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

GATE-2008

60. If the P-wave velocity is twice that of S-wave velocity in a medium , the Poisson's ratio of the material is 

A) 0.50                  

B)0.33                             

C)0.25                              

D)0.12

Solution:

The relation between P-wave velocity, S-wave 

velocity and Poisson's ratio (υ) is given by
 

(V_p/V_s)² = 2 (1 -υ) / (1- 2υ) ----------(1)

From above problem 

 

S- wave velocity = x

P- wave velocity = 2x

From eq(1)

(1- 2 υ)* (V_p/V_s)² = 2- 2υ

(V_p/V_s)² – 2υ * (V_p/V_s)² = 2 -2υ

(V_p/V_s)² - 2  = -2υ + 2υ (V_p/V_s)²

                    = 2υ ((V_p/V_s)² -1)   --------(2)

Revere the above equation (2)

 

2υ ((V_p/V_s)² -1) = (V_p/V_s)² - 2 

 

2υ =((V_p/V_s)² - 2) / ((V_p/V_s)² - 1 ) 

 

    =(4 – 2) / (4 – 1)
 

    = 2/3

υ = 2 / (3*2)

υ = 1/ 3

Poisson's ratio(υ) = 0.33