The other quantity usually evaluated in refraction data is 
the angle the crests make with the shore at the point of observa- 
tion near the shore. This angle is identically eaual to op which 
is the direction toward which the elemental crest is traveling. 
8p equal to zero designates a crest traveling directly toward the 
coast. These values can be plotted as a function of # and Ope 
The function, @p = O( 4 ,e,*), can thus be shown as isopleths of 
8, as a function of w and On" As# becomes large, ©, equals 
o,* asymptotically. These relations are defined in equation 
(12.33). © (#,0,*) will be called the direction function. 
The inverse of 0, = © CH en”) is also needed. That is, values 
of 6, isoplethed on ap 4,0, polar coordinate system, are needed. 
This inverse function is defined by equation (12.34) as 6,*= O*(,e,). 
From the isoplethed values of equation (12.34) it is possible 
to evaluate lr ( # yep) as given by equation (12.35). The function, 
['(#,0,), is the change of @,* per unit of change of 6, expressed 
as a dimensionless number in radians per radian or degrees per de- 
ereee r(p ,Op) is a measure of the crowding together of the power 
spectrum due to refraction and its significance will be discussed 
later. It is the Jacobian of the inverse of the direction function. 
Steps in wave refraction 
Given the functions described above and their definitions, three 
steps are required to find Eee Cae from [Ap Mt # 50,*) 1°. The 
function, eae ils could also be the power spectrum of any 
system observed immediately offshore in deep water. At this stage, 
then, the functions defined by equations (12.29), (12.32), (12.33), 
(12.34) and (12.35) are known. 
48 
