13 



particles occupying an average position at distance z below the 

 surface are obtained by differentiating the displacements with respect 

 to t. 



L/j=->^=2^ sin 2i 



TF.=^^=-^cos27r(^|-|) (96) 



The average velocities over one-half cycle are then: 



and the maximum orbital velocities are: 



In these equations, a and 6 are the full amplitudes, while Tis the period 

 of one full cycle. Computation of these velocities for particular 

 values of h, T, and d, are effected by equations 5 and 6 or figures 

 3 and 4, using the value of L for the specified d and T from figure 2, 



These theoretical equations apply only to waves of small amplitude 

 passing over water of constant depth but "the numerical results show 

 that a variation in the depth will have no appreciable influence, 

 provided that the depth everywhere exceeds (say) half the wave 

 length" (5). 



At any instant the total energy per unit width of crest in one wave 

 length is obtained as the sum of the potential and kinetic energies. 

 The kinetic energy is the summation from bottom to surface of the 

 individual kinetic energies corresponding to the orbital velocities as 

 given by equations 9a and 96. 



The potential energy is computed (6) from the elevation or depres- 

 sion above the still water surface as: 



Ep=w I I z dzdx 



Jx=o Jz=o 



Integrating with respect to z under the surface curve, where y is the 

 particular value of z at the free surface, we obtain: 





I y^dx 



For a small wave of sinusoidal form described by the equation: 



h 

 2 





