DIFFRACTED SEISMIC WAVES 113 





|2 + ^2' 1 |2_|_,^ 



to the chord equation 



In the elHptical region, by substituting the variables according to the formulas 



_ s ,^^'>/i^v(f:+'/') (13) 



|2 + ^2 . 1 ^2 + ^2 



it is reduced to the Laplace differential 



'^^"+^-0. (14) 



9a^ ' 9tj^ 

 Using this and introducing the complex variable 



"1" ^ ..2 I /.. 7,\2 ' U"-*-' 



" .x2 + (^_/j)2 ;»;2 _|. (^ _ /^)2 



we shall seek the solution in the region ABA^CO^ A (Fig. 9) which is filled 

 mth a diffracted wave in the form 



n = Re^{0,)^ (16) 



where ^{6^ is the analytic function in the region to which the region 

 ABA^CO^A passes; and R^0{6^ is the real part of the function. 



Let us see what happens to points on the circumference ABA^CO^A 

 after conversion (15). 



It can be readily seen that O^ = a^ corresponds to the point A {x = t/aQ, 

 y = A} ; ©0 = Oj to the point B {x =^ (^/c^o) ^i^ <^o ' 7 — (*/<^o) ^^^ ^o} '■> 

 0Q = — Qj to the point A^ {x = —t/aQ, y — h} and 0q = oo to the point 

 Oj_{x=0, y=h]. 



The lower semicircle is converted into a lower plane, while the whole 

 circle O^ABA^ CO^ passes by means of the conversion indicated into the plane 

 of the complex variable Oq with a cross-section along the real axis Qq^ ~ ciq . 



The boundary condition (8) is written in the form 



Re {V a,' -6,^0' (do)} =0. (17) 



when we use the variable 6q. 



Applied geophysics 8 



