and 



Doctors 



L jf(t) } = / f(t) exp(-qt) dt 





 6 + i oo 



f(t) =-^r- /Z-)f(t)} exp(qt)dt (15) 



5 — i oo 

 h being a positive constant. 



The Fourier transform of Eq. (7) is first taken, giving 



2 

 $ - k $ = (16) 



zz 



where <J> is the Fourier transform of 4> , and 



2 2 2 , , 



k = w + u (17) 



The solution of Eq. (16) subject to the transformed bed condition, 

 Eq. (13), is 



4>(w, u; z, t) = A (w, u; t) . cosh {k (z + d )[ (18) 



Eq. (18) is now substituted into the Fourier transform of the 

 free surface condition, Eq. (12), giving 



A + 7 2 A = - — sech (kd) P (19) 



tt P t 



where P is the Fourier transform of p s and 



y 2 = gk« tanh(kd) (20) 



We now take the Laplace transform of Eq. (19) : 

 2 



Z 7 1 



(q + 7 ) L\A\ = sech(kd) 



q L{ P(w,u;t) (- P(w,u;0) 



The inverse Laplace transform is taken, using the convolution theo- 

 rem on the first term : 



42 



