for many density distributions p(2) and may be handled numerically for any dis- 

 tribution p(z). The result is a surface wave iiJg*^* ^""^ ^ ^^^ of internal waves 



CO 



/ IV i^^^\ each one being mutually independent and traveling with a phase 



n — 1 



n= 1 



velocity given by the eigenvalues of the problem. 



The same procedure is used for deriving an equation for k;*2> from (26) to 

 (28): The operator V/,- applied to (26) together with (28) yields 



aV 



Vp''* = -V/,-(p'''^ .p-^lWlT;,'!' 



' dtdz " ■ ■" V dt 



Differentiating with respect to z and t we have 



(35) 





i2„(2) 



-V 



d^p 



2 "" " ^Ap'''— ^P''-'^^h''' 



dz dt'^dz '' dt^dz^ '" dzdt dzdt "\ dt 



Combining the vertical component of (26) with (27) gives 



a2J2) ^2p(2) ^- 



dt^ dzdt dz 



dt\ dt 



dw^ 



which after applying Vi ^ and substituting p*2) in both equations above yields 



d 



-,4 (2) .3 (2) 



d w d w 



r 



2-1,2 



dt-dz 



dfdz 



Ml) ^ (1) 

 gu '\ p 



+ ^ P 



dt \ dt 



H ^ / p dzdl f^X dt ^ ^ / 



(36) 



Combining the boundary conditions (29) and (23) together with (35) and eliminating 

 _ / 1 \ 



in the inhomogeneous part by means of the vertical component of (26), we 



dp 

 dz 

 find 



.3„,(2) 



d'^W 

 dt'^dz 



'(1) ;i7r(l) 



■ + gV;,V2>= -V^^.^^ — +u- 'xufj' j+- yi^- 



tt ^ I dt 



grc 



(1)2 



S^h^[^''~ -I^;^^-V,<'l'jatz = 



P dt /'' 

 (37) 



and (31) remains unaltered 



„(2) 



= at z = H 



(38) 



14 



