48 THE APPROXIMATE CALCULATION OF WAVE RESISTANCE AT HIGH SPEED 
steps. The first step is to neglect the wave motion and consider the fluid 
motion produced by the ship assuming the water surface to remain plane ; 
the next step is to obtain the wave disturbance produced by this fluid motion 
while ignoring the presence of the ship. A third step would then be to evaluate 
the influence of the ship on the waves so calculated, and so on by successive 
steps. Meantime the theory has not in fact proceeded further than the first 
two steps. 
Equivalent Source Distribution 
3. The first step in the process may be expressed in another form. Consider 
a double ship formed of the immersed volume of the ship and its inverted 
reflection in the water plane, and suppose this complete solid entirely immersed 
in water and moving forward with uniform velocity v. Over the fore part of 
the ship the water is moving forwards and outwards, and over the after part 
it flows in to follow the motion of the ship. This fluid motion can be repre- 
sented completely by a definite continuous distribution of sources and sinks 
over the surface of the ship at each instant and moving with the ship. Let o 
be the source strength per unit area at any point of the ship’s surface, o being 
positive over the fore part and negative over the after part for a normal form. 
(The notation used in this paper is that if m is the strength of a point source, 
4xm is the volume of liquid flowing out in unit time). It is clear that, since 
the total volume of water is unaltered, the integrated value of o over the whole 
surface is zero, or the sum of the positive sources is equal to the sum of the 
negative sources. On the other hand, if x is the distance of any point from 
some transverse reference plane, say the mid-ship section, the integrated value 
of cx taken over the whole surface is a definite amount and is the moment M of 
the distribution. A simple expression for M can be derived from general 
Ree: without knowing the actual distribution. It can be shown that 
=(1+k) Vo/4r . : : : ol) 
In ae expression V is the volume of ne bodes armel ea iS the inertia coefficient 
for longitudinal motion ; that is, gpkV is the added mass due to the motion 
of the water. 
If v, is the component of the velocity v normally outwards at any point 
of the ship’s surface, it is convenient to write the corresponding source distri- 
bution in the form co = (1 + k*)v,/4r. In general, k* varies from point to 
point, but for an ellipsoid it is constant and equal to k. The added mass for 
longitudinal motion is not very important in ship problems and there are few 
estimates of its value. It is of interest to note that W. Froude investigated 
this effect in his well-known experiments on H.M.S. Greyhound. He made 
two sets of experiments, one with retarded motion and the other with accelera- 
tion ; the former gave a coefficient of about 20% and the latter of about 7%, 
and on experimental grounds Froude attached more value to the larger esti- 
mate. Whatever may be the interpretation of experimental results, we are 
concerned here with the theoretical coefficient for non-viscous fluid motion ; 
and there is reason to regard the lower value as more appropriate for normal 
ship forms. Although this correction should be noted for future examination, 
we may meantime regard it as relatively small, at least for the so-called narrow 
models to which the wave theory has so far been limited. The usual approxi- 
mation amounts, in fact, to neglecting the inertia coefficient k for longitudinal 
motion ; and, in what follows, we take the source strength per unit area to 
be given by o = v,/4x. We can easily verify the total moment M of the distri- 
bution in this case. Imagine a horizontal cylinder of small cross section cutting 
the midship section in an area dS, and cutting out an. area dS, at a point P,; 
on the fore part of the ship’s surface and an area dS, at a point P, on the after 
part. Then we have 
6, dS, = 2, dS,/4n = vdS/4x (2) 
6, dS, = — vdS/4x Os . . . . . . . . 
Hence 
M = |vp,P,dS|4n = vV/4r Mery en aA eR ra Tel ce tran) 
501 
