406 Lord Kayleigh on the 



U being the group-velocity. By Taylor's theorem when 

 k — K is very small, 



k[x-tf(k)]=KHfXK)+±t(k-Ky{-Kf>>(K)-2f'(K)}. 



Using this in (9) and integrating with the aid of 

 da cos a 2 = da sin a 2 = V(^7r), 



we find as an approximate value 



n/ (2tt) . cos {t *■/ ' («) + 1^} 



^W{-^/"W-2/'W}' ' * (12) ' 



As a particular case, for deep-water gravity waves 



/(*)=•(#*), /'(*) = -&**-*, 



and finally with use of (11) 



v = ^.-l00B(g-j) . . . (13)*. 



This gives the effect of the impulse at (0, 0). If the 

 impulse be at x\ i', we are to write x—x ! for x and t— t' 

 for t. For our purpose of finding the effect of a travelling 

 force, we are to make x'=Yt' and integrate with respect to 

 t' from tot', t' being the duration of the force. The integral 

 will depend mainly upon the part where 



x-Vt" 



under the cosine, is stationary. This occurs when 



2x=Y(t+t') (14), 



and then 



g(t-tj _ g(yt-x) 

 4(a-W) V 2 * • • • [10) ' 



Omitting the variation of the other factors as less important, 

 we see that, when sensible, the effect is proportional to 



cos 



{^^-1} ■ ■ ■ ■ d6), 



representing simple waves of velocity V. But this is limited 

 to such values of x and t as make t' in (14) lie between 

 and t' . Thus if t be given, the range of x is from ^Sft to 

 jfVt + ^Vt'; so that the train of waves covers a length ^W, 



* An almost equally simple formula applies when more generally 



