This represents the expenditure of 



ft-lb sec , . ,, 



5120 X 3600 — = 18.55 X 10* ft-lbs , 



sec hr 



of energy each hour on each foot of beach. 



************************************* 



The mean rate of energy transmission associated with waves propagating 

 into an area of calm water provides a better physical description of the 

 concept of group velocity. An excellent treatment of this subject is given 

 by Sverdrup and Munk (1947) and is repeated here. 



Quoting from the Beach Erosion Board Technical 

 Report No. 2, (1942) : "As the first wave in the group 

 advances one wave length, its form induces correspond- 

 ing velocities in the previously undisturbed water and 

 the kinetic energy corresponding to those velocities 

 must be drawn from the energy flowing ahead with the 

 form. If there is equipartition of energy in the wave, 

 half of the potential energy which advanced with the 

 wave must be given over to the kinetic form and the 

 wave loses height. Advancing another wave length 

 another half of the potential energy is used to supply 

 kinetic energy to the undisturbed liquid. The process 

 continues until the first wave is too small to identify. 

 The second, third, and subsequent waves move into water 

 already disturbed and the rate at which they lose height 

 is less than for the first wave. At the rear of the 

 group, the potential energy might be imagined as moving 

 ahead, leaving a flat surface and half of the total 

 energy behind as kinetic energy. But the velocity 

 pattern is such that flow converges toward one section 

 thus developing a crest and diverges from another 

 section forming a trough. Thus the kinetic energy is 

 converted into potential and a wave develops in the 

 rear of the group." 



This concept can be interpreted in a quantitative 

 manner, by taking the following example from R. Gatewood 

 (Gaillard 1904, p. 50). Suppose that in a very long 

 trough containing water originally at rest, a plunger 

 at one end is suddenly set into harmonic motion and 

 starts generating waves by periodically imparting an 

 energy E/2 to the water. After a time interval of n 

 periods there are n waves present. Let m be the posi- 

 tion of a particular wave in this group such that m=l 

 refers to the wave which has just been generated by 

 the plunger, m={n+\)/2 to the center wave, and m=n to 

 the wave furthest advanced. Let the waves travel with 

 constant velocity C, and neglect friction. 



2-31 



