30) A 10 Hz seismic wave Propagates for 40 km through a material with a P-wave velocity of 5 km /sec and Quality factor (Q) of 100. The percentage of the initial amplitude retained in the attenuated wave________ (Round off to 1 decimal place)
(Thanks to Chandrasekhar, ANU)
Solution:
Given that Frequency (f) = 10 Hz
Distance from the source (r) =40 km = 40,000 meters
Velocity of the P-wave= 5 km/sec= 5000 m/sec
Quality factor (Q) = 100
The Percentage of initial amplitude retained in the attenuated wave $\frac{A}{A_{0}}$=?
We know that the Damped amplitude of a seismic wave at a distance of r from its source is
$A = A_{0}e^{-\frac{\pi}{Q}\frac{r}{\lambda}}$
$\frac{A}{A_{0}} = e^{-\frac{\pi}{Q}\frac{r}{\lambda}}$-------------------------------(1)
Here $\lambda$ can be calculated from the equation $\lambda=\frac{c}{f}$
$\lambda=\frac{5000}{10}$
$\lambda=500m$
Substitute all the given values in the equation (1)
$\frac{A}{A_{0}} = e^{-\frac{\pi}{Q}\frac{r}{\lambda}}$
$\frac{A}{A_{0}} = e^{(-\frac{3.14}{100})(\frac{40000}{500})}$
$\frac{A}{A_{0}} = e^{-\frac{3.14\times4}{5}}$
$\frac{A}{A_{0}} = e^{-2.512}$
$\frac{A}{A_{0}} = 0.0811$
$\frac{A}{A_{0}}$=8.11%
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