570 Prof. T. H. Havelock 
The solution can be obtained by various methods ; for 
example, by combining suitable elementary solutions of (1) 
and (4). The usual solution for free progressive waves is 
found from 
eae aes 5 a 10 oo () 
There is also another elementary solution, 
pe ™(Kcosxz—Ky sin kz), . . « - (7) 
where « may have any real positive value. 
We can generalize these solutions by means of the 
following integral theorem, which may easily be verified : 
f= 
EP pele eee sin x2) a 
da 
Ke + Ko" 
fal 
MORAN IP (ah\e-Be dis 5 6 oo | @) 
0 
Here f(z) is given for all positive values of 2, and it should 
be remarked that the proof involves the Fourier integral 
theorem, and that f (z) is subject to suitable limitations. 
We may now write down a solution which satisfies the 
condition (5). It is clear that, on the forced vibrations so 
obtained, we may superpose any free oscillations for which 
0¢/Ox is zero over the plane x=; we shall choose the 
latter so that the complete solution represents waves 
travelling outwards tor large positive values of a. This 
solution is given by F 
: ane 2 
b= 2e-** sin (ot — wut) F (we "erda + moos atx 
0 
K COS KZ — Kg SIN KZ) X 
( 
er tae x (kK COS Ka—Ky SIN KH) _ 
x ii \ T (@) ACE) e "de da. . (9) 
This gives a normal velocity f(z) cos ot over the plane 
a=0, and reduces to a positive wave for large positive values 
of a. The corresponding surface elevation is 
ans) 
= 2a koa 
b= Zoos (atx) | f (a)e-"o0 dx 
0 
as 20 A, ( \ f(a) KCOS KA Ko SINK, wot Ty. (10) 
7g Jo Jo 
Ke + Ko 
The first term of (10) is a plane progressive wave of the 
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