412 T. H. Havelock 
we see that the source distribution is made up of vertical line sources of this 
type, and we obtain for the complete surface elevation 
€ = thetttka) _ Lishe' H® (kr) ! e5 dk, (7) 
Tn (7), the first term represents the incident waves; further, r? = (~—)?+ y?, 
and the integration extends over the axial length of the form. It should be 
noted that this result is comparatively simple because we have taken the 
obstacle to be of infinite draft; for a ship of finite draft there would be terms 
representing a local surface elevation in addition to the diverging waves 
from each element. Further, the result is only an approximation and 
assumes, in fact, that the additional surface elevation is relatively small. 
3. We shall apply (7) to one case only, so as to estimate the magnitude 
of the effect due to the scattering of waves by a narrow ship of great draft 
and of form similar to those for which previous calculations of wave 
resistance have been made. 
The model is of symmetrical form with straight sides, of total length 21, 
beam 26, and with a parallel middle body of length 2a; the bow and stern 
are equal wedges of axial length /—a and of semi-angle «, where 
tana = b/(l—a). We take the origin at the centre of the axis, with the 
positive direction of Ox from stern to bow. Thus we have 
dy/0& = a, for —l<&é<-—a 
= 0, for —a<&<a 
==, ioe <<), (8) 
From (7), the surface elevation at any point (2, y) is given by 
r—a 
l 
C = thetottxa) _ Likhace'| A? (Kr) et§ dé + bikhacta H® (xr) e*d&. (9) 
a 
=U 
We shall use this only for the elevation along the axis y = 0, as in the 
corresponding calculation of wave profiles for a ship. We note that in these 
expressions the quantity r is essentially positive. Asan example, for a point 
in the bow wedge, that is for a<a<l, we have 
—a 
Cet! = the*® — Lixha AHP {K(x— &)} e d& 
x 1 
+ hikha| H2{K(a—&)t e§dé + tixha| H{K(E—x)e*S dé. (10) 
a =x 
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