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 343 + 0.6 (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,...
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.
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.
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?
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?