Wave Resistance. 26 
If R is the equivalent wave resistance, Re is equal to the rate of dissipation 
of energy ; this gives, following Lamb, 
ip sal jae ag, (4) 
On 
taken over the free surface. Thus in the two-dimensional case we have 
R = Lim yup | soe dz, (5) 
Peto =e! GB 
with z = 0. 
The surface values of ¢ and 0¢/0z can be obtained from (3) ; after applying 
well-known transformations we obtain, at z= 0, 
oo . 
m s } mf _ 
eae 2M | (m+ p) sin mf + Kp» Cos m e-™ dm, 
(m+ By + KQ 
od —_—e 4Maf == Wr M [ (par) 08 Ty gS ac dm 
mB Cape. (mt uP Ke 
forz > 0; and 
hb = 4rxyMe**— "J cos (ko + uf) 
cf 2g | (m— p) sin mf — Kp COs mf me dm, 
(m — p)? + K9" 
od 
ao ArcicyM e#*—"F {cg cos (gt + wf) + w sin (Kow + uf} 
hal Maf A Oe MI (m — wp) cos mf — Kp sin mf nem Bin, (@) 
(a? = or (m — p.)? + Kk, ; 
ce 
fora <0. 
These expressions are continuous at z= 0. It is easily seen that the only 
terms which give any contribution to (5) in the limit are the first terms in the 
expressions for ¢ and 0¢/dz when « is negative. These are the terms which 
arise from the train of regular waves established in the rear of the moving doublet 
and so this method is connected with the alternative calculation of wave 
resistance by means of group velocity. The dissipation of energy when there 
is a frictional term is represented in the limit, when y is made zero, by the propa- 
gation of energy away from the system in the train of regular waves. To com- 
plete the calculation from (5) and (6) we have 
0 
R = Lim pe | L 67023 M2e7#*— 2F cos? yx dx 
—o 
= 4r29x,3M2e~ 2, (7) 
280 
