Height equation 



17. The conservation of energy is given by 



3V 

 3E 



!+-!— e(~V. + (C).~||+S. . -5-^= , i = 1, 2 (18) 



where 



2 

 E = wave energy density and is equal to pgH /8 



p = density of water 



H = wave height 



S- • = "radiation stress" 



as defined by Longuet-Higgins and Stewart (1964). Writing Equation 18 in 

 terms of H and assuming a steady-state wave field (3E/3t = 0) gives 



|^'(U + C cos 9) + |5 (V + C sin 9) + 4t" <U + C cos 9) 

 3x g 3y g 2 3x g 



+ | ■§- (V + C sin 



2 3y g 



H /- 3U - 3U - 3V - 3V] _ MQ , 



2 V xx 3x yx 3y xy 3x yy 3y/ 



The dimensionless radiation stresses are given by 



S 



(20) 



a xx = nT =(2" -|)cos 2 9 + (n -i)sin 2 9 



= !pi = ( 2 n -^)sin 2 9 + (n - |) cos 2 9 (21) 



a 

 yy 



a =a = !^ = !p = ^sin 29 (22) 



xy yx E E 2 



where 



n - i = I [l + _^M 1 ( 2 3) 



n C 2 L sinh (2kd)J V ; 



Finite difference equations 



18. Equations 3 and 5 are substituted into Equation 17, and all terms 

 are written in centered finite difference form. 



14 



