Collecting terms and integrating 



■y/2 2 d f 7 ^ 2 



ry/2 2 d f 



-'o S sm z a, •'a 



This gives 



y 



S sin 2 a 1 •'a 



1"^ a 1 sin 2a 1 "J 

 a, |_4 " 2~ ~4 J 



sin 2 a da (242) 



2 S sin 2 M| 



(243) 



These equations are limited to the particular case of a long, straight 

 coastline, but may provide a first approximation for solutions on some 

 sections of continental shelves. Refraction diagrams would be required 

 to obtain exact solutions for irregular coastlines. If the waves travel 

 for long distances over a shelf, it may be desirable to use wave refrac- 

 tion equations in spherical coordinates such as the equations given by 

 Chao (1970) (see Sec. IV, 3). 



************** EXAMPLE PROBLEM 17************** 



GIVEN : A wave ray reflects from a straight shoreline at an initial angle 

 a j = ir/4 radians. The water depth at the toe of the shoreline slope 

 d s = 30 meters and the shelf at the toe of the shoreline slope has a 

 uniform seaward slope S 2 = 0.003. 



FIND : The distance the wave ray will travel away from the shoreline, and 

 the distance along the shoreline to the point where the reflected wave 

 ray will impinge upon the shoreline. 



SOLUTION: 



From equation (237) 



d d 



s s 



S sin 2 a. 



30 30 



r\ nr>7 ■ 2 It 0.003 



0.003 s.in* — 

 4 



Xp = 10,000 meters (6.214 miles) 



From equation (243) 



v 2d f, a, sin 2a, ' 



y s m i i 



Oj 1.4 2 



S sin 2 a, |_4 2 4 



sin 

 y 2(30) 



2 0.003(0. 707) 2 L.4 8 



y 



- = 25,700 meters (15.97 miles) 



2 



112 



