(2) 

 The second term S can be shown equal to 



,(2) 



-pw^dz 



(5) 



Longuet-Higgins and Stewart (1964) considered the time mean flux of vertical 

 momentum across a horizontal plane balanced by the weight of water above that 

 plane. This is equivalent to integrating and time averaging the vertical mo- 

 mentum equation over the Stillwater column. This term is obviously generally 

 negative. The third term S^' poses some difficulties when ri is below z = 0. 

 To get around this, Honguet-Higgins and Stewart (1964) assumed the pressure to 

 fluctuate in-phase with the surface elevation, i.e., a hydrostatic distribution 

 for pressure, p. This gave 



.(3) 1 ; 



^XX =2PS^ 



(6) 



which is also generally positive. Combining gives 



^XX "^ 



p(u^-w2)dz + -2pgn^ 



J 

 -d 



(7) 



for the principle radiation stress. Equation (7) is completely general. 

 Further simplification depends upon what classical wave theory is used for the 

 three variables involved, n, u, and w and the averaging time employed. How- 

 ever, in deep water the particle orbits are almost circular so that the inte- 

 gral term approaches zero. Conversely, in shallow water u>>w so that the hori- 

 zontal velocity stress dominates the integral term. 



(2) Transverse Stress Component . Analogous to equation (1) the 

 transverse radiation stress, S , is defined 



YY 



(p+pu^)dz - 



p dz 



(8) 



-d 



-d 



where v is the wave orbital velocity in the YY-direction parallel to the wave 

 crests. For long-crested gravity waves, V equals zero. The transverse stress 



S thus becomes 



YY 



-pw^dz + -2pgri^ 



-d 



(9) 



if the same assumptions are the same as for S 



XX 



68 



