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

118. An isotropic, homogeneous sample is subjected  to uniaxial stress along the x-axis. The absolute  value of the axial strain is found to be thrice the  normal strain. The ratio (Vp/Vs) of the P wave  and S wave velocity is 

1. 1.5 

2. 1.75 

3. 2.0 

4. 2.25

Solution:

Poisson's ratio is defined as the negative of the ratio of the lateral strain to the axial strain for a uniaxial stress state.

$Poisson's ratio (ν ) = \frac{normal strain}{axial strain}$

From the above problem, axial strain = 3 normal strain

  
ν =$\frac{1}{3}$

$\frac{1}{3} = \frac{normal strain}{axial strain}$

The relation between velocities of P and S-waves and Poisson's ratio is given below 

$\frac{V_{p}}{V_{s}} =\sqrt\frac{2(1-ν)}{(1-2ν)}$  

$\frac{V_{p}}{V_{s}}=\sqrt\frac{2(1-\frac{1}{3})}{(1-2*\frac{1}{3})} $

$\frac{V_{p}}{V_{s}} =\sqrt4$

$\frac{V_{p}}{V_{s}} =2$