THE APPROXIMATE CALCULATION OF WAVE RESISTANCE AT HIGH SPEED 49 
the integral being taken over the midship section, and V being the immersed 
volume. 
To sum up, with this approximation, the source distribution on the ship’s 
surface is specified as follows: the total source strength on any portion of 
the surface is given by v/4m times the area of the projection of that portion 
on to the midship section, with an obvious rule for determining the sign of the 
projected area. 
Formule for Wave Resistance 
4. We give now expressions which will be used for the calculation of wave 
resistance ; for general formule for any distribution of sources reference 
may be made to Roy. Soc. Proc. A. 138, p. 339 (1932). We take the origin O 
on the centre line of the form and in the water plane, Ox in the direction of 
motion, Oz vertically downwards, and Oy horizontally at right angles to the 
other two axes. We shall be concerned here with sources only in the longi- 
tudinal zx-plane. If we have any distribution of which a typical source is 
of strength m, at the point (x,, 0, z-), the corresponding wave resistance -is 
given by 
co 
Res Niuha (Pspdpiceieds 5 5s oe a © 
where 
I = Imye~ Ker cosh*u sin (kx, cosh uv) 
J = Imye— Ker Cosh’ cos (kx, cosh u) 
where & = g/v? and the summation extends over the given system of sources. 
(5) 
If we make the assumptions for a narrow ship, outlined in the previous sections 
from a somewhat different point of view, it can easily be verified that these 
expressions lead to the usual integrals for the wave resistance. We have the 
same form for R with 
l= meal (dy/dx) sin (kx cosh u)e7kz cosh®u dadz 
ee ; (6) 
J= ae i i (dy/dx) cos (kx cosh u)e~kz cosh®u dadz 
The integrals are taken over the longitudinal section of the ship, and (3y/¢x) 
is taken from the equation of the surface of the ship. 
First Approximations 
5. After this preliminary survey we proceed to the immediate problem, 
namely dividing the ship into a finite number of sub-divisions. Although of 
no practical value, we begin with the most extreme simplification to illustrate 
the point of view of the present study. We have seen that the total moment 
of the source distribution is Vv/4m where V is the immersed volume and v the 
speed. We now suppose this moment to be concentrated at a point as a source 
and sink doublet with its axis in the direction of motion. The longitudinal 
location of this doublet is immaterial so far as the resistance formula is con- 
cerned and we may suppose it to be in the midship section. For its depth 
we use here, and throughout the work, the principle that for a first approxi- 
mation we replace any system of sources by a source of the total strength placed 
at the centroid of the system. Since the source strength on any element of 
the ship’s surface is proportional to the projection of that element on the mid- 
ship section, it follows at once that the depth of the centroid of the distribution 
is the depth h of the centroid of the midship section. Thus the first approxi- 
mation is a horizontal doublet of moment Vv/4r at a depth fh. Putting these 
values into the expression for the wave resistance of a doublet, which may 
be deduced from (4), (5), we obtain 
ee) o 
R = (go/n) ks v5 e —2khcoshttcoshtudu . . . ... @) 
502 
