crests would be curves in the refraction zone. Such a solution 
would be exact (in a linear sense) everywhere, and would agree 
well with reality until non-linear effects near the breaker zone 
caused it to fail. Apart from the difficulty of evaluating the 
result, (and it is difficult enough for a pure sine wave), very 
few linear sloping beaches are found in nature. As soon as the 
depth becomes a complicated function, wave refraction theory must 
be used. 
The solution to the wave refraction problem in the transition 
zone is found in practice by graphical methods. The orthogonal 
method as presented by Johnson, O'Brien, and Isaacs [1948] and most 
recently by Arthur, Munk and Isaacs [1952]* is the best procedure 
because errors are not cumulative and the method discovers caustic 
curves. It would now appear that it is possible at a sufficient 
distance beyond the caustic to use the usual formulas for the value 
of KD based on the separation of the orthogonals at the point of 
In general, for a pure sine wave in deep water, the crests in 
the transition zone are curved. All of the systems discussed so 
far consist of elemental straight crests. The equations for the 
crests in the transition zone are very complicated and they have 
rarely been formulated mathematically except for extremely simple 
bottom configurations. Some examples in which the crests can be 
found explicitly (since the orthogonals are given) can be found in 
papers written by Arthur [1946], Pierson [195la] and Pocinki [1950]. 
*The abstract of the paper by Arthur, Munk and Isaacs[1952] can be 
found in the American Geophysical Union's vrogram for its May 5-7, 
1952 meetings in Washington. A preliminary copy provided by the 
authors shows that errors in previous methods can be eliminated 
by a more refined application of Snell's law. 
40 
