Now, if the numerator of the left side of equation (227) is expanded, 

 the term can be written 



sin 2 (a + da) = — [1 - cos 2 (a + da)] 



= — [1 - cos(2a) cos(2da) + sin(2a) sin(2da)] 

 2 



= — fl - (cos 2 a - sin 2 a) (cos 2 da - sin 2 da) 

 2 



+ 2 sin a cos a(2 sin da cos da)] (228) 



But, where da -> 0, 



cos da -»■ 1 



sin da -> da 

 sin 2 da -> (da) 2 - 

 Then equation (228) can be written as 



sin 2 (a + da) = — fl - (cos 2 a - sin 2 a) (1) + 4 sin a cos a da] 



2 



= — [2 sin 2 a + 4 sin a cos a da] (229) 



and equation (227) becomes 



2 sin 2 a + 4 sin a cos a da _ (d + dd) 

 2 sin 2 a d 



(230) 



which reduces to 



1 + 2 cot a da = 1 + — (231) 



d 



2 cot a da = — (232) 



d 



Now, integrating along the wave ray from the shoreline to the point where 

 it turns parallel to the shoreline, taking dp as the water depth where 

 the ray is parallel to the shoreline and x~ as the distance from the 

 shoreline at that point, aj as the initial direction of the wave ray at 

 the shoreline (Fig. 35), and noting that a = tt/2 radians for a straight, 

 uniform coastline at the point where the wave ray turns parallel to the 

 bottom contour, 



110 



