Radiation and Dispersion of Internal Waves 



is the Heaviside step function. By applying the Fourier transform (denoted with 

 a tilde over w) with respect to x, and the Laplace transform (denoted with a bar) 

 with respect to t , as defined by 



w(k, z, s 



/•OO - 00 



) = e-"* dt e-^"^" w(x,z,t) 



•'0 J - 00 



dx , 



(72) 



Eq. (71) becomes 



w(x,z,t)= (— - 2/3jG(x,z,t), 



4- 2/^f-- kMl + 

 Ldz^ dz \ o2 



8(z) 



(73) 

 (74) 



The solution of G of (74) which is continuous at z = and satisfies the required 

 jump in dc/dz across z = o is 



G = - 4 



^/3z-ml z I 



2 (s + iaj)m 



and 



m(k, s) = 



,.^,.,. 



1/2 



(75a) 



(75b) 



The function m(k, s) has two branch points in the complex k -plane. An appro- 

 priate branch cut is to be introduced such that m-|k| ask->±co along the real 

 k -axis in order to ensure the convergence of the Fourier inverse integral. By 

 using the translation s j = s + iw in the Laplace inversion, we have 



G(k,z,t) = ^ e^^-i'^* 



^ ' ' ' 27T1 



/^•■' 



ds J 

 F(k,z,Si)— - , 

 ^1 



(76) 



where 



F(k,z,Sj) = exp[- |z| m(k, Sj- ia))]/[- 2m(k, Sj- ioj)] 



and r is an integration path parallel to the imaginary Sj-axis in the right half 

 plane. The large time solution of G(k,z, t) can be evaluated by applying the 

 Tauberian theorem which states that 



lim G(k,z,t) = e^'-'"-"^ lim F(k,z,Si), 



(77) 



provided that a certain necessary and sufficient condition is satisfied (which can 

 be verified separately). Now for small positive s ^ the branch points of m are 



567 



