This represents an expenditure of 



39,195 ^^ X 3600 ^ = 14.11 x lo'' J 

 s h 



of energy each hour on each meter of beach (31.72 x 10^ foot-pounds 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. Sverdrup and Munk (1947) provide an excellent 

 discussion of this subject. Quoting from Technical Report No. 2, by the Beach 

 Erosion Board (U.S. Army, Corps of Engineers, 1942): 



"As the first wave in the group advances one wave length, its 

 form induces corresponding 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 subse- 

 quent 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+l)/2 to the center wave, and m=n to the wave furthest 

 advanced. Let the waves travel with constant velocity C, and 

 neglect friction. 



"After the first complete stroke one wave will be present and 

 its energy is E/2. One period later this wave has advanced 

 one wave length but has left one-half of its energy or E/4 



2-29 



