and from equations (159) and (160) 



E E + /V 1/2 - d} /2 \ 2 / 2d n 1/2 \ 2 d 1/2 



r t i l 2 1/ l \ 2 



E. E. \d 1//2 + d 1/2 / \d 1/2 + d 1/2 / d 1/2 

 ^ % \ \ 2/ \l 2/1 



which reduces to 



E + E , 

 r t 



E. 



(161) 



(162) 



Cochrane and Arthur (1948) extended Lamb's work to consider waves 

 approaching a shelf at varying angles of incidence. They give the ratio 

 of reflected wave height to incident wave height as 



H /d~ cos 8, - /d7 cos 0. 



r _ _1 1 2 2_ 



H. /dT cos 6, + VcL cos 0. 



% 1 12 2 



(163) 



for an abrupt change in water depth. The water depths d L and d 2 , 

 and the angles X and 2 , are defined in Figure 16. This equation 

 also applies to a single wave with a reflected component and a trans- 

 mitted component. 



For a given incident wave angle Q\, the value of 2 can be 

 determined using Snell's Law so that 



sin 0„ = (sin 0^1 — J (164) 



Equation (163), as written, applies to shallow-water waves; wave disper- 

 sion on the shelf is not considered. The solutions to equations (163) 

 and (164) are presented graphically in Figures 17 and 18, respectively. 



The ratio of transmitted wave height H to the incident wave height 

 is given by 



H . 2 /d7 cos 9 n 



(165) 



H 7 - /d" cos 0, + /d7 cos 0„ 



u 1 1 2 2 



or, alternatively, 



as before. 



ti t 



2 Jd~ cos 1 



ti i 



/d~ cos + /d7 cos 2 



H * 



H 

 1 + -£ 



H. 

 % 



H. 



^ 



69 



