the transmission coefficient is therefore given by 



T = -i- + 0(k£)^ . (35] 



In obtaining the reflection coefficient from equation (31) it is 

 seen that the real part of the denominator, S+n^, may be neglected as 

 being small relative to the imaginary part -i (f + 2n/kQJl) since 

 k i << 1. In the numerator, however, the term S-n" must be retained 

 since it is of the same order as f unless it is assumed that f >> 1. 

 Thus, the simplified solution for the reflection coefficient is obtained 

 from equation (31) as 



i / f 



which shows that R = X/(l+A) if f > 1. Thus, for f > 1, which is 

 usually the case, the transmission and the reflection coefficient are 

 independent of the value of the coefficient S. This supports the finding 

 discussed in Section II. 1 where it was concluded that the value assigned 

 to S was of minor importance. 



For later use, the simplified expression for the horizontal velocity 

 within the structure is found from equation (32) to be 



'^' ^ 0(k£)^ , (37) 



/— 1+A 

 a /^ 







i.e, the velocity within the structure is identical to the velocity 

 associated with the transmitted wave. 



The simplified formulas derived here are limited to small values 



of nk^l by virtue of the nature of the approximation. The equations 



for T and R (eqs. 35 and 36) may be shown to be in good agreement with 



the general solutions presented in Figures 2 and 3 for values of 



nk £ < 0.2. 

 o 



The simplified formulas for the transmission and reflection 

 coefficient may be derived from very simple considerations. Thus, if 

 an incident long wave of amplitude, a^, is considered normally incident 

 on a structure the maximum free surface elevation in front of the 

 structure may be taken as 



k^l = ^l^R) ^i (38) 



26 



