o90 Prof. H. Lamb on Waves due 



In this case we have 



tr 2 =^tanhM, (24) 



and 



•^-ifsS) • • • • (25) 



Hence for the effect o£ a travelling disturbance, we find 

 _2 sinh 2 /eh _ . t 



pc 2 ' sinh kJi cosh kJl — tch 



This, again, is in accordance with a known result *. 



The critical case of U = c occurs when we have water- 

 waves subject to gravity and capillarity combined, if the 

 velocity c of the travelling disturbance is equal to Kelvin's 

 minimum wave-velocity. Since 



a 2 =gk+Tk\ (27) 



where T' is the ratio of the surface-tension to the density, 

 we find, in the critical case, 



Taking the value of <j)(k) from (20), and substituting in (17), 

 we have 



K* f \ 



V= + -r— ^sinl/e^-l- -r7r). . . . (29)t 

 IX' 2 pa \ 4 / 



The critical values of k and c are given by 



«=a/(^/T'), c 2 =2 */(</!'). . . . (30) 



4. The formula (12) supplies all that is essential for 

 calculations of wave-resistance. If we denote the potential 

 energy of the waves,, per unit length of the axis of #, by 



where C is a coefficient depending on the nature of the 

 medium, the mean energy, both potential and kinetic, in 

 the wave-train will be 



E _i c iM!L (»n 



JjJ "2 t (c-uf {61) 



absolute value being indicated in case </)(«) is imaginary. 

 Supposing, for deh'niteness, that U<c, let a fixed point 



* Lamb, 'Hydrodynamics,' 4th ed., Art. 245. 



t Of. Rayleigh, Proc. Lond. Math. Soc. (1) vol. xv. p. 75 (1883) ; 

 ' Scientific Papers/ vol. ii. p. 264. 



