515 T. H. Havelock. 
pattern advancing steadily with velocity c in the direction Ox, we may write 
(1) in the form 
2 
CER OE (2) 
with ky = g/c?. 
A simple plane wave advancing in a direction making an angle @ with Oz 
is given by 
¢ = asin {ky sec? 6 (x cos 8 + y sin @ — ct cos @)} 
= ac cos Be" °"? cos {xq sec? 8 (x cos 8 + y sin 6 — ct cos 8)} 
We may generalize this to obtain a free wave pattern made up of plane waves 
advancing in all directions, so that the pattern itself moves steadily with 
velocity ¢ in the direction Ox; we have then 
C= ie f (8) sin {xy sec? 6 (x cos 8 + y sin 6 — ct cos 8)} dé. (4) 
—ir 
We shall suppose in the first place that the pattern is symmetrical with respect 
to Oz, so that we have 
dar 
G—2 | f (9) sin (kp 2’ sec 8) cos (Ky y sin @ sec? 6) dO 
0 
» (5) 
tn 
¢ = 2c | f (8) et" * cos (icgx’ sec 8) cos (Ky y sin 6 sec? @) cos 6 d6 
0 
with a’ = « — ct. 
Consider a fixed vertical plane «= constant. The rate of flow of total 
energy across this plane is given by 
iS 0 é 0 2 
tee | de | {(@ ay = ( a ( sey } dy + hgee | Ody. (8) 
—7o —~o ¥ dz =e 
The variable part of the fluid pressure being e0d/0t, or — pcdd/ex, the rate at 
which work is being done across the same plane is 
of ef Bm ° 
We shall assume that the wave pattern is such that these quantities are finite 
and determinate. 
To evaluate these expressions with the values (5) for ¢ and ¢ we use the 
following theorem ; 
