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 s^nie speed precluding the alternate 

 reinforcing and cancelling of components. In deep and transitional water, 

 wave celerity 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 mediioriy 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, and an observer moving along with a group 

 of waves at the 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 con- 

 sideration of wave energy. Therefore an additional explanation of group 

 velocity is provided on wave energy and energy transmission. 



2.238 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 mean water level, 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 



Subscripts k and p refer to kinetic and potential energies. Total 

 average wave energy per unit surface area, termed the specific energy or 

 energy density, is given by 



E pgH* 

 E = - = ^ . (2-39) 



Wave energy flux is the rate at which energy is transmitted in the 

 direction of wave propagation across a vertical plane perpendicular to the 

 direction of wave advance and extending down the entire depth. The average 



2-27 



