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CSIR-NET 2019 JUNE

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CSIR-NET
96) The travel time differences between the P and S arrivals at two stations A and B are 3 sec and 4 sec Respectively, while the difference in the P- wave arrival time between the same stations is 1sec. What is the ratio of the P-wave velocity to that of S-wave velocity?
(Thanks to Chandra sekhar, ANU)


Solution:
Given that,
Station A travel time difference given that (t_{s}-t_{p})_{A}= 3 seconds

Station B travel time difference given that (t_{s}-t_{p})_{B}= 4 seconds 

Difference in the P-wave arrival time between the station B to station A is given that,
(t_{p})_{B}-(t_{p})_{A}= 1 seconds

\frac{V_{p}}{V_{s}}=?


we know that,


t_{s}-t_{p}= D(\frac{1}{V_{s}}-\frac{1}{V_{p}})


(i.e t_{p}=\frac{D}{V_{p}})


t_{s}-t_{p}= t_{p}V_{p}(\frac{1}{V_{s}}-\frac{1}{V_{p}})


t_{s}-t_{p}= t_{p}(\frac{V_{p}}{V_{s}}-1)-----------(1)
From the above equation,we can derive equations for station B and A respectively.


(t_{s}-t_{p})_B= t_{p_B}(\frac{V_{p}}{V_{s}}-1)----------------(2)

(t_{s}-t_{p})_A= t_{p_A}(\frac{V_{p}}{V_{s}}-1)----------------(3)

Subtract equation (3) from eq (2),
we get (t_{s}-t_{p})_{B}-(t_{s}-t_{p})_{A}={[(t_{p})_{B}(\frac{V_{p}}{V_{s}}-1)]-[(t_{p})_{A}(\frac{V_{p}}{V_{s}}-1)]}


4-3=(\frac{V_{p}}{V_{s}}-1)[(t_{p})_{B}-(t_{p})_{A}]


1=(\frac{V_{p}}{V_{s}}-1)


\frac{V_{p}}{V_{s}}=1+1


\frac{V_{p}}{V_{s}}=2





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