•^n ~ W/ f^n ^*" "^n "*" ' ^n '^os'^n^ exp [i(cjt-kx)] , 

 where 



The components of motion and stress in the n^'^ layer are, therefore, 



'^ ^n " ^n ^°^ Pn ~ ' ^n ^^" Pn "^ ""jSn *^n '^°^ ^n ~ '^ ""iSn ^n ^'" ^n ' 

 cWj^ = -i r^,^ A,^ sin p,^ + r^,^ B,-, cos P^, + i C^ sin q,^ - D,^ cos q^^ , 



"'n " ^n^'Yn"' ^ '^n ^°^ Pn ~ ' -^n^l'n"^ * ^n ^'" Pn 



+ ^n Tn % Cj^ ^os q,^ - i p,^ 7n >-j3n Dn sin Qn , 

 [n = 'Pn Tn 'an \ ^in p,^ - p^ 7„ r^n ^n ^os ?„ 



-iPn(T„-l) Cj^ sin q,^ +Pn*Tn~l' ^n ^os q^ • 



In the above, c is the horizontal phase velocity (c = w/k), Uj^ and Wj^ are the horizontal and 

 vertical components of particle velocity, o is the normal (vertical) stress and t^ is the 

 tangential (horizontal) stress. 



By separating 0q into an incident and reflected wave it is easy to show that the 

 plane wave reflection coefficient R is given by 



R = (Ao-Bo)/Ao + Bo) • 



For convenience set the value of Ag = 1 . The three interface conditions at the water/ 

 sediment layer 1 interface can be written as 



rQ,QBQ = rQ,] Bj-D| (continuation of W) 



-Pq = Pj(7j-1)A] +Pj 7] roj Cj (continuation of ct) 



= -p J 7j r^, J Bj + P](7j-1) D| (continuation of t) 



Divide the last two above equations by pj and form the matrix of coefficients of Bq, Aj, 

 Bi,Ci,D, 



_B0_ ^\_ _^\_ _^\_ _Dl_ 



>-aO -'a\ 1 =0 



(7]-l) 7j r^i = -PqIPi 



-7] r^i (71-1) = 



54 



