Relation between nepers and decibels
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| 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 2.17
The natural logarithm of the ratio of two amplitudes is measured in nepers. Show that one neper = 8.68 dB.
Background
By definition, if $ E_{1} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): E_{2} are energies, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \log_{10} \left(E_{2} /E_{1} \right) is the value of the ratio in bels. One bel = 10 decibels (dB), and energy is proportional to (amplitude)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{2} , so
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{dB}=10\log_{10} \left(E_{2} /E_{1} \right)=20 \log_{10} \left(A_{2} /A_{1} \right). \end{align} ()
Solution
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): N={} value measured in nepers, dB = same value in decibels. Then,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} N= \ln \left(A_{2} /A_{1} \right)=\left(\log_{e} 10\right)\log_{10} \left(A_{2} /A_{1} \right)=2.3026\log_{10} \left(A_{2} /A_{1} \right). \end{align}
Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{dB}=20 \log_{10} \left(A_{2} /A_{1} \right) .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} N=\left(20/2.3026\right)\ \mathrm{dB} = 8.686\ \mathrm{dB}. \end{align}
Continue reading
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| Tube-wave relationships | Attenuation calculations |
| Previous chapter | Next chapter |
| 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
- Far- and near-field effects for a point source
- Rayleigh-wave relationships
- Directional geophone responses to different waves
- Tube-wave relationships
- Attenuation calculations
- Diffraction from a half-plane
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