Wave Resistance: Effect of Parallel Middle Body. 79 
for a distribution of doublets in the plane y = 0, the moment per_unit area 
being wt (h, f) and the integrations extending over the whole distribution. The 
function ) must be such that the integrals are convergent, as well as the 
corresponding expressions for the velocity potential and the surface disturbance. 
We now integrate (2) by parts with respect to h and h’, and as we shall deal 
with distributions which are of finite extent in the x co-ordinate, we obtain 
7 
RB = 16ng%o0-4 if df a le dh | “a i Dw/ah. OU far’ . sec? 
x EW WETHICF see* > eos [£g (h—h’)/c?} sec 6] 0d. (3) 
The fluid motion is symmetrical with respect to the plane y=0; we may 
therefore confine our attention to the fluid on one side of this plane and we may 
interpret (3) in terms of the distribution of normal fluid velocity over the plane 
y=0. For, from the definition of |, the normal fluid velocity at the point 
(h, 0, f) is2x0/dh. Substituting in (3) we should then have the wave resistance 
for a given distribution of normal fluid velocity over the plane y= 0. From the 
latter point of view the solution can also be obtained by methods of harmonic 
analysis; the expression for the wave resistance, used in a former paper,* 
agrees with (3) found by the method of sources and sinks. 
3. In the application to ship waves the same assumptions are made as in the 
paper just quoted. The plane y = 0 is the fore and aft median plane of the 
ship, and the inclination of the ship’s surface to this plane is supposed small. 
The ship is then replaced by an equivalent distribution of normal fluid velocity 
over its section by the plane y = 0, namely the component of the stream velocity 
c over the actual surface of the ship ; thus if 
y = F(a, 2) (4) 
is the equation of the ship’s surface, we use in (3) 
ou oF 
9 —  — — 
ana = on. (5) 
A difficulty which may arise in the general solution should be mentioned, but 
need not be considered further in the present applications. A mathematical 
infinity may occur in some of the expressions ; this may be removed by intro- 
ducing a suitable factor to ensure convergency, but in any case it only occurs 
in those parts of the velocity potential and surface disturbance which represent 
the local symmetrical disturbance. The integrals for the wave disturbance, 
and consequently expression (3) for the wave resistance, remain finite. 
* «Roy. Soc. Proc.,’ A, vol. 103, p. 574 (1923). 
216 
