation extends over the real positive roots 
of the eqnation (29). We find that the coefficients are 
given by 
4k 
h 
ee a ee ah 
A= 2Kyh-+ sinh 2xoh t) J (a) cosh Ky(a—h) da, 
4k BA 
B= spate |, fo «(a—h) da. } - (31) 
If at z=0 we are given 
2 Si ()COG 5s o 6 » (x) 
then the velocity potential for positive values of x, such 
that the motion at a distance is a plane progressive wave, 
is given by 
g=Axy* cosh x,(z—h) sin (ot ~ x2) 
+ ZBe'e-** cosx(z—h) cosat. . (33) 
Suppose, for instance, that one end of a long tank is 
made to execute simple harmonic vibrations of small 
amplitude a, then we have f(z)=oa. The values of A and B 
follow from (31), and from (33) we deduce the surface 
elevation in this case : 
2o7%a sinh 2xoh 
= gko(2Koh + sinh 2h) IS Oli) 
2o°ae-*"* sin 2eh 
— sin o> (OREO) , 
(34) 
309 
