Far- and near-field effects for a point source
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Series | Geophysical References Series |
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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 |
Contents
Problem 2.13a
Show that for harmonic waves of the form
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi =\left(A/r\right)\cos [\omega (r/V-t)], \end{align} } ( )
the displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u\left(r,\; t\right)} is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align} } ( )
Background
If we set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\chi} =0} in equation (2.9a), we obtain the result Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\zeta} =\nabla \mathrm{\phi}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\phi}} is a solution of the P-wave equation. Furthermore if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {r}} . If we take the x-axis along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {r}} , equation (2.9d) shows that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align} }
Problem 2.13b
Show that the two terms in equation (2.13b), which decay at different rates, are of equal importance at distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathbf{r=\lambda /2\pi}}} .
Solution
The two terms are of equal importance when the two amplitudes are equal, that is, when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A/r^{2} =\left(A\omega /rV\right)} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r=V/\mathrm{\omega} =\mathrm{\lambda} /2\pi} .
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Disturbance produced by a point source | Rayleigh-wave relationships |
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Introduction | Partitioning at an interface |
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