For example, Snell's Law is either assumed or proved from very 
Simple considerations. Also the shrinking in the wave length as 
the wave progresses into shallower water is not shown to be a con- 
tinuous process; that is, the length in deep water is Ly and the 
length in water of depth, H, is given by equation (12.10), but no- 
where in the theory is the exact profile along an orthogonal given. 
Luneberg started with Maxwell's equations and showed how the 
theory of geometrical optics for light or any other form of electro- 
magnetic radiation could be derived rigorously from the equations. 
In addition, the systematic approach which he used has permitted 
attempts to refine the theory to the level of physical optics. Con- 
siderable success along these lines has been obtained by Keller, 
Kline, and Friedman of New York University.* 
Similarly, it ought to be possible to derive wave refraction 
theory with the original hydrodynamic equations as a start. Were 
this done, the results would possibly indicate better relations for 
the wave height in the neighborhood of a caustic made possible by 
the consideration of higher order effects. 
One fundamental assumption of wave refraction theory is that 
the dimensions of the refracting bottom contour systems must be 
large compared to the wave length of the waves on the surface. As 
has been pointed out by Pierson [195la], in many practical cases 
*The author in this section is indebted to Professor Joseph Keller 
for his series of lectures on geometrical optics given at the Math 
Institute during the past year. Wave refraction theory for 
Gaussian waves has an analogue in the problem of colored light 
scattered in two dimensions passing through a medium with a con- 
tinuously varying index of refraction such that the index of re- 
fraction is a function of the wave length of the light and of 
only two space variables. 
38 
