394 Prof. H. Lamb on Waves due 



The potential energy, per unit length of the axis of x\ being 



lffPV 2 +l>9(p'-p)v'\ ....... (59) 



the mean energies are 



^= 2 ^{p + {p'-p)e-^}Aax i ), .... (60) 



The total wave-resistance is 



R = R 1 + R 2 , (62) 



where 

 Hence 



R^^-^Ei, R,= ^-iZ?B 2 . . . . (63) 



Ri=5{p + (p'-p)«- 2 "*}Ai 2 (« l ), (64) 



6. As regards the determination of A^k), A 2 (k), two 

 special cases may be considered. 



If an impulsive pressure coskx be applied to the upper 

 surface at the instant £ = 0, we have, initially, 



p(j> = cos kx (66) 



for ?/ = 0, and 



p£=/>'f (67) 



for y=.—li i the latter condition securing the continuity of 

 the initial distribution of impulsive pressure. Hence from 

 (52) and (53) 



p(A 1 + A 2 ) = l, (68) 



/ (To 2 o'e kh a 2 \ 



( / o / -p)^- M A 1 -p(cosh^--^sinhH + 4— -4 }A 2 =0. (69) 



With the help of (44), we find 



p+{p—p)e ~ kh p{p + {p'-p)e 2kh \ 



