Graphs or overlays may be made up from this equation, or somewhat more 

 easily, from the solution for cc s 



t^,. ^^(K^Jll^ (12) 



One such overlay is sha^n in Figure 2a. 



If the orthogonal is being drawn from shallow to deep water (e.g. 

 past a shoal, or from shore seaT/ard), a 'may be defined as the angle^ 

 between the wave crest and the first contotir crossed (at which the velocity 

 Gj^ vrj.ll be less than 02)0 In this case Aoc will be turned in the opposite 

 direction, and a derivation similar to that above gives 



jia^^l^j2l£rLCc±Aa _ (2~K)- 1/(2 -K) ^- 8K ian ^a ' ' ^13) 



and 



An overlay based on this solution is shown in Figure 2b. This overlay 

 should be used whenever the orthogonal is being drawn from shallower to 

 deeper T/ater . 



If a derivation by the Isaacs' method is made for an orthogonal 

 being drawn from shallow to deep water the form of the solution for Aa is 



Aa^ K -tan a ' 



For a particular orthogonal, (X=CK-A(X but since Isaacs' original derivation 

 gave for A(X^\ 



this could never hold. 



Indeed, a measure of the accuracy of the new derivation is that for 

 equa-tion (11) and (13), or (12) and (I4) O:' very nearly equals or-/^cr , 

 arid, to the limit of accuracy of the overlays, oc' does equal c^^^c^ , 



The solutions presented will be applicable in all ordinary cases. 

 Hovvever, the application of these solutions is limited by the necessity of 

 kaoping ^a small so that the distance between contours becomes small as <x 

 approaches 90 degrees. Equation -(8) also represents a valid solution,, 

 however, and, for practical reasons of workable distances between contours, 

 it is desirable to use' this equation wherever c^ exceeds about 80 degrees. 

 For large values of ^, sin a and cos (X do not vary appreciably and the 

 values for a = 85° may be chosen as representative of the range betvreen 

 80° and 90°. Equation (8) maybe solved to give 



Aa=. ^^^^^^^ ^ (15) 



Z-^K -J Cos oc 

 26 '^ 



