in which i = /^ and the amplitude functions ? and u are functions of x 

 only. These amplitude functions will generally be complex, i.e., consist 

 of a real and an imaginary part. The magnitude of the amplitude function, 

 |c| or |u|, expresses the maximum value, i.e., the amplitude, of this 

 variable. Only the real part of the complex solutions for n and U 

 constitutes the physical solutions. 



Introducing equations (7) and (8) in equations (1) and (2) the 

 general solution for the motion outside the porous structure may be ob- 

 tained as discussed in Appendix A 



-ik X ik X 



o 



r - a. e + a e 

 1 r 



J — -ik X ik X 



u=/— -fa.e -ae J J 

 /hi r 

 o 



a < 



(9) 



■ik (x-Jll 

 -' 



u = /r a^e 







■ik (x-£) 

 o 



> X > £ 



(10) 



in which a. is the amplitude of the incident wave, which without loss 

 in generality may be taken as real. The reflected and transmitted 

 complex wave amplitudes are a^. and a^, respectively. The magnitudes 



a^|, express the values of the physical 



of a-p and a^, i.e., | a^, | and 



wave amplitudes. The wave number, k =2Tr/L, is given by the familiar 



long wave expression 



(11) 



The preceding expressions show that we expect an incident wave, 

 a., propagating in the positive x-direction to coexist with a reflected 

 wave, a^., propagating in the negative x-direction in front of the 

 structure, x ^ 0. Behind the structure, x >_ £, only a transmitted wave, 

 a , is expected to propagate in the positive x-direction. 



The general solution for the flow within the structure is found 

 (App. A), by introducing equations (7) and (8) in equations (3) and (4). 

 The solution, which consists of a wave propagating in the positive 



