1506 
we shall illustrate by means of a simple example how these 
approximations must fall down for angles of incidence near the 
critical angle. 
5.2 The example is sketched in Figure 4. Path ABC is the normal 
path of a reflected ray, making equal angles of incidence and 
reflection ( 4] ) with the bottom. The angle @ would be used in 
calculating the phase shift E for this wave. Path AB'B'C is the 
path of a ray entering and leaving the bottom at the critical 
angle of incidence ( 6. ), and traveling just within the bottom 
from B' to B", 
We seek to calculate the difference in time of 
propagation along the two paths and also to show that AB'B"C 
is the least time path for all rays from A to C which touch the 
bottom. To this end we make the following constructions: In 
order to show that AB'B is a least time path, consider a varied 
path ABB. Extend AB and drop a perpendicular bD to it from b. 
Then we have: 
( 
SIA &. = Bd = ao, 
B’b c 
This expression shows that propagation from B* to d 
(58) 
at velocity C, takes the same time as propagation from B* to b 
at velocity Co. Hence the difference in time between the two 
wo 30 = 
