Department of Physics United States Naval Academy. Lecture 39: Sound Waves

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1 Department of Physics United States Naval Academy Lecture 39: Sound Waves Sound Waves: Sound waves are longitudinal mechanical waves that can travel through solids, liquids, or gases. The speed v of a sound wave in a medium having bulk modulus B and density ρ is B v = ρ The speed of sound in a gas depends on its composition and also on temperature. The most important case is air near room temperature and pressure; v sound (T 20 C) In air at 20 C, the speed of sound is 343 m/s. A sound wave causes a longitudinal displacement s of a mass element in a medium as given by s = s m cos (kx ωt) where s m is the displacement amplitude (maximum displacement) from equilibrium, k = 2π/λ, and ω = 2πf, λ and f being the wave-length and frequency of the sound wave. The wave also causes a pressure change p from the equilibrium pressure: p = p m sin(kx ωt) where the pressure amplitude is p m = (vρω)s m. Thus, sound is a pressure wave which is created by a vibrating object. Sound Interference: The interference of two sound waves with identical wavelengths passing through a common point depends on their phase difference φ at that point. If the sound waves were emitted in phase and are traveling in approximately the same direction, φ is given by φ = 2π L λ where L is their path length difference (the difference in the distances traveled by the waves to reach the common point). Fully constructive interference occurs when φ is an integer multiple of 2π, φ = m(2π), for m = 0, 1, 2, 3,... and, equivalently, when L is related to wavelength λ by L λ = 0, 1, 2, 3,... Fully destructive interference occurs when φ is an odd multiple of π, and, equivalently, when L is related to λ by φ = (2m + 1)π, for m = 0, 1, 2, 3,... L λ = 1 2, 3 2, 5 2, 7 2,...

2 Exercise 1.0 Show that the type of interference undergone by two sound waves with the same amplitude and wavelength and traveling in the positive direction of an x axis depend on their phase difference φ. If the two sound waves travel different distances to reach a detector, show that their phase difference also depends on the path length difference.

3 Exercise 2.0 (a) A battleship is using sonar (reflected underwater sound signals) to detect the presence of nearby submarines. It is found that a sonar reflection has a round-trip travel time (from the battle-ship to the submarine and back) of 17 s. How far away from the battleship is the submarine? (b) A child drops a rock into a vertical mineshaft that is precisely 406 m deep. The sound of the rock hitting the bottom of the shaft is heard 10.3 s after the child drops the rock. What is the temperature of the air in the shaft? Assume the temperature is the same throughout the mine.

4 Exercise 3.0 In the figure below, two speakers separated by distance d 1 = 2.00 m are in phase. Assume the amplitudes of the sound waves from the speakers are approximately the same at the listener s ear at distance d 2 = 3.75 m directly in front of one speaker. Consider the full audible range for normal hearing, 20 Hz to 20 khz. (a) What is the lowest frequency f min,1 that gives minimum signal (destructive interference) at the listener s ear? By what number must f min,1 be multiplied to get (b) the second lowest frequency f min,2 that gives minimum signal and (c) the third lowest frequency f min,3 that gives minimum signal? (d) What is the lowest frequency f max,1 that gives maximum signal (constructive interference) at the listener s ear? By what number must f max,1 be multiplied to get (e) the second lowest frequency f max,2 that gives maximum signal and (f) the third lowest frequency f max,3 that gives maximum signal?

5 Exercise 4.0 The figure below shows the output from a pressure monitor mounted at a point along the path taken by a sound wave of a single frequency traveling at 343 m/s through air with a uniform density of 1.21 kg/m 3. The vertical axis scale is set by ps = 4.0 mpa. If the displacement function of the wave is s(x, t) = s m cos(kx ωt), what are (a) s m, (b) k, and (c) v?