346 T. H. Havelock. 
Moving Solid. 
7. An obvious application of these results is to the uniform motion of a sub- 
merged solid when we replace the solid by a distribution of sources or doublets 
over its surface ; for a first approximation we may take the distribution to be 
that appropriate to the motion of the solid in an infinite liquid. This will, of 
course, give the same result as if we had used the system of sources and sinks 
which is the image of a uniform stream in the solid, or, in fact, any equivalent 
surface or volume distribution on or within the surface of the solid. Simple 
forms, such as the sphere or ellipsoid, for which the wave resistance has already 
been found, have been calculated from the known image system. For instance, 
the sphere was replaced by a doublet at the centre ; it can be verified, after 
some reduction of integrals, that the expressions (16) and (17) with the proper 
value of c over the surface of the sphere, lead to the same result for the wave 
resistance. In general, the expressions (16) and (17) allow the wave resistance 
to be calculated for solids for which an image system is not known, but for 
which the distribution of surface density can be determined by known methods 
of approximation. 
Consider now an open plane distribution of sources and sinks over the 
vertical zz-plane. In this case the normal fluid velocity at a point on either 
side is 2mc, where o is the source density at the point. For a ship of slender 
form, and small beam, symmetrical about the za-plane, the normal velocity 
is taken to be approximately c dy/ox if the surface of the ship is given by an 
equation y = f(z, x). From (16) and (17), the usual expression for the wave 
resistance follows : 
2. pin 
Ree “oe | (P2 + Q2) sec? 6 d0, (33) 
—}hr 
as | | EY greet OF (cyr sec 0) de de (34) 
x 
the latter integrations being taken over the vertical longitudinal section. 
For the other extreme case, a ship of flat form and small draught, comparison 
is usually made with a suitable distribution of pressure applied to the surface 
of the water, with the wave resistance given by, say, (31) and (82). 
The similarity between the expressions for the resistance in these two extreme 
forms has been remarked upon by Weinblum,* and more recently by Hogner.t 
In an attempt to cover both cases by a single expression, Hogner has proposed 
*G. Weinblum, ‘Z.A.M.M.,’ vol. 10, p. 458 (1930). 
TE. Hogner, ‘Jahrb. Schiffbautech. Ges.’ (1932). 
374 
