356 Prof. T. H. Havelock. Wave Resistance: Some 
water surface by means of this cover; hence the corresponding wave resist- 
ance is simply the total resolved pressure in the direction Oz.* With the 
usual limitation that the slope of the surface is everywhere small, this leads 
to 
Re [F (w) Eas, (7) 
IX 
taken over the whole surface. 
The evaluation of the steady wave resistance for an assigned pressure 
F (aw) is to be carried out by means of (3), (5), (6), and (7). However, we 
may obtain simpler expressions before applying them to particular cases. 
3. For this purpose we analyse the wave disturbance (5) into simple con- 
stituents, in fact into one-dimensional disturbances ranged at all possible 
angles round Oz, the line of advance. We have 
mIJo[K{(w+euP+y?}?] = [ie (z+ew) cos $ eos (Ky sin h) dd. (8) 
0 
Substitute in (6) and we can now carry out the integration with respect to w; 
for we have 
@o 
2 | e~ aut grxcucos sin (Vu) du 
0 
= (xccosP+KV+hpi)1—(xecosp—KVt+hpi) 3. (9) 
We simplify this expression further by using the fact that m was 
. introduced only to keep the integrals determinate, and is eventually to be 
made infinitesimal; we can therefore reject terms in « which are super- 
fluous for this purpose. The process receives its justification in the course of 
the analysis. This being understood, we can use, instead of the right-hand 
side of (9), the expression 
—2 (V/c?) sec? p/ {«—x sec? 6 +7 (w/c) sec p}, (10) 
where «= g/c?. Using these results in (6), and making a slight trans- 
formation, we can express the surface elevation in the form 
Ko mlz 2 a p 
a 2779p \o ae (, ef (1) { K— ky Sec? +2 (p/c) sec b 
e—t« (cos p+ysin >) ha 11 
Omen 
etx (zcos p+y sin >) 
g 
In (11) we have the surface elevation analysed into plane wave con- 
stituents, each element moving in a line making an angle @ with Oz. Carry- 
ing out the integration with respect to «, we can express each constituent 
* ‘Roy. Soc. Proc.,’ A, vol. 98, p. 244 (1917). 
148 
