where T, a factor of proportionality, is an equivalent depth to be 



determined. Since <t> = ap-^h-^ + /3P2^2' ^^^^ ^^^ "^^^ ^® valid for all 

 combinations of h^^ and h2 if 



aH-^ + /3H2 = Ta, and (10a) 



aE^<ip-^/P2) + /3H2 = r^. (10b) 



It can be readily shown that (6) and (7) can be written in the form 



9M/9t + fkxM + grV<l> - B = G, (11) 



b<t>/dt + V-M - C = , 



(12) 



where 



B = gVip^h-^Va + p^^&) , and 



C = Mj^'Va + M2'Vj3, 



(13) 



For the case of constant layer depth H^, ^2' ^^® °- ^^^ /3 are 

 constant and B, C vanish. In general these terms produce coupling of 

 the modes in the presence of bottom topography. 



Eqs. (10a,b) can be written in matrix form as 



Hi-r 



_(Pj^/P2)Hi H2-r 



(14) 



and hence the eigenvalues of T are the roots of the characteristic 

 equation 



