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

CSIR NET – DEC 2014

65) A compressional wave traveling through a medium of seismic velocity V₁, when incident at the interface of an underlying medium with seismic velocity V₂, gets refracted at an angle 30⁰. If the velocity in the underlying medium is 2V₂, the wave incident at the same angle

1) Is critically refracted

2) totally reflected

3) Is refracted at an angle less than 30⁰

4) Is refracted at the same angle

(Thanks to Chandrashekar, ANU)

(Thanks to Neeraja, AU)

 

Solution:

From Snell’s law,

$\frac{\sin i}{\sin r}=\frac{V_1}{V_2}---(1)$

Refracted angle(r) =30 degrees 

Substitute in eq(1)

$\frac{\sin i}{\sin30^o}=\frac{V_1}{V_2}$

$sin30^o=\frac{1}{2}$

$2sini=\frac{V_1}{V_2}$

$sini=\frac{V_1}{2V_2}$

$incident angle, (i)=\sin^{-1}(\frac{V_1}{2V_2})--- (2)$

Given that, velocity of the underlying medium is 2V₂ and wave incident at the same angle,

$\frac{sini}{sinr}=\frac{V_1}{V_2}---(3)$

eq(2) substitute in eq(3)

$\frac{sin(sin^{-1}\frac{V_1}{2V_2})}{sinr}=\frac{V_1}{V_2}$

$\because \sin(\sin^{-1})=1$

 $\frac{V_1/2V_2}{\sin r}=\frac{V_1}{V_2}$

$\sin r = \frac{V_1}{2V_2}\times\frac{2V_2}{V_1}=1$

$\sin r=\sin 90^0$

r=90⁰ (critically refracted)