APPENDIX E 



MISCELLANEOUS DERIVATIONS 



I, Refraction Diagrajns; derivation of method and design of template* 



A. Derivation . The parametric equations for an orthogonal given by 

 Arthar, Munk and Isaacs are: 



dx 

 dt 



C co.^ ;g . C sin^ ! f-ln^ |C . co., ^ ||. (,.„ 



These may be solved by simple separation of variables for a velocity field 

 which is a function of y alone. (Here ^ = 90° - df where flf= the angle 

 between a tangent to a contour and a norma], to an orthogonal, see 

 Figure E~lj. In particular , for a field which varies linearly with y, 

 G = Cq (l - ay), the solutions for x and y are 



(cos OC - cos QC ) 

 e 



y = I (sin a - slnCO (E-2) 



a sinoCg o 



which are the parametric equations of a circle of radius 



a sinOtg 



= 1^ 



sin^ (2Cg«) ■*• h and center at (E-3) 



a sin (2 flf^) 



"" = " a sin (2aJ ' ^ = a 

 The solution for j may be put in the form 



(E-l;) 



sinCC= sin (a^-AQj = (1 - AC. ) sinOC^ (E-5) 



^o 



= ^ sin Qf which is Snell's law. 

 - Co '^O 



From these, exact values of AOCand x at any point in the field may be 

 found , 



B. Template Design . Referring to Figure E-1, if from the point of inter- 

 section P of the mid-contour with an incoming orthogonal, a perpendicular 

 is dropped to an arbitrary point R, then the line RQ perpendicular to the 

 tangent to the raid-contour = sin Ot, x PR_( angle RPQ = <X, ). If another 

 line from R equal in length to C-[^/G2 x PR (where C-^ and ^2 ^^® ^^^ velocities 

 at contours 1 and 2 respectively) is drawn to intersect the tangent to the 

 mid-contour, the following relationships hold: 



E-1 



