DIFFRACTED SEISMIC WAVES 111 



We choose a system of co-ordinates so that the ::c-axis runs along the 

 interface and the j-axis up towards the upper medium (we are considering 

 the plane problem). 



Along the half -line x > 0, y = h let a. cross-section be taken, the edges 

 of which are firmly attached; that is to say, the displacements on them 

 will be zero. Such a cross-section will be an approximate representation 

 of a thin tapering stratum with high velocity in seismic prospecting. 



Let t <. and let the following wave system 



n = nfiit - GqX sin ccq - a^y cos ^o), 7o = — • (5) 



Gq cos (Xq 



<Pi = A/lit -a^x)~Byf^'{t~a^x}, A = ^^li«_ , B = 2a^. (6) 



Oo Qx COS Oq i. \ / 



where Oq = arc sin Oj/ao t>e propagated along the interface in the direction 

 of increase of x. 



The potentials cpQ and (p-^ satisfy the wave equations in the upper and 

 lower media. As has been shown in (^\ formulas (5) and (6) give the local 

 representation of a head and a shear wave near the interface. At a distance 

 from the boundary we can regard (5) as an ordinary plane wave. We shall 

 give the function f^ in the form 



\ i when 5 > 0. 



Since the components of the displacements are expressed by the displace- 

 ment potential as partial derivatives of this potential along the corresponding 

 co-ordinates, the boundary condition on the boundaries of the cross-section 

 X '> 0, y = h \% expressed by the equation 



Now let the wave (5) meet the edge 0^ of the cross -section at a moment 

 of time f = 0. Fig. 9 shows the wave fronts and the values of the displacement 

 potential in front of and behind the wave fronts at some moment of time 

 i > 0. It is required to find the diffraction disturbance, at a moment of 

 time ^ > 0, which is concentrated inside the region bounded by the contour 

 ABA^CA (Fig. 9). 



With high values of h, the problem as formulated is in essence the classic 

 problem of the diffraction of an ordinary plane wave from a rectilinear edge 



