a certain direction of forward progress at the point and a wave 
length determined by pu I and H. Finally, the crests will have a 
new amplitude and phase at that point, which can be determined from 
tracing the family of orthogonals near the point of study. These 
features are all incorporated in equation (12.24). Any is the new 
height at the new point of observation which can be determined from 
A by wave refraction theory and equation (12.25). Op is the new 
direction of progress of the crests. 6 is a phase lag due to the 
Slowing down of the crests. 
Equation (12.24) does not hold everywhere in the transition 
zone. In fact it holds only at one point; namely, Xp = 0; Ya = 0. 
However, in the vicinity of the point, the equation approximates 
the local state of affairs. The degree of approximation is somewhat 
crude but actually to develop the formulas with curved crests which 
would apply to greater distances away from the point of observation 
would be far too difficult. 
The problem of the refraction of a short crested Gaussian sea 
surface can be solved by showing how it is possible to extend the 
application of the refraction data already obtained for pure sine 
waves to an infinite sum of infinitessimally high sine waves in ran- 
dom phase. It can be done easily to the degree of approximation 
just described above. In this way, the sea surface is approximated 
in the vicinity of the point under study by a Lebesgue Power Integral 
quite similar to the one discussed above and in previous chapters. 
A wave record taken as a function of time at the point of interest 
will be quite accurately given but the slight curvature of the 
individual crests in the neighborhood of XpoVp will not be represented. 
