Acoustics

Fundamental Attributes

$$ c = \lambda f $$

$$ f = \frac{1}{T} $$

$$ \omega = 2 \pi f $$

$$ k = \frac{2 \pi}{ \lambda} $$

Where \( c \) is the speed of sound in the medium, \( \lambda \) is the wavelength, \( f \) is the frequency, \( T \) is the period,\( \omega \) is the angular frequency, \( k \) is the wave number.


Power

$$ P = \frac{E}{t} $$

The sound power per unit area, \( p \times u \) can be found by

$$ p \times u = \frac{p^2}{\rho_0 C} = \frac{p_{max}^2 \cos^2 (\omega t - kx)}{\rho_0 C} $$

Where \( p_{max} \) is the peak pressure:

$$ p_{max} = \rho_0 C^2 k y_{max} $$

Intensity

The intensity, \( I \), of the sound, is the time-average of the power per unit area:

$$ I = \frac{1}{T} \int^{T}_{0} (pu) dt = \frac{p^2_{max}}{\rho_0 C} \frac{1}{T} \int^{T}_{0} \cos^2 (\omega t - kx)dt $$

$$ I = \frac{p^2_{max}}{2 \rho_0 C} $$

$$ I = \frac{P}{A} $$

Where \( A \) is the area.

Sound Pressure
The instantaneous sound pressure measured in Pascals ( \( Pa \) ) can be found via the wave equation

$$ p = A \sin ( \omega t - kx ) + B \cos ( \omega t - kx ) $$

To measure the sound pressure experimentally, the instantaneous sound pressure is squared to give continuous positive values, then summed, averaged and square rooted to find the RMS value.

For a sinusoidal wave:

$$ p_{rms} = \frac{p_{max}}{\sqrt{2}} $$

This can be used to calculate intensity:

$$ I = \frac{p^2_{rms}}{\rho_0 C} $$

Sound Pressure Level

The sound pressure level (SPL) is measured with the logarithmic decibel (dB) scale relative to a reference value.

$$ SPL = 20 \log_{10} (\frac{p}{p_0}) $$

Where the reference value, \( p_0 \) is usually the threshold of human hearing, \( 20 \mu Pa \).


Point source
Soundwaves spread evenly in a sphere around the source, the intensity obeys the inverse square law:

$$ I = \frac{P}{4 \pi r^2} $$

The sound pressure variances with distance can then be determined from the intensity differences.

Line source
Soundwaves spread in planes perpendicular to the line source.

$$ I = \frac{P}{2 \pi r^2} $$


Medium Dependancy

The speed of sound depends on the properties of the medium:

$$ c = \sqrt{\frac{k}{\rho}} $$

Where \( k \) is the elastic modulus, \( \rho \) is the density. For solids, Young's modulus ( \( E \) ) is used instead of the elastic modulus.

For ideal gasses:

$$ k = \gamma p $$

Where \( \gamma \) is a gas constant, \( p \) is the atmospheric pressure.

Since \( PV = mRT \):

$$ c = \sqrt{ \gamma RT } $$

Soundwaves can be described by a wave equation with its pressure or amplitude against time or displacement:

$$ p = A \sin ( \omega t ) $$

or

$$ p = A \sin ( k x ) $$

Any waveform can be mapped against time or displacement via

$$ p = A \sin ( \omega t - kx ) + B \cos ( \omega t - kx ) $$

$$ \omega = 2 \pi f $$ $$k = \frac{2 \pi}{ \lambda} $$


Spectra and the Fourier Transform

The Fourier decomposition spectra for a pure tone can be defined by

$$ P(f) = \frac{A}{2} \delta (f - f_0) $$

Where \( A \) is the amplitude, \( f_0 \) is the frequency of the original composite wave.

For a pure tone of amplitude \( 2 Pa \) and a frequency of \( 200 Hz \);

$$ p(t) = A \cos (2 \pi f t) $$
$$ P(f) = A \delta (f - f_0) $$
$$ p(t) = 2 \cos (400 \pi t) $$
$$ P(f) = 2 \delta (f - 200) $$

Bands can be defined by their central frequency. An octave band defined by its central frequency will have its lower band limit at half the frequency of its upper limit.

For a band \( f_0 Hz \):

Lower band limit: \( \frac{f_0}{\sqrt{2}} \)
Upper band limit: \( \sqrt{2} f_0 \)


Filters

Constant bandwidth: bandwidth is a constant frequency range.

Proportional bandwidth: constant ratio between the upper and lower limits.

Constant percentage bandwidth: constant percentage of the central frequency becomes the bandwidth.