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CSIR_NET_2018_ DECEMBER

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CSIR_NET_2018_ DECEMBER

115) A P-wave is incident at an angle of 300 upon a horizontal interface separating two media which are both poissions solid. If the P-wave velocity of the first layer is 3 km/sec and the critical angle is 600 , then the angle of reflection of the shear wave is 
 
A) \arcsin(\frac{1}{6}) 
B) \arcsin(\frac{1}{4})
C) \arcsin(\frac{1}{3}) 
 D) \arcsin(\frac{1}{2})

(Thanks to Chandrasekhar, ANU)

Solution:

Here we use two formulas 
 
  1. \frac{V_{P}}{V_{S}}= \sqrt{\frac{2(1-\nu)}{(1-2\nu)}}

  2. \frac{Sin Ip}{Vp1}=\frac{Sin Is}{Vs1}=\frac{Sin Rp}{Vp2}=\frac{Sin Rs}{Vs2}

Poissions solid value (\nu) =0.25

Velocity of the P-wave (V )= 3 km/s
 
\frac{V_{P}}{V_{S}}= \sqrt{\frac{2(1-\nu)}{(1-2\nu)}}

\frac{3}{V_{S}}=\sqrt{\frac{2(1-0.25)}{(1-2*0.25)}}

\frac{3}{V_{S}}=\sqrt{\frac{2*0.75}{0.5}}

\frac{3}{V_{S}}=\sqrt{3}

V_{S}=\frac{3}{\sqrt{3}}

V_{S}=\sqrt{3}km/sec

i.e V_{s_{1}}=\sqrt{3}km/sec

now we use the 2nd formula

\frac{Sin Ip}{Vp1}=\frac{Sin Is}{Vs1}=\frac{Sin Rp}{Vp2}=\frac{Sin Rs}{Vs2}

Ip - incident  and reflected angles  of P wave
Is - reflected angle of S-wave
Vp1 - P-wave velocity of the upper layer
Vs1 - S wave velocity of the upper layer
Vp2 - P wave velocity of the lower layer
Vs2 - S wave velocity of the lower layer
 
from this we can write \frac{Sin Ip}{Vp1}=\frac{Sin Is}{Vs1}

Sin Is=Sin Ip(\frac{Vs_{1}}{Vp_{1}})

Sin Is=Sin 30(\frac{\sqrt{3}}{3})

Sin Is=\frac{1}{2}(\frac{\sqrt{3}}{3})

Sin Is=\frac{1}{3.4}

I_{s}=\arcsin(\frac{1}{3})


References : Fundamentals of Geophysics , William Lowrie



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