Seismic waves in a quarter plane

1969 ◽  
Vol 59 (1) ◽  
pp. 347-368
Author(s):  
Z. S. Alterman ◽  
A. Rotenberg
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Finite Difference ◽  
Elastic Wave ◽  
Surface Waves ◽  
Seismic Waves ◽  
Initial Pulse ◽  
Surface Conditions ◽  
Quarter Plane ◽  
Boundary Lines

Abstract The equations for elastic wave propagation are solved by a finite difference scheme for the case of an elastic quarter plane. A point-source emitting a compressional pulse is located along the diagonal inside the quarter plane. Free-surface conditions are assumed on the boundary lines, so that the problem is nonseparable. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The effect of different finite difference formulations for the boundary conditions and the effect of different mesh sizes are studied. Various reflected volume and surface waves are identified, corner-generated surface waves are clearly seen in the seismograms and their particle motion is studied. The amplitude of the pulse observed at the corner is three times the amplitude of the initial pulse.

Seismic waves in a wedge

1975 ◽  
Vol 65 (6) ◽  
pp. 1697-1719
Author(s):  
Z. Alterman ◽  
R. Nathaniel
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Particle Motion ◽  
Seismic Waves ◽  
Wedge Angle ◽  
Compressional Wave ◽  
Elastic Wedge ◽  
Compressional Wave Velocity ◽  
Quarter Plane ◽  
The Waves

abstract The equations for elastic-wave propagation caused by an explosive point source are solved, by a finite difference scheme, for the case of an elastic wedge, with free boundary. Varying the wedge angle shows that the amplitude of the motion, at the corner, increases as the wedge angle is decreased. The results indicate that for wedges with angles varying from 0° to 180°, the amplitude decreases with decreasing β/α (shear- to compressional-wave velocity). The corner of the wedge generates surface waves and the elliptical particle motion in the waves is analyzed. The particle motion is elliptic and the major axes of the ellipses are inclined at half the wedge angle to the free surface. The surface wave travels to the corner from where it is “transmitted” and reflected. Surface waves are shifted by 180° - θ after transmission. For the case of a quarter plane, we get the same result as Alterman and Loewenthal (1970).

Hybrid modeling of elastic‐wave propagation in two‐dimensional laterally inhomogeneous media

Geophysics ◽  
1987 ◽  
Vol 52 (6) ◽  
pp. 765-771 ◽  
Author(s):  
B. Kummer ◽  
A. Behle ◽  
F. Dorau
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Hybrid Scheme ◽  
Finite Difference Techniques

We have constructed a hybrid scheme for elastic‐wave propagation in two‐dimensional laterally inhomogeneous media. The algorithm is based on a combination of finite‐difference techniques and the boundary integral equation method. It involves a dedicated application of each of the two methods to specific domains of the model structure; finite‐difference techniques are applied to calculate a set of boundary values (wave field and stress field) in the vicinity of the heterogeneous domain. The continuation of the near‐field response is then calculated by means of the boundary integral equation method. In a numerical example, the hybrid method has been applied to calculate a plane‐wave response for an elastic lens embedded in a homogeneous environment. The example shows that the hybrid scheme enables more efficient modeling, with the same accuracy, than with pure finite‐difference calculations.

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