Chapter Summary

14.1 Sound

The relationship of the speed of sound [latex]v_{w}[/latex], its frequency [latex]f[/latex], and its wavelength [latex]\lambda[/latex] is given by

[latex]v_{\text{w}} = fλ ,[/latex]

which is the same relationship given for all waves.

In air, the speed of sound is related to air temperature [latex]T[/latex] by

[latex]v_{\text{w}} = \left(\text{331} \text{m}/\text{s}\right) \sqrt{\frac{T}{\text{273} \text{K}}} .[/latex]

[latex]v_{\text{w}}[/latex] is the same for all frequencies and wavelengths.

Intensity is the same for a sound wave as was defined for all waves; it is
[latex]I = \frac{P}{A} ,[/latex]

where [latex]P[/latex] is the power crossing area [latex]A[/latex]. The SI unit for [latex]I[/latex] is watts per meter squared. The intensity of a sound wave is also related to the pressure amplitude [latex]Δ p[/latex]

[latex]I = \frac{\left(\left(\right. Δ p \left.\right)\right)^{2}}{2 \rho\text{v}_{w}} ,[/latex]

where [latex]\rho[/latex] is the density of the medium in which the sound wave travels and [latex]v_{w}[/latex] is the speed of sound in the medium.
Sound intensity level in units of decibels (dB) is
[latex]\beta \left(\text{dB}\right) = \text{10} \text{log}_{\text{10}} \left(\frac{I}{I_{0}}\right) ,[/latex]

where [latex]I_{0} = 10^{–12} W/ \text{m}^{2}[/latex] is the threshold intensity of hearing.

The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer.
The actual change in frequency is called the Doppler shift.
A sonic boom is constructive interference of sound created by an object moving faster than sound.
A sonic boom is a type of bow wake created when any wave source moves faster than the wave propagation speed.
For a stationary observer and a moving source, the observed frequency [latex]f_{\text{obs}}[/latex] is:
[latex]f_{\text{obs}} = f_{s} \left(\frac{v_{w}}{v_{w} \pm v_{s}}\right) ,[/latex]

where [latex]f_{s}[/latex] is the frequency of the source, [latex]v_{s}[/latex] is the speed of the source, and [latex]v_{w}[/latex] is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away.
For a stationary source and moving observer, the observed frequency is:
[latex]f_{\text{obs}} = f_{s} \left(\frac{v_{w} \pm v_{\text{obs}}}{v_{w}}\right) ,[/latex]

where [latex]v_{\text{obs}}[/latex] is the speed of the observer.

Sound interference and resonance have the same properties as defined for all waves.
In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics.
The resonant frequencies of a tube closed at one end are:
[latex]f_{n} = n \frac{v_{w}}{4 L} ,\textrm{ }\text{ } n = 1, 3, 5 . . .,[/latex]

[latex]f_{1}[/latex] is the fundamental and [latex]L[/latex] is the length of the tube.
The resonant frequencies of a tube open at both ends are:
[latex]f_{n} = n \frac{v_{w}}{2 L} ,\textrm{ }\text{ } n = 1, 2, 3 . . .[/latex]

The range of audible frequencies is 20 to 20,000 Hz.
Those sounds above 20,000 Hz are ultrasound, whereas those below 20 Hz are infrasound.
The perception of frequency is pitch.
The perception of intensity is loudness.
Loudness has units of phons.

The acoustic impedance is defined as:
[latex]Z = ρv ,[/latex]

[latex]\rho[/latex] is the density of a medium through which the sound travels and [latex]v[/latex] is the speed of sound through that medium.
The intensity reflection coefficient [latex]a[/latex], a measure of the ratio of the intensity of the wave reflected off a boundary between two media relative to the intensity of the incident wave, is given by
[latex]a = \frac{\left(Z_{2} - Z_{1}\right)^{2}}{\left(Z_{1} + Z_{2}\right)^{2}} .[/latex]
The intensity reflection coefficient is a unitless quantity.

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