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Math
CalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

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Chapter 4, Problem 21P
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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.

Chapter 4, Problem 21P, The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the

(a)

To determine

To express: the values of times T1,T2 and T3 in terms of distance D thickness of the rock h speed of the sound in upper layer c1 and speed of the sound in lower layer c2 and the angle θ .

Explanation

Given:

A figure showing the arrangement for the experiment is given.

Calculation:

Time T1 is the time required for a signal to travel from P to Q.

T1=Distance between P and QSpeed of the sound in upper layer=|PQ|c1=Dc1

Therefore T1=Dc1 .

Time T2 is the time required for a signal to travel from point P to R then from R to S in lower layer and then from S to Q.

T2=[Distance between P and RSpeed of the sound in upper layer+Distance between R and SSpeed of the sound in lower layer+Distance between S and QSpeed of the sound in upper layer]=|PR|c1+|RS|c2+|SQ|c1

From the given figure observe that PR=SQ using in above equation,

T2=|PR|c1+|RS|c2+|SQ|c1=2|PR|c1+|RS|c2

Observe that,

|PR|=hcosθ=hsecθ

Similarly,

|RS|=D−htanθ−htanθ=D−2htanθ

Substitute both values in the equation for T2

(b)

To determine

To show: The time T2 is maximum when sinθ=c1c2 .

(c)

To determine

To find: The values of speeds of sound c1 and c2 and the value of height h.

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