14 WAVE RESISTANCE THEORY AND APPLICATION 
a succession of small steps, at each step an impulse 
being applied to the surface of the water. Each 
impulse starts a series of ring-waves traveling 
out in all directions; and to get the total effect 
at any time we have simply to sum up the effects 
due to all the previous elementary steps, the well- 
known wave pattern emerging from the mutual 
interference of these elementary ring-waves. The 
process can be expressed mathematically to give 
the complete solution of this problem. 
Returning to the submerged solid, we regard 
the continuous motion as the limit of elementary 
steps and examine what happens at any given 
step. We picture the solid as suddenly started 
from rest with a given velocity and then stopped 
after a short interval of time. For this impulsive 
motion ¢o is the potential as if the solid were 
started from rest in an infinite liquid. But the 
form of the surface condition for this step is that 
there shall be no impulse at the free surface and 
we must add the appropriate function ¢,. This 
may be written down directly as a reflected po- 
tential, but we may picture it in this way. Sup- 
pose the water continued above the free surface 
and place in it the image of the given solid. When 
the solid is moved through its elementary step, 
we move the image suddenly through an equal 
small step in the opposite direction. The poten- 
tial for these two motions in an infinite liquid 
gives the required approximation ¢» + ¢1. We may 
notice, in passing, that gravity does not come into 
play during this impulsive motion. We now cal- 
culate the vertical velocity of the free surface, 
and the result of the step from rest to rest is that 
the free surface is left with a known elevation. 
The subsequent motion due to this elevation can 
be worked out, the elevation spreading out in all 
directions in the form of free gravity waves. 
Finally, for any continuous motion of the solid 
we sum up the total effect of all the previous ele- 
mentary steps in the motion. The process can be 
set out in mathematical form, and so we obtain 
the first approximations for the assigned motion; 
it may be remarked that further approximations 
are possible by generalizing this process. An in- 
teresting point is that this formulation of the 
problem automatically leads to the so-called prac- 
tical solution with the solid advancing into still 
water, and with the main wave pattern to the 
rear. This result is connected with the fact that 
for water waves the group velocity is less than the 
wave velocity; if the contrary had been the case, 
we should have arrived at a steady state with the 
solid pushing the wave pattern in advance in- 
stead of leaving it to the rear. It will be seen 
also from this description that this impulse 
method can be applied equally well to nonuniform 
motion or to motion of any kind in a curved path. 
Although not necessary, it is convenient often 
to introduce the idea of sources and sinks. The 
potential ¢» due to the motion of the solid as if in 
an infinite liquid can be regarded as due to a dis- 
tribution of sources and sinks, or other singulari- 
ties, on or within the boundary of the solid, and 
an elementary step in the motion corresponds to 
establishing this distribution for a short interval 
of time. Consider in Fig. 1, a point source of 
strength m established at time 7 at the point (0, 0, 
—f) in the liquid, where we have taken the origin 
O in the free surface with OZ vertically up- 
wards. During the short interval of time 67 we 
have the velocity potential 
Fic. 1 
with 
Ae = Gee Case aes ta = ae ap ah sp (B= i }F 
The initial elevation left by the elementary step is 
Qmf br 
* _ mor (= ee) 
oper py = S#f pe 
x cos [k(x cos 6 + y sin 0)]k dk 
and the motion at any subsequent time ¢ due to 
this elevation is given by 
= mg'/2 or (Ee dé ihe ek S—-) x 
Tw —* 0 
x cos [k(x cos @ +y sin @)] sin [g'/2k!/2(t — 7)]k'/2 dk 
In particular, suppose the source starts from rest 
at time ¢ = 0, is of constant magnitude, and 
moves with uniform velocity ¢ in a horizontal line 
parallel to OX. The velocity potential at time 
tis given by 
1 t C) 
on mm 4 MEE far f" ao f eH) X 
Ty Te Tv 0 —7 0 
cos [k{(x + ct — cr) cos 6 + ysin 8}] X 
sin [g'/2k'/x(t — r)]k'/2 dk 
564 
