where 



Miles 



Z|| = Z22= - (j/bc) cot ki, Z|2 = - (j/bc) esc ki , 



and 



(5.4) 



- Z 



12 



= Z22- Z,2 = (j/bc) taniki. 



The four-terminal network implied by (5.3) and (5.4) is sketched in 

 Fig. 3, wherein the arms (Z,, - Z,2) and pillar (Z,,) are inductive 

 and capacitative, respectively, for ki < tt (i less 

 length) . 



than a half- wave - 



The preceding results remain valid for a canal of arbitrary 

 (but constant) cross section S if h = S/b, where b is the breadth 

 of the canal of variable depth in the sense that the effects of the cross 

 waves (y- dependent modes) that are generated by a change in depth 

 are negligible in the shallow- water approximation (see Lamb, §176 

 for a qualitative argument and Bartholomeusz [ 1958] for a proof). 



Inserting the equivalent circuit for the canal between the 

 equivalent circuits for the harbor mouth (at x = 0) and the harbor 

 (at X = i), we obtain the equivalent circuit shown in Fig. 4a. Cal- 

 culating I2 and the corresponding voltage drop across Zp and 

 invoking (4. 3a) for the modal amplification factor, we obtain 



Z„-Z,. 



too'tv 



(0) 



WA,— 



C = /b4= 



C = A=r 



t 



1 Vo 



(b) 



Fig. 4, Equivalent circuit for harbor connected to coast through 

 canal: (a) general case; (b) Helmholtz mode (k A « 1, 

 ki « i). 



108 



