(3) Shear Component . Since v is zero for long-crested waves, the 

 shear component resulting from the cross products uv is identically zero 

 everywhere, therefore 



XY 



n 



puvdz - 



-d 



(10) 



for wave propagation in the X-direction. Consequen 

 cipal stresses. It must be emphasized that S and 

 per unit width acting normal to and parallel to the 

 (see Fig. 19). Subsequently these principal radiat 

 formed to a more convenient coordinate system orien 

 coordinate) and shore-normal (x-coordinate) directi 

 making some angle, ct with the shoreline, Sxy is not 

 transformed coordinate system (Fig. 19). For addit 

 tions of these stress components see Svendsen and J 



tly, S^ and S^y are prin- 

 S are horizontal forces 

 wave crests, respectively 

 ion stresses will be trans- 

 ted in the alongshore (y- 

 ons. In general, for waves 



identically zero in this 

 ional physical Interpreta- 

 onsson (1976). 21 



b. Components Using Linear Wave Theory . For small-amplitude waves of 

 sinusoidal form, 



a cos(kx-a)t) 



(11) 



where 



k=^ • 



2Tr 



and the particle orbital velocities u and w are defined in the usual manner . 

 Inserting this definition and equations for ri , u and w into equation (7) for 

 S and equation (9) for S , using the linear theory dispersion relation co = 

 gK tanhkh, performing the integration, and time-averaging over one wave period 

 yields 



Sxx = E 



2kd 

 sinh2kd 



4 



^YY = ^ 



kd 





sinh2kd 





(12) 



(13) 



where E is the usual total energy density of the waves given by 



21SVENDSEN, I.A. , and JONSSON, I. G. , "Hydrodynamics of Coastal Regions," Den 

 Private Ingenirfond, Technical University of Denmark, Lyngby, 19 76 (not in 

 bibliography) . 



See, e.g., WIEGEL, R.L., Oceanographical Engineering, Prentice-Hall, Inc,, 

 1964, Englewood Cliffs, N.J., p. 15, for equations when equation (11) is in 

 terms of the sine function (not in bibliography) . 



69 



