x-direction, of complex amplitude a , and a wave propagating in the 

 negative x-direction, of complex amplitude a_, is given by 



-ikx ikfx-Ji) 

 C = a e + a e 



'o v^^If * 



(a e -a e ^ ■' ) 



y < X < I 



(12) 



with the complex wave number, k, given by 



k = 



/S=^ 



k /s^n 







(13) 



Equation (13) shows the wave number to be complex, i.e., to have a real 

 as well as an imaginary part. The solution of equation (13) should be 

 chosen such that the imaginary part is negative since this will lead to 

 a wave motion exhibiting an exponentially decreasing amplitude in the 

 direction of propagation as discussed in Appendix A. 



The general solutions for the motions in the three regions given 

 by equations (9) , (10) , and (12) show the problem to involve four unknown 

 quantities. These unknowns are the complex wave amplitudes a,,, a^, a+ , and 

 a_ and they may be determined by matching surface elevations and veloci- 

 ties at the common boundaries of the various solutions. Thus, we obtain 

 at x=0 from equations (9) and (12) : 



■ik£ 



(14) 



and 



n , -ik£, 

 (a^ -a e ) 



+ — 



and at x=£ from equations (10) and (12) 



(15) 



■ikl 



(16) 



19 



