?The speeds of sound \(c_{1}\) in an upper layer and \(c_{2}\) in a lower layer of rock

Chapter 15, Problem 21

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The speeds of sound \(c_{1}\) in an upper layer and \(c_{2}\) in a lower layer of rock and the thickness \(h\) of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite charge is detonated at a point \(P\) and the transmitted signals are recorded at a point \(Q\), which is a distance \(D\) from \(P\).

The first signal to arrive at \(Q\) travels along the surface and takes \(T_{1}\) seconds. The next signal travels from \(P\) to a point \(R\), from \(R\) to \(S\) in the lower layer, and then to \(Q\), taking \(T_{2}\) seconds. The third signal is reflected off the lower layer at the midpoint \(O\) of \(RS\) and takes \(T_{3}\) seconds to reach \(Q\). (See the figure.)

Express \(\mathrm{T}_{1}, \mathrm{~T}_{2}\), and \(T_{3}\) in terms of \(D, h, c_{1}, c_{2}\), and \(\theta\)

Show that \(T_{2}\) is a minimum when \(\sin \theta=c_{1} / c_{2}\).

Suppose that \(\mathrm{D}=1 \mathrm{~km}, \mathrm{~T}_{1}-0.26 \mathrm{~s}, \mathrm{~T}_{2}=0.32 \mathrm{~s}\), and \(\mathrm{T}_{3}=0.34 \mathrm{~s}\). Find \(\mathrm{C}_{1}, \mathrm{C}_{2}\), and \(h\).

Note: Geophysicists use this technique when studying the structure of the earth’s crust— searching for oil or examining fault lines, for example.

Equation Transcription:

Text Transcription:

c_1

c_2

Q

D

P

T_1

R

S

T_2

O

T_3

h

theta

Sin theta =c_1/c_2

D=1km, T_1-0.26 s, T_2=0.32 s, T_3=0.34 s