GATE-2018 (24) :Wyllie – Time average equation

 Q) Over the distance between two seismic recording sites at different ranges from a seismic source, seismic waves have been attenuated by 5dB. What is the ratio of the wave amplitudes observed at the two sites?

 (Thanks to Chandrasekhar, AKNU)

Solution:

The Dynamic range is an expression of the ratio of the largest measurable amplitude A_max to the smallest measurable amplitude A_min in a sample function.

 Dynamic range is normally expressed in the decibel (dB) scale used to define electrical power ratios.

$dB=10 log(\frac{P_1}{P_2})$

 we know that $ power \propto (Amplitude)^2$

 

  then from the above equation we get

  $dB=10 log(\frac{A_1}{A_2})^2$

  $dB=20 log(\frac{A_1}{A_2})$----------------------------------------(1)

 

 Now using this equation, we get our solution for the problem

 

Given that

 Dynamic range = 5 dB

 Then what is the ratio of the wave’s amplitudes?

From the equation 1

  $dB=20 log(\frac{A_1}{A_2})$

 

  $5=20 log(\frac{A_1}{A_2})$

 

  $log(\frac{A_1}{A_2})=\frac{5}{20}=\frac{1}{4}$

 

  $\frac{A_1}{A_2}=10^{\frac{1}{4}}=1.8$

Reference: An Introduction to Geophysical Exploration by Philip Kearey, Michel Brooks, and Ian Hill.