LS SE 
Consider the short crested Gaussian sea surface given by 
equation (9.47) and apply equations (10.41) and (10.42) so that 
the sea surface can be studied as a function of x'. The result 
is equation (10.47). For convenience © equals zero should be 
picked to be the dominant direction of travel of the apparent 
crests, and the direction © equals zero is therefore along the 
x axis. The angle, 0©*, then measures the angle between the x 
axis and the line of flight. The observed spectral wave number, 
Vo? then depends upon the spectral frequency and the cosine of 
the difference between 6 and 6*. Angular directions above and 
below @©* are determined by an angle, 85° 
The procedure is now to transform the / ,@ polar coordinate 
system and the integration over [a5 (um ,0)1° to a v.40, polar 
coordinate system and an integration over a new function 
LA. ( 1598590") 1°. The variables, v, and ©, are defined in terms 
fo) 
of # and © by equations (10.48) and (10.49). The inverse trans- 
formation which defines # and © in terms of ve and oF is given by 
equations (10.50) and (10.51). The Jacobian of the transformation 
is given by equation (10.52). Substitution of (g y,/cos 5 one for 
@ and ©, + @*for © in equation (10.47), and the use of the Jacob- 
ian to preserve the mapping then yields equation (10.53). The 
Jacobian is needed because the function [A> (m5) ]° has been dis- 
torted by the mapping, and in order to preserve the total power 
in the wave system it must be amplified for low values of vie and 
cos 6, and decreased for high values of v.. Stated another way, 
an increment, Am , maps into a much smaller Av if it is at the 
ie) 
= On 
