38 



bending of the waves so that the crests tend to conform to the bottom 

 contours results from the reduction of wave velocity as the depth 

 decreases ; the inshore end of the wave travels at a lower velocity than 

 the portion in deeper water. The phenomenon is known as wave 

 refraction because of its similarity to the refraction of light and 

 sound. 



Consider the problem of the refraction of a wave approaching diag- 

 onally onto a straight shore line (all bottom contours parallel to 

 shore). It is evident that each section of wave crest passes through 

 the same sequence of angles and velocities and that the time history 

 of the waves passing all points at equal distances from the shore will 

 be the same except for the phase lag. The problem of determining 

 the angle of the crest v/ith the shore line at any point may be analyzed 

 by a method suggested by C. K. Bagby. 



Consider points A and B, figure 14, separated by an infinitesimal 

 distance ds. The velocity at B exceeds that at A and the wave front 

 swings aroimd as it advances during a short time interval dt. The 

 change in angle of the wave front is given by 



, . , dCdt dC dy J, 



t^n(-da) = -da=-^=-^-f/t 



but: ^^^^^ " ^^^ 'I'f^~^ ^^^ "' ^'^^ *^® 



differential equation may be expressed as 



cot a (ux=-p 



log sin a=log (7+ constant 



The boundary conditions are expressed in terms of the direction, ao, 

 and velocity, Co, in deep water as: 



constant = log sin ao— log Co 



C 



sin Q:=sin <Xo~n' (47) 



a=sm" 



'(^^sina„) 



]jt is noteworthy that the change in angle is not dependent upon the 

 slope of the bottom, provided that the bottom contours are parallel 

 to the shore and the slope gradual. 



An auxiliary diagram will be convenient in determining the velocity 

 as a function of depth and period (see fig. 3). 



The crest length will increase as a result of the refraction, the increase 

 being in the ratio 



cos {ao—a) 



