Far- and near-field effects for a point source

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	2
Pages	7 - 46
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 2.13a

Show that for harmonic waves of the form

${\begin{aligned}\phi =\left(A/r\right)\cos[\omega (r/V-t)],\end{aligned}}$

(2.13a)

the displacement $u\left(r,\;t\right)$ is

${\begin{aligned}u\left(r,\;t\right)=-\left({\frac {A}{r^{2}}}\right)\cos \left[\omega \left(r/V-t\right)\right]-\left({\frac {A}{r}}\right)\left({\frac {\omega }{V}}\right)\sin[\omega (r/V-t)].\end{aligned}}$

(2.13b)

Background

If we set $\mathrm {\chi } =0$ in equation (2.9a), we obtain the result $\mathrm {\zeta } =\nabla \mathrm {\phi }$, where $\mathrm {\phi }$ is a solution of the P-wave equation. Furthermore if $\mathrm {\phi }$ is independent of latitude and longitude, the wave equation reduces to equation (2.5c), and the solution of problem 2.5c shows that equation (2.13a) is a P-wave solution of equation (2.5c).

Solution

Since $\mathrm {\phi }$ in equation (2.13a) is a solution of equation (2.5c), it represents a spherically symmetrical P-wave and therefore the only displacement is along ${r}$. If we take the x-axis along ${r}$, equation (2.9d) shows that

${\begin{aligned}u\left(r,\;t\right)={\frac {\partial \phi }{\partial r}}=-\left({\frac {A}{r^{2}}}\right)\cos \left[\omega \left(r/V-t\right)\right]-\left({\frac {A}{r}}\right)\left({\frac {\omega }{V}}\right)\sin[\omega (r/V-t)].\end{aligned}}$

Problem 2.13b

Show that the two terms in equation (2.13b), which decay at different rates, are of equal importance at distance ${\mathbf {r=\lambda /2\pi } }$.

Solution

The two terms are of equal importance when the two amplitudes are equal, that is, when $A/r^{2}=\left(A\omega /rV\right)$ or $r=V/\mathrm {\omega } =\mathrm {\lambda } /2\pi$.

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Introduction	Partitioning at an interface
Table of Contents (book)

Also in this chapter

• The basic elastic constants
• Interrelationships among elastic constants
• Magnitude of disturbance from a seismic source
• Magnitudes of elastic constants
• General solutions of the wave equation
• Wave equation in cylindrical and spherical coordinates
• Sum of waves of different frequencies and group velocity
• Magnitudes of seismic wave parameters
• Potential functions used to solve wave equations
• Boundary conditions at different types of interfaces
• Boundary conditions in terms of potential functions
• Disturbance produced by a point source
• Rayleigh-wave relationships
• Directional geophone responses to different waves
• Tube-wave relationships
• Relation between nepers and decibels
• Attenuation calculations
• Diffraction from a half-plane

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