In Section 17.2, we derived the speed of sound in a gasusing the impulsemomentum theorem

Chapter 17, Problem 72

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In Section 17.2, we derived the speed of sound in a gasusing the impulsemomentum theorem applied to thecylinder of gas in Figure 17.5. Let us find the speedof sound in a gas using a different approach based onthe element of gas in Figure 17.3. Proceed as follows.(a) Draw a force diagram for this element showing theforces exerted on the left and right surfaces due tothe pressure of the gas on either side of the element.(b) By applying Newtons second law to the element,show that2'1DP 2'xA Dx 5 rA Dx'2s't 2 (c) By substituting DP 5 2(B 's/'x) (Eq. 17.3), derivethe following wave equation for sound:Br'2s'x2 5'2s't 2(d) To a mathematical physicist, this equation demonstratesthe existence of sound waves and determines theirspeed. As a physics student, you must take another stepor two. Substitute into the wave equation the trial solutions(x, t) 5 smax cos (kx 2 vt). Show that this functionsatisfies the wave equation, provided v/k 5 v 5 !B/r.