The pressure of water waves upon a fixed obstacle 411 
point to be determined by the horizontal fluid velocity in the primary 
motion and by the gradient of the surface at the point. We then replace the 
obstacle by a plane distribution of sources over the vertical section by the 
za-plane. The primary fluid motion in the present case is that of the plane 
waves. Thus, if (€, 0, —f) is a point on the vertical section, and if dy/0z 
is the gradient at the corresponding point on the contour, the required 
distribution of sources over the vertical section is of strength per unit area 
given by 
= ixgh oy etlottKe)—Kf (3) 
270 0& 
Consider now a point source m cos ct in the liquid at the point (0, 0, —f). 
The velocity potential was obtained by Lamb (1922) and we use his result 
with a slight change of notation. The surface elevation ¢ is given by 
g6 = 0p/0et with z = 0; we have 
Ziom __| 
_giat 
ST GEE 
— inxetH® (kr) 
Pe [2 
ry sin fv + K cos fu 
F F € 
0 KER 
—vr cosh u du av} ; (4) 
where 7? = x?+y?, kK = 0?/g, H?) = J,—7Y), and the real part of the expres- 
sion is to be taken. 
Let there be a vertical line source extending from the origin downwards, 
the source strength per unit length at depth f being me~*’. We substitute 
this value for m in (4) and integrate with respect to f from 0 to 00. For the 
last term in (4) this integration gives 
De [2 (2 eur cosh uw OK (°K (vr 
x e dudv = = Mol dy 
> By 2 29 
TJoJo K2+v? CHO Kae 
2 co e-kt df (5) 
—_— Ae TT lo o 
I aE 
Hence the terms in (4) which represent the local oscillations disappear 
from the integrated result for this particular vertical source distribution; 
and we obtain the simple result 
ace elt) (Kr) = el(at—xr cosh u) du, 
g i J 
aN 
mom Ziom [ ° 
Le LG | (6) 
0 
representing circular waves diverging from the origin. Returning to (3) 
472 
