# 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.