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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 and the transmitted signals are recorded at a point p, 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.

(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 = 0.26s, T2 = 0.32 s, and T3 = 0.34 a. 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.

D Speed of sound =c h R S Speed of sound = = cz