116 T. I. Oblogina 



12 T{At'- {yjAt -yj At-T) -^At [At^l^-{At~T)^i^ + 

 1 



+ — [At^l^-{At-Tyi^]} + 8 {At^ {]/At -YAt-T) - 

 -At^Af^!^-{At-T)^i^] + ^At[At^i^-{At-Tyi^- 



_^ 1 [Jf7/2 _ (J i_ 2^)7/2] j^ 



At = t — aQr. 



To calculate the dynamic hodographs from these formulas we must 

 first find the values of At at which the function /(f — aQ^) has its first ex- 

 treme value. Assuming T — 0.03 sec and calculating values for the function 

 f (t — aQr) at a series of points, we find that when At — 0.01 sec it has its 

 first maximum, at a value of 0.857. 



We shall further assume that the velocity in the upper medium 

 l/oj = 4 km/sec, and that the bedding depths of the interface and the 

 tapering stratum are y = 1 km and jl — ^^ = 0.5 km respectively. 



We select observation points on the profile at intervals Ax = 200 m and 

 calculate the first onset times at these points, using the kinematic hodograph 

 equation. 



For each pair of values of x and ig "^^ compute values for 0q and 0' {0q) 

 according to formulas (21) and (18). We then calculate the functions u^ 

 and v^ according to formula (24) and obtain a value for the wave amplitude 

 at each point on the hodograph. 



Figure 10 shows the dynamic hodographs for u^^ and Vi calculated for twenty - 

 one observation points on the profile. The following regular features can be 

 noticed: the amphtudes of the diffracted wave are different at different 

 points of the profile; and the nearer the observation point is to the point 

 of contact between the hodographs of the diffracted and the head waves, 

 the greater the amplitudes of the diffracted wave. In the example given, the 

 point of contact has an abscissa x = 0.288 m. The vector of the displace- 

 ment [ui, fj) changes sign near this point of contact. The horizontal com- 

 ponent of displacement u^^ becomes zero at the point on the profile x = 0, 

 which is a projection of a point on the diffracting edge on to the profile. 

 This is obvious even physically, without calculation: the displacement 

 vector runs along the radius of a circle which is the front of the diffracted 

 ■wave; at the point on the profile x = 0, y — 1 km it is perpendicular to the 

 A' — axis, and consequently its horizontal component is nil (Fig. 10). 



