with slightly different wavelengths and periods, 

 water surface is given by: 



The equation of the 



H /27rx 



+ n, = — cos I 



' 2 \ L. 



-— , (2-33) 



where n, and r) are the contributions of each of the two components. 



They may be summed since superposition of solutions is permissible when 

 linear wave theory is used. For simplicity, the heights of both wave 

 components have been assumed equal. Since the wavelengths of the two 

 component waves, L-,^ and L2, have been assumed slightly different, for 

 some values of x at a given time, the two components will be in phase 

 and the wave height observed will be 2H; for some other values of x, 

 the two waves will be completely out of phase and the resultant wave 

 height will be zero. The surface profile made up of the stun of the two 

 sinusoidal waves is given by Equation 2-33 and is shown in Figure 2-5. 

 The waves shown on Figure 2-5 appear to be traveling in groups described 

 by the equation of the envelope curves: 



^envelope 



= ± H cos 





^2 - T, 



(2-34) 



It is the speed of these groups, i.e. the velocity of propagation of 

 the envelope curves, that represents the group velocity. The limiting 

 speed of the wave groups as they become large, i.e., as the wavelength, 

 L^, approaches L2 and consequently the wave period T-^ approaches T^ 

 is the group velocity and can be shown to be equal to: 



s = 



1 h 



2 T 



1 + 



47Td/L 



sinh (47rd/L) 



= nC 



(2-35) 



where 



1 



n = — 

 2 



1 + 



47rd/L 



sinh (47rd/L) 

 In deep water, the term (47rd/L)/sinh(4Trd/L) is approximately zero and, 



1 L^ 1 

 ^g " 2 Y ^ 2 ^° (deep water) 



(2-36) 



or the group velocity is one-half the phase velocity. 

 sinh(47rd/L) ^ 4TTd/L and, 



In shallow water. 



C = - = C =« 



S T 



gd (shallow water) 



(2-37) 



2-25 



