WAVE PATTERNS AND WAVE RESISTANCE. 11 
resultant fluid pressure on the ship. Another method is to introduce an artificial kind of 
fluid resistance, calculate the rate of dissipation of energy, and so ultimately arrive at 
expressions for the wave resistance. All these methods must lead to the same results 
if carried out correctly; but perhaps the most natural method is that outlined above and 
embodied in the general expression (18). 
It has been shown recently that the necessary calculations can readily be extended to 
wave patterns of the general type which occur in ship waves.* The results may be given 
here, without going into the detailed analysis. 
Suppose first that we have a free wave pattern given by 
2 
r= [fo sim (OG o co o o 6 « o (i) 
and suppose that the amplitude factor f(#) is an even function of 0, so that (19) is 
equivalent to 
2 
C= afi (0) sin (« a sec 0) cos (x ysin @sec?@)d6 . . . (20) 
0 
We can write down the velocity potential of the fluid motion for the wave form (19) and 
so obtain the pressure and velocity at any point of the fluid. The quantities E and W 
of (18) can then be calculated, with suitable limitations on the function f (@) which amount 
to ensuring that E and W are in fact finite and calculable. Under these conditions it is 
found that E — W for the pattern (19) is given by a remarkably simple expression, namely, 
BW = mpet| {F(0}* 008 6a ee ANIC) 
0 
Hence the wave resistance of a body moving with velocity ¢ and leaving in its rear 
a pattern (19) would be given by 
2 
R=7pe ly OVP CoPOd 5 6 5 2 5 o () 
“0 
12. Suppose, for illustration, that the amplitude factor is independent of @ and that 
we have 
7 
2 
c= | sin ya Bie oa ied ten aie Ang 23) 
a simple sine pattern, with h possibly a function of the velocity c. This is certainly a 
hypothetical case; (23) is like the first term of (13) or (15), so presumably the sort of 
body which would produce this wave pattern would be the bow of a ship of great draught, 
but without any sides or stern. However, without inquiring any further into that, if the 
wave pattern is (23), then from (22) the corresponding wave resistance would be 
Rew p iit |eoebde an pe Bh i) REINS (24) 
0 
* Proc. Roy. Soc. A., 144, p. 514, 1934. 
387 
