where 



Radiation and Dispersion of Internal Waves 



^= ^^"(^>l^^^(^>|; 



(24) 



Now p^, ui, vj can be readily eliminated from Eqs. (21), (22), and (23) (e.g., 

 first eliminate p^ in (21) and (22) by substitution of (23), then eliminate 

 (Buj/Bx + Bvj/By) calculated from (22) by using (21)), giving 



='is-rii-"' 



(D^ + N^) Wj 



,' - ^ d2 



h^ii-^' — )Pi 



Pi + DQ 



3z c. 



where 



V,2.ii+li, D' = U'A+V'^, N2 



' Bx2 By2 



3x 



By 



^0 \"-o 



(25) 



(26) 



(27) 



N(z) being known as Vaisala frequency. Further elimination of Pi or Vj can be 

 effected by cross differentiation and by using again (25) or (26), giving, after 

 some manipulation. 



(D^ + N^) [V^^ - -5^) Wj + D 

 •^0 



-$)(^-^)-^ 



(Lwj-DQ) = , (28) 



where 





If a horizontal free surface is present, represented by z= ^x,y,t), then 



Wj(x,y,0, t) = DC ■ 



The dynamic condition dp/dt = on z = ^ is linearized to give 



DPi - gPo *i = ^ on z = , 



which becomes, upon using (25) or (26), 



(29) 



(30) 



or 



D^ — - gV^-DD' 



dz 



T^U^l-^P^ 



Bz pQ 



w, = on z = 



p J = on z = 



(31) 



(32) 



555 



