32 



at a rate sufficient to develop the full wave form in one wave period 

 and all energy is being withdrawn from the water behind in the same 

 period, leavmg a horizontal surface and zero motion. That such a 

 situation is difficult to imagine is perhaps an intuitive proof that the 

 full wave energy does not advance with the wave velocity. 



Consider next the problem of beat frequency in natural waves and 

 assume that two infinite wave trams of equal amplitude and very 

 nearly equal wave length are rmrning in the same direction. The 

 elevation of the water surface is the summation of the elevations of 

 the individual waves. Limiting the analysis to waves of small 

 amplitude, which are very nearly sinusoidal, the surface curve would 

 be described by (17) 



y=a smi-j^ j,- ]-\-a smi-j^ -jr ) (39) 



Rewriting the equation m equivalent form, 



2/=2acos(xx(^--^-)-w(^-^)}sin{x:r(^^^ 



"(39a) 



If ii and L2 are very nearly equal, the wave form at any time t con- 

 sists of a series of sine waves of length approximately L and ampli- 

 tudes which increase from zero to 2a and then decrease to zero. 

 Fixing attention on the surface form, the equation states that the 

 group advances a distance 



in a time 



The group velocity is then, with L—CT 



_Ax_aa(T^-Tr) 



Passing to the limit, 



Cg= J=j^^ = C-X^=group velocity (41) 



For waves of small amplitude in any depth of water 



^=Vi+^^ (2) 



and 



^+"774^) 



sinh-y-/ 



4:Trd 

 ^«-=^|l + — ^1 (42) 



