114 T. I. Oblogina 



Making use of the fact that in the region ahead of the front of the plane 

 Avave (5) 99q = 0, in the region ACD (pQ = 2'yQ and in the region to the left 

 of the hne NBA^CDM (p^ = y^, we introduce the boundary conditions for 

 the function 0'{Q^. 



In the interval Oq < 0q < oo into which the segment 0-^A^ passes, Q^ 

 always satisfies the inequality 0q > Cq. In the boundary equation (17) the 

 radical l/(ao~^o) ^^^ have an imaginary value, and therefore to fulfil this 

 boundary condition the condition ImO' {B^ — must be satisfied. 



It is sufficient to solve the problem for the upper half -plane. At the point 

 Qq = a^ the function 0'{Q^ will have a pole with its main part — [iyo/^ 

 (0Q — %)]. Let us now construct such a function so that its material part is 

 converted to zero. We multiply the function ^'{d^ by the radical }/{ao~^o)- 

 For the upper half -plane when Qq > a^ we choose a minus sign ; that is, 

 Im]/{aQ-Q^<Q. It will be seen that R^{0' {Q^]f{aQ — d^ =0 on the 

 entire material axis. 



Making use of this, and also of the fact that at the point ^q = % there 

 will be a pole with the main part we have indicated, we obtain 





The components of the function required are equal to 



u = R40'{Q,)^ 



. = i?.(m)f) 



(19) 



Formulas (18) and (19) give the solution to our problem. We thus find 

 the displacements in a region filled with a diffracted wave. 



The Dynamic Hodographs — To determine the dynamic hodograph 

 of a diffracted wave we must find an expression for the components 

 of the displacement of this wave near its front. First we find asymptotic 

 formulas for the displacement of the diffracted wave in the vicinity 

 of its front for the case of a Dirac pulse as the shape of the incident 

 wave, then we convert to an alternating smooth pulse for the displace- 

 ment. 



For the region near the front of the diffracted wave when r-> ^/oq? '^^ 

 obtain from (15) and (19) the following asymptotic expressions for the hori- 



