Sound Fields

🎯 Learning Objectives:

In this chapter, we introduce the physical foundations of linear acoustics, focusing on the derivation of the wave equation, its solutions, and the generalization from one-dimensional to three-dimensional spaces. Understanding these principles is essential for describing sound propagation in fluids, solids, and gases, and provides the basis for practical acoustics applications such as audio engineering, architectural acoustics, and noise control.

Sound: A Microscopic Perspective

Before delving into equations, it is useful to visualize what sound really is at a microscopic level. Imagine a snapshot of a medium—air, water, or a solid—at a particular moment in time. The molecules or atoms in the medium are not static; they oscillate around their equilibrium positions, independent of the constant random motion due to thermal energy. These oscillations are induced by sound waves traveling through the medium.
In a sound wave, each particle follows a displacement vector that varies with both space and time:

\vec{s} = