56. For an earth material with both the lame parameters equal (λ=μ), the ratio of P-wave to S-wave velocities is
(Thanks to Pragnath, AU)
Solution: we know that
the velocity of the P-wave is
$\alpha=\sqrt{\frac{\lambda+2\mu}{\rho}}$
The velocity of S-wave is
$\beta=\sqrt{\frac{\mu}{\rho}}$
Given that (λ=μ) so on
substituting that value we get
$V_{p}$ or $\alpha$=$\sqrt{\frac{3\mu}{\rho}}$
And the ration is
$\frac{V_{p}}{V_{s}}= \frac{{\sqrt{\frac{3\mu}{\rho}}}}{\sqrt{{{\frac{\mu}{\rho}}}}} $
$\frac{V_{p}}{V_{s}}=\sqrt{\frac{3\mu \rho}{\mu \rho}}$
Therefore,
$\frac{V_{p}}{V_{s}}=\sqrt{3}$
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