III-71 



Most authors share the same general approach to the problem. They 

 regard the scattered wave as a superposition of plane waves emanating from the 

 surface and having the same frequency UJ as the incident wave. If they are to 

 satisfy the wave equation, these plane waves must have the form: 



ik(A. x+Ll v+vz) -iUJt , n p a . , 



p, =<Je ^ ^^ ^ where ^^ +|j^ +v^ = 1 (III-47) 



'^plane 



In particular, for given \ and la , we must choose the root v such that 

 the radiation condition is satisfied. This means that we desire: 



TT (i . e . , V negative) if v real 



arg V = arg . 1 - >t - H 



^ 



— (i.e., iv positive) if V imaginary (III- 48) 



The scattered wave is now written as a superposition of such plane waves: 



: f\ x+\J. y+z JlTx^^ia^ j - 



^^ J ..n ^ ik(^.x+My+^ Jl-X= -^^ - iUDt ,„^ ^„, 



d>^ d|a(Zli(X ,1-1) e V ^ \ J (III-49) 



Finally, we apply the boundary condition that the total pressure vanishes at the 

 surface, and obtain an integral equation for (X ,|j), given S(x, y): 



^ik(ax+Bi^YS(x,y))+ [[ dX d^0(X , u )e^4 ^''^ ^^^^^'^^^ ^ " ' ' -^')= 



(III-50) 



Different authors use different schemes for obtaining approximate solutions to 

 (III- 50). All are especially interested in the value of (a, 8), i.e. , the strength 

 of the scattered wave in the specularly reflected* direction. Most authors confine 

 themselves to sinusoidal surfaces, although their approaches are generally valid 



*The direction in which a plane wave is reflected by a plane interface. 



jarthur Sl.littlcJnir. 



S-7001-0307 



