The speeds of sound c1 an upper layer and c12 lower layer of rock and the thickness 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, from R to 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 = 1km, T1 = 0.26x, T2 = 0.32s, T3 = 0.34s. Find c1, c2 and h.
Note: Geophysicists use this technique when studying the structure of Earth€™s crust, whether searching for oil or examining fault lines.

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