Davis and Zarnick 

 To illustrate this result, suppose that 



T7(Xj, t) = COS OJ^t . (9) 



The Fourier transform of this wave height is given by 



where /x^( • ) represents the unit impulse. The transform of t7(x2, t) is given by 

 We have for r](x2,t), then, the relation 



r]( 



00 



I I Ct) I x/g j ^t 



■^ 1 e e + e 



cos (w^t - o)^ x/g) 



(12) 



which is a well-known result from linear wave theory. 



The remarks of the previous section apply to wave height pairs in the sense 

 that the latter are both "effects" rather than "cause" and "effect." However, 

 with knowledge of wave height at one point, the corresponding time history at 

 another point can be determined by convolving the first wave height with the in- 

 verse transform of e" J'^''^'''/s as is well known from linear systems analysis. 

 This inverse transform is computed in the Appendix and yields the "weightii^ 

 function" or "impulse response" of water 



w(t) - a cos -^ a t' 



where 



and 



^ + -p- C( ar) 



+ a sin — a'' r' 



1 + -1- S(aT) 



(277xj 



1/2 



(13) 



(14) 



C(aT 



) = I dm cos -ly m-^ , S(aT) = dm sin — m'^ . 



(15) 



This weighting function is shown in Fig. 4, where 



512 



