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

CSIR-NET 2019 Dec

Q) In a Homogenous isotropic elastic medium, the P-wave travels 1.5 times faster than the S-wave. The ratio of $\frac{\lambda}{\mu}$ of the lames constant $\lambda$ and $\mu$ will be __________.

(Thanks to Chandrasekhar, ANU)

 

Solution:

Given that $V_p = 1.5 V_S$

$\frac{\lambda}{\mu}$=?

$Poissions ratio(\nu)=?$

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

$\frac{1.5 \times V_{s}}{V_{s}}=\sqrt{\frac{2(1-\nu)}{(1-2\nu)}}$

$1.5=\sqrt{\frac{2(1-\nu)}{(1-2\nu)}}$

$2.25=\frac{2(1-\nu)}{(1-2\nu)}$

$2.25-4.5\nu=2-2\nu$

$0.25=2.5\nu$

$\nu=0.1$

we know that

$\nu =\frac{\lambda}{2(\lambda+\mu)}$

$1+\frac{\mu}{\lambda}=\frac{1}{2\nu}$

$1+\frac{\mu}{\lambda}=\frac{1}{2\times 0.1}$

$1+\frac{\mu}{\lambda}=5$

$\frac{\mu}{\lambda}=5-1$

$\frac{\lambda}{\mu}=0.25$