Calculation of Wave Resistance. 518 
Hence, from (17), the wave resistance is 
us 
R = 16rpr iM? | “sees 0 en 2 "do, (19) 
0 
which is the known result for this case. 
4. Before generalizing these results we may put (6) and (7) into an explicit 
form for the wave resistance. 
The kinetie energy of the liquid in a strip between two parallel vertical 
planes at a distance dz apart is 
0 00 rs) 2 
| ae [” (a4 ey +(2 ao) + (22) ay. (23) 
= x oy 0z 
Transform (23) into the equivalent form of a surface integral over the boundaries 
of this portion of fluid, assuming the wave pattern to be such that the various 
integrals are convergent. Thus we obtain the rate of flow of kinetic energy 
across a vertical plane as 
too |” (98) tut teof el” [oSt+ (yh. em 
Further, we may transform the other terms in (6) and.(7) by using the surface 
condition (2) together with gf = — c@d/éx at z = 0. 
Finally, equating the difference between (6) and (7) to Re, we obtain for the 
wave resistance 
ET URI —#E) a af (GS -a8e 
(25) 
0x 0a? ee, 
In this expression ¢ is the velocity potential of the free wave pattern to which 
the disturbance approximates at a great distance in the rear. Considering the 
disturbance produced by a body of any form, it appears that this free wave 
pattern must be expressible, in general, in the ferm 
e— F F (8) sin {xy sec? 8 (a cos 8 + y sin 6)}d0 
=jr 
+ FP F (8) cos {iy sec? 0 (x cos 8 + y sin 6)}d8, (26) 
—4r 
that is, in the form 
chr 
{=| "(Pisin A cos B + Py cos A sin B + Py cos A cos B + Py sin A sin B) a8, 
0 
(27) 
where A = xyz’ sec 6, B = «yy sin 9 sec? 0. 
394 
