Problem 2.8
The magnitudes of period $ T $, wavelength $ \mathrm {\lambda } $, wavenumber $ \mathrm {\kappa } $, frequency $ f $, and angular frequency $ \mathrm {\omega } $ are important in practical situations. Calculate $ T $, $ \mathrm {\lambda } $, and $ \mathrm {\kappa } $ for 15 and 60 Hz for the following velocity situations:
- weathering, 100 and 500 m/s (minimum and average values);
- water, 1500 m/s;
- sands and shales, 2000 (poorly consolidated) and 3300 m/s;
- limestone, 4300 (porous) and 5500 m/s;
- salt, 4600 m/s;
- anhydrite, 6100 m/s.
Solution
The period $ T $ equals $ 1/{\it {f}} $, 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/":): \it{T}=0.067
s for 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/":): \it{f}=15
Hz 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/":): \it{T}=0.017
s for 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/":): \it{f}=60
Hz. Also, 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{\lambda} =V/f
, 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{\kappa}=2\mathrm{\pi} /\mathrm{\lambda} =2\mathrm{\pi} f/V
. Using these equations we get the values of 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{\lambda}
and $ \mathrm {\kappa } $ in Table 2.8a.
Table 2.8a. Magnitudes of 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/":): T
, 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{\lambda}
, 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/":): \mathrm{\kappa}
.
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For 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/":): \it{f} = 15
Hz
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For 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/":): \it{f} = 60
Hz
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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/":): V
(km/s)
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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/":): T
(s)
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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{\lambda}
(m)
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$ \mathrm {\kappa } (\mathrm {m} ^{-1}) $
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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/":): T
(s)
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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{\lambda}
(m)
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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{\kappa}(\mathrm{m}^{-1})
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| Weathering (min.)
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0.1
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0.067
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7
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0.7
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0.017
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2
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4
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| Weathering (avg.)
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0.5
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0.067
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30
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0.2
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0.017
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8
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0.8
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| Water
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1.5
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0.067
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100
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0.063
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0.017
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25
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0.25
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| Poorly consolidated sandstone-shales at 0.75 km
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2.0
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0.067
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130
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0.047
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0.017
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33
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0.19
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| Tertiary clastics at 3.00 km
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3.3
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0.067
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220
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0.029
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0.017
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55
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0.11
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| Porous limestone
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4.3
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0.067
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290
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0.022
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0.017
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72
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0.088
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| Dense limestone
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5.5
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0.067
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370
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0.017
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0.017
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92
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0.069
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| Salt
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4.6
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0.067
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310
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0.020
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0.017
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77
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0.082
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| Anhydrite
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6.1
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0.067
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410
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0.015
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0.017
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100
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0.062
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Continue reading
Also in this chapter
External links
find literature about
Magnitudes of seismic wave parameters
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