Sound Sources

🎯 Learning Objectives

By the end of this topic, you should be able to:

Describe and mathematically model monopole, dipole, and distributed sources.
Derive the spherical wave solution of the acoustic wave equation.
Understand volume velocity, sound power, and radiation impedance.
Calculate directivity factors and far-field sound pressure levels.
Apply these principles to practical sound source characterization and simulation.

In the chapter on fundamentals in acoustics and for the discussion of plane waves, we concentrated on the sound field at arbitrary observation points. At this point, sound sources were not yet taken into consideration. The characterization of sound sources, however, is one of the most important tasks in simulation and auralization. Solving the wave equation in polar coordinates is the first step to obtain a physical description of a source. The solution will lead to fundamental properties of spherical waves, to sound source directivities and to mathematical methods for acoustic signal processing of radiation problems.

Spherical Waves

A spherical wave models sound radiation from a point-like, omnidirectional source, also called a monopole.

The source is assumed to be much smaller than the wavelength: $a \ll \lambda$
The source injects a volume flow (volume velocity) into the medium: $Q(t) \ [\mathrm{m^3/s}]$

This represents the rate at which the source moves air, and due to omnidirectionality:

There is no angular dependence on $\theta$ or $\phi$
The pressure depends only on radius $r$ and time $t$

Thus, the problem is spherically symmetric.

Reduction of the Wave Equation

In full 3-D, the Laplace operator is:

\Delta = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r}\right) + \frac{1}{r^2 \sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta} \frac{\partial^2}{\partial \phi^2}

With spherical symmetry, all angular derivatives vanish:

\Delta = \frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}{\partial r}

The inhomogeneous acoustic wave equation becomes:

\frac{\partial^2 p}{\partial r^2} + \frac{2}{r} \frac{\partial p}{\partial r} = \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} - \rho_0 \dot{Q}(t)

where:

$p(r,t)$ = sound pressure
$c$ = speed of sound
$\rho_0$ = air density
$Q(t)$ = volume velocity of source

Solution Using d’Alembert Form

We assume a solution of the form:

p(r,t) = \frac{f(r,t)}{r}

Substituting reduces the equation to:

\frac{\partial^2 f}{\partial t^2} - c^2 \frac{\partial^2 f}{\partial r^2} = \rho_0 c^2 \dot{Q}(t)

The d’Alembert solution gives:

f = f(r - ct)

Hence, the general solution for a spherical wave is:

p(r,t) = \frac{1}{r} f(r - ct)

This shows the pressure decays with distance as $1/r$ and propagates outward from the source.

The relation between $f(r-ct)$ and the source volume velocity $Q(t)$ is:

p(r,t) = \frac{\rho_0}{4 \pi r} \dot{Q}\left(t - \frac{r}{c}\right)

Note: The pressure at the observation point is proportional to the time derivative of the volume velocity.
This explains why small loudspeakers exhibit poor low-frequency output.

Harmonic Monopole Source and Sound Power

For a harmonic excitation:

\underline{Q}(t) = \hat{Q} \cdot e^{j \omega t}

The corresponding sound pressure:

\underline{p}(r,t) = \frac{j \omega \rho_0 \hat{Q}}{4 \pi r} \cdot e^{j (\omega t - kr)}

The particle velocity:

\underline{v} = \frac{1}{j \omega \rho_0} \frac{\partial p}{\partial r} = \frac{\hat{Q}}{4 \pi} \left(jk + \frac{1}{r}\right) e^{j (\omega t - kr)}

The wave impedance is:

\frac{\underline{p}}{\underline{v}} = \frac{j \omega \rho_0}{jk + \frac{1}{r}} = Z_0 \frac{1}{1 + \frac{1}{jkr}}

The sound intensity:

I = \frac{\rho_0 \hat{Q}^2}{32 \pi^2 c} \frac{\omega^2}{r^2} = \frac{\bar{p}^2}{\rho_0 c}

Total radiated sound power:

P = \int\int I(r) r^2 dr \sin\theta d\theta d\phi = \frac{\rho_0 \hat{Q}^2}{8 \pi c} \omega^2

Decibel representation:

L_W = 10 \log_{10}\frac{P}{P_0} \quad , \quad P_0 = 10^{-12} W

These results hold for small, volume-moving sources.

Pulsating Sphere and Radiation Impedance

A spherical surface of radius $a$ vibrating harmonically generates a spherical wave.

Radiation impedance:

4 \pi a^2 \underline{p}(a,t) = \underline{Z}_r \underline{v}(a,t)

with:

\underline{Z}_r = \frac{\rho_0 c S}{1 + \frac{1}{jka}} , \quad S = 4 \pi a^2

The radiated power:

P = \frac{1}{2} |v(a,t)|^2 \cdot Re[\underline{Z}_r]

Low- and high-frequency limits:

W_r = \frac{\rho_0 c S}{1 + 1/(k^2 a^2)} \approx \begin{cases} S \rho_0 c k^2 a^2 & ka \ll 1 \\ S \rho_0 c & ka \gg 1 \end{cases}

Admittance form:

\underline{Y}_r = \frac{1}{\rho_0 c S} + \frac{1}{j \omega m_s} , \quad m_s = 4 \pi a^3 \rho_0

Python Example: Spherical Monopole Source Simulation

Press Run Code: Output will appear here.

