1 9 1 - 



E = -^^ga'^ = -^gH^ 



(14) 



for linear theory. Use of the ratio n between group celerity, c , and wave 

 celerity, c, defined as ^ 



= _& = I 

 c 2 



1 + 



2kd 



sinhkd 



(15) 



gives the commonly found forms 



^XX = ^(2n - 2) 



S^Y = E(n - |) 



(16) 

 (17) 



where the units of radiation stresses as force per unit length are now obvious, 



1 



In deep water, k.d>>l and n ^ -^ so that 



S =-iE 

 ^XX 2 ^ 



Syy = o 



(18a) 

 (18b) 



In shallow water, kd<<l and n ->■ 1 so that 



XX 



S =^E 



(19a) 

 (19b) 



c. Coordinate Transformations . The usual strength of material method 

 of plane stress analysis is applicable to transform the principle stresses 

 into equivalent stresses on any other orthogonal coordinate system. The most 

 convenient is a y-direction parallel to the coastline and positive to the 

 right facing seaward, and the x-direction normal to the coast and positive 

 toward the shoreline (Fig. 19). From either the Mohr's Circle or the stress 

 element shown in Figure 20, the mathematical transformation equations become 



c . ^XX ^ ^YY . ^ /XX ^YY , „ 



S = ( 7: ) + ( ^ ) cos 2a 



XX 2 2 



(20) 



70 



