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.) (a) Express T 1 , T 2 , and T 3 in terms of D , h , c 1 , c 2 , and θ . (b) Show that T 2 , is a minimum when sin θ = c 1 / c 2 . (c) Suppose that D = 1 km, T 1 , = 0.26 s, T 2 = 0.32 s, and T 3 = 0.34 s. Find c 1 , c 2 , and h . Note: Geophysicists use this technique when studying the structure of the earth’s crust, whether searching for oil or examining fault lines.
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.) (a) Express T 1 , T 2 , and T 3 in terms of D , h , c 1 , c 2 , and θ . (b) Show that T 2 , is a minimum when sin θ = c 1 / c 2 . (c) Suppose that D = 1 km, T 1 , = 0.26 s, T 2 = 0.32 s, and T 3 = 0.34 s. Find c 1 , c 2 , and h . Note: Geophysicists use this technique when studying the structure of the earth’s crust, whether searching for oil or examining fault lines.
Solution Summary: The author calculates the time required for a signal to travel from point P to R, from R to S in lower layer, and then from S to Q.
The speeds of sound c1 in an upper layer and c2 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 T1 seconds. The next signal travels from P to a point R, from R to S in the lower layer, and then to Q, taking T2 seconds. The third signal is reflected off the lower layer at the midpoint O of RS and takes T3 seconds to reach Q. (See the figure.)
(a) Express T1, T2, and T3 in terms of D, h, c1, c2, and θ.
(b) Show that T2, is a minimum when sin θ = c1/c2.
(c) Suppose that D = 1 km, T1, = 0.26 s, T2= 0.32 s, and T3 = 0.34 s. Find c1, c2, and h.
Note: Geophysicists use this technique when studying the structure of the earth’s crust, whether searching for oil or examining fault lines.
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