1^ and consequently the wave period T, approaches T2) is the group veloc- 

 ity and can be shovm to be equal to 



c =ii 



g 2 T 



where 



1 + 



4Trd/L 



n = 



1 + 



sinh(4Trd/L) 



4TTd/L 

 sinh(4Trd/L) 



= nC 



(2-35) 



In deep waters, the term (4iTd/L)/sinh(4iid/L) is approximately zero and 



L 

 1 o 1 



C = = — C (deep water) (2-36) 



g 2 T 2 o 



or the group velocity is one-half the phase velocity. In shallow water, 

 sinh(4ird/L) «; 4Trd/L and 



C = — = Ci* xjgd (shallow water) 

 g T 



(2-37) 



hence, the group and phase velocities are equal. Thus, in shallow water, 

 because wave celerity is fully determined by the depth, all component waves in 

 a wave train will travel at the same speed precluding the alternate reinforc- 

 ing and canceling of components. In deep and transitional water, wave celer- 

 ity depends on the wavelength; hence, slightly longer waves travel slightly 

 faster and produce the small phase differences resulting in wave groups. 

 These waves are said to be dispersive or propagating in a dispersive medium', 

 i.e., in a medium where their celerity is dependent on wavelength. 



Outside of shallow water, the phase velocity of gravity waves is greater 

 than the group velocity; an observer that follows a group of waves at group 

 velocity will see waves that originate at the rear of the group move forward 

 through the group traveling at the phase velocity and disappear at the front 

 of the wave group. 



Group velocity is important because it is with this velocity that wave 

 energy is propagated. Although mathematically the group velocity can be 

 shown rigorously from the interference of two or more waves (Lamb, 1932), the 

 physical significance is not as obvious as it is in the method based on the 

 consideration of wave energy. Therefore an additional explanation of group 

 velocity is provided on wave energy and energy transmission. 



h. Wave Energy and Power . The total energy of a wave system is the sum 

 of its kinetic energy and its potential energy. The kinetic energy is that 

 part of the total energy due to water particle velocities associated with wave 

 motion. Potential energy is that part of the energy resulting from part of 

 the fluid mass being above the trough: the wave crest. According to the Airy 

 theory, if the potential energy is determined relative to SWL, and all waves 

 are propagated in the same direction, potential and kinetic energy components 

 are equal, and the total wave energy in one wavelength per unit crest width is 

 given by 



2-25 



