Effect of reflector curvature on a plane wave

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Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 6 181 - 220 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

Problem 6.4

Redraw Figure 6.4a for a plane wave incident on the reflector, and explain the significance of the changes which this makes.

Background

Figure 6.4a assumes a point source at the surface, whereas for an incident plane wave in Figure 6.4b the source is at infinity. The plane wave reflected by the plane reflector produces a plane wavefront (R), i.e., the reflected wavefront has zero curvature. For a point diffractor the virtual source is at the diffractor and the wavefront has the maximum curvature (D). The curvature of the wavefront from the anticlinal reflector (A) is intermediate between those of ${\displaystyle R}$ and ${\displaystyle D}$ and the curvature of the wavefront from the synclinal reflector (S) is negative (assuming the center of curvature is below the surface).

Figure 6.4a.  Reflector curvature effects for point source.
Figure 6.4b.  Reflection curvature for incident plane wave.

Solution

By Huygens’s principle (problem 3.1), a diffracting point acts as a point source whenever a wave falls upon it; hence, the diffraction response ${\displaystyle D}$ to a plane wave (Figure 6.4b) is the same as that in Figure 6.4a. A plane wave incident on a plane reflector gives rise to a reflected plane wave ${\displaystyle R}$. For a plane wave incident on an anticline or syncline of circular cross-section of radius ${\displaystyle r}$, we can use the mirror formula, namely,

{\displaystyle {\begin{aligned}1/f=2/r=1/u+1/v,\end{aligned}}}

where ${\displaystyle f}$ is the focal length ${\displaystyle =r/2}$, ${\displaystyle u}$ and ${\displaystyle v}$ are the distances of the source and reflected image from the apex of an anticline or trough of a syncline; ${\displaystyle r}$ is positive for a syncline and ${\displaystyle u}$ is infinite for a plane wave, so ${\displaystyle v=r/2}$. For a syncline, the reflected wave comes to a focus a distance ${\displaystyle r/2}$ above the trough. For an anticline, a virtual image (see problem 4.1) is at ${\displaystyle r/2}$ below the high point.