Calculation of true thickness of the bed

12) A 10Hz P-wave traveling at 5km/s propagates for 1000m through a rock formation with an absorption coefficient of 0.2 dB/λ (that means it loses 0.2 dB per wavelength traveled by wave). What is the wave attenuation in decibel due solely to absorption? If the initial amplitude A₀ =1, what is the final Amplitude?

Solution:

Amplitude loss (or) gain in $dB=20\log_{10}\left(\frac{A_1}{A_2}\right) ---(1)$

 

Given that, 

absorption coefficient $x=0.4 dB/\lambda$

 

Substitute the x value in the above equation (1)

 

$0.4=20\log_{10}\left(\frac{1}{A_2}\right)$

 

$\frac{0.4}{20}=\log_{10}\left(\frac{1}{A_2}\right)$

 

$0.02=\log_{10}\left(\frac{1}{A_2}\right)$

 

$\frac{1}{A_2}=10^{0.02}$

 

$\frac{1}{A_2}=1.047$

 

$A_2=\frac{1}{1.047}$

 

$A_2=0.9547$

 

Therefore, the final amplitude is $A_2=0.9547$