Multipoles and Extended Sources

Multipoles or extended sources are constructed by summing or integrating multiple monopoles.

Example: Dipole formed by two out-of-phase monopoles:

\hat{Q}_2 = -\hat{Q}_1 = \hat{Q}

Superposition formula:

\underline{p} = \frac{j \omega \rho_0}{4 \pi} \sum_n \underline{\hat{Q}}_n \frac{e^{j (\omega t - k r_n)}}{r_n}

For closely spaced sources ( $kd \ll 1$ ):

\underline{p} \approx \frac{\rho_0 k^2 d \hat{Q}}{4 \pi r} \cos\theta e^{j (\omega t - kr)}

The directional factor gives a figure-of-eight pattern.
Summation over extended sources defines the total amplitude and directivity:

\hat{Q}_{tot} = \sum_n \hat{Q}_n , \quad \underline{p} = \frac{j \omega \rho_0 \hat{Q}_{tot}}{4 \pi r} e^{j (\omega t - kr)} \Gamma(\theta, \phi)

The far-field intensity:

I_n = \frac{P}{4 \pi r^2} |\Gamma(\theta, \phi)|^2

Sound pressure level from directivity factor:

L_p = 20 \log_{10} r - 11 + L_D , \quad L_D = 10 \log_{10} \iint_{\Omega} |\Gamma(\theta, \phi)|^2 d\Omega

Python Example: Dipole Source Simulation

Press Run Code: Output will appear here.

Python Example: Quadrupole Source Simulation

Press Run Code: Output will appear here.

Example Animations

Dipole Source Radiation

Dipole radiation — **Figure.** Dipole source radiation pattern.

Quadrupole Source Radiation

Quadrupole radiation — **Figure.** Quadrupole source radiation pattern.

🧪 Interactive Examples

Advanced Acoustic Source Simulator

Microphone Pressure: 0.000

Visualization ModeSource TypeFrequency: 500 HzDistance Scaling: 1.00×Simulation Speed: 1.00×

📝 Key Takeaways

A monopole source is simplest acoustic radiator, characterized by a time-varying volume flow $Q(t)$
It radiates spherical waves with pressure: $p(r,t)=\frac{\rho_0}{4\pi r}\dot{Q}(t-r/c)$ .
A dipole source consists of two out-of-phase monopoles close together. Its radiation pattern is figure-8, strongly directional, and pressure decays as $\propto \frac{1}{r}\cos(\theta)$ .
A quadrupole source is formed from two dipoles and produces a four-lobe radiation pattern.
It appears naturally in turbulence and high-speed flows.
Directivity increases from monopole → dipole → quadrupole, meaning higher-order sources radiate less energy forward and more in structured spatial patterns. Higher-order sources also roll off faster at low frequencies, making monopoles efficient for bass sound reproduction and quadrupoles inefficient.
For harmonic (sinusoidal) sources, pressure fields scale with frequency, distance, radiation order, and medium impedance.
Moving farther away from the source always increases the spherical spreading loss, where sound pressure ∝ $1/r$ in the far field.

🧠 Quick Quiz

1) Which of the following statements about monopole and dipole sound sources is correct?

A) A monopole radiates directionally and a dipole is omnidirectional.
B) A monopole’s pressure decays as 1/r, but a dipole also decays as 1/r and adds directional dependence.
C) A dipole radiates spherical waves identical to a monopole but at higher amplitude.
D) A monopole’s radiation pattern is figure-8 shaped like a dipole.

2) The quadrupole source produces:

A) An omnidirectional pattern.
B) Two-lobed pattern aligned with its axis.
C) Four symmetric lobes from two orthogonal dipoles.
D) Zero sound radiation in all directions.

3) A dipole source has maximum radiation at which angle?

A) 0° and 180° (front and back)
B) 90° only
C) 45°
D) All angles equally

4) Why are monopoles efficient at radiating low frequencies?

A) They have no directional pattern.
B) They do not require particle acceleration gradients.
C) Higher-order sources (dipole, quadrupole) suffer low-frequency cancellation due to opposite phases.
D) Their size is always very large.

5) Increasing the distance between listener and a monopole source:

A) Does not affect the pressure amplitude.
B) Increases sound pressure proportionally.
C) Decreases sound pressure as 1/r due to spherical spreading.
D) Decreases sound pressure as 1/r².

Chapter - 10

Key Words: Sound sources, Spherical waves, Monopole, Dipole, Radiation, Acoustics