given by equation (2-2) or (2-3) . For waves propagating in deep or transi- 

 tional water with gravity as the primary restoring force, the group velocity 

 will be less than the phase velocity. (For those waves propagated primarily 

 under the influence of surface tension, i.e. capillary waves, the group 

 velocity may exceed the velocity of an individual wave.) 



The concept of group velocity can be described by considering the 

 interaction of two sinusoidal wave trains moving in the same direction with 

 slightly different wavelengths and periods. The equation of the water surface 

 is given by 



H llitx 2-nt\ H /Zirx 2Trt\ 

 n = Hi + n2 = -J cos \— ;;;-| +- COS {— ;^ | (2-33) 



where ti^ and n2 are the contributions of each of the two components. They 

 may be summed since superposition of solutions is permissible when the 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, Lj^ and l^, 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 sum of the two sinusoidal waves is given by 

 equation (2-33) and is shown in Figure 2-5. The waves shown in Figure 2-5 

 appear to be traveling in groups described by the equation of the envelope 

 curves 



n - = ± H cos 

 envelope 



'L2 - L^^ 



X - IT 



'T2 - Ti^ 



Tl T2 



(2-34) 



17 = 77, + 7^2 



^envelope 



■0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3 

 "2 I L, L2 ^ ~ T It, T2 / (after Kinsmon.l 



,1965) 



Figure 2-5. Formation of wave groups by the addition of two sinusoids 

 having different periods. 



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 



2-24 



