Sections#

Review & Summary#

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 \(\rho\) is

(110)#\[ v = \sqrt{\frac{B}{\rho}} \]

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 the medium as given by

(111)#\[ s = s_m \cos(kx - \omega t) \]

where \(s_m\) is the displacement amplitude (maximum displacement from equilibrium), \(k = 2\pi/\lambda\), and \(\omega = 2\pi f\), with \(\lambda\) and \(f\) the wavelength and frequency of the sound wave. The wave also causes a pressure change \(\Delta p\) from the equilibrium pressure:

(112)#\[ \Delta p = \Delta p_m \sin(kx - \omega t) \]

where the pressure amplitude is \(\Delta p_m = \rho v \omega s_m\).

Interference#

The interference of two sound waves with identical wavelengths passing through a common point depends on their phase difference \(\phi\) there. If the sound waves were emitted in phase and are traveling in approximately the same direction, \(\phi\) is given by \(\phi = (2\pi/\lambda)\Delta L\), where \(\Delta L\) is their path length difference (the difference in the distances traveled by the waves to reach the common point). Fully constructive interference occurs when \(\phi\) is an integer multiple of \(2\pi\), i.e., \(\Delta L = n\lambda\). Fully destructive interference occurs when \(\phi\) is an odd multiple of \(\pi\), i.e., \(\Delta L = (n + \frac{1}{2})\lambda\).

Sound Intensity#

The intensity \(I\) of a sound wave at a surface is the average rate per unit area at which energy is transferred by the wave through or onto the surface:

(113)#\[ I = \frac{P}{A} \]

where \(P\) is the time rate of energy transfer (power) of the sound wave and \(A\) is the area of the surface intercepting the sound. The intensity \(I\) is related to the displacement amplitude \(s_m\) by \(I = \frac{1}{2}\rho v \omega^2 s_m^2\). The intensity at a distance \(r\) from a point source that emits sound waves of power \(P_s\) is

(114)#\[ I = \frac{P_s}{4\pi r^2} \]

The sound level in decibels is \(\beta = 10 \log_{10}(I/I_0)\), where \(I_0 = 10^{-12}\) W/m² is a reference intensity. For every factor-of-10 increase in intensity, 10 dB is added to the sound level.

Standing Wave Patterns in Pipes#

Standing sound wave patterns can be set up in pipes. A pipe open at both ends will resonate at frequencies

(115)#\[ f = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots \]

where \(v\) is the speed of sound in the air in the pipe. For a pipe closed at one end and open at the other, the resonant frequencies are \(f = nv/4L\) for \(n = 1, 3, 5, \ldots\)

Beats#

Beats arise when two waves having slightly different frequencies \(f_1\) and \(f_2\) are detected together. The beat frequency is

(116)#\[ f_{\mathrm{beat}} = |f_1 - f_2| \]

The Doppler Effect#

The Doppler effect is a change in the observed frequency of a wave when the source or the detector moves relative to the transmitting medium (such as air). For sound, the observed frequency \(f'\) is given in terms of the source frequency \(f\) by

(117)#\[ f' = f \frac{v \pm v_D}{v \mp v_S} \]

where \(v_D\) is the speed of the detector relative to the medium, \(v_S\) is that of the source, and \(v\) is the speed of sound in the medium. The signs are chosen such that \(f'\) tends to be greater for motion toward and less for motion away.

Shock Wave#

If the speed of a source relative to the medium exceeds the speed of sound in the medium, the Doppler equation no longer applies. In such a case, shock waves result. The half-angle \(\theta\) of the Mach cone is given by

(118)#\[ \sin\theta = \frac{v}{v_S} \]

where \(v_S\) is the speed of the source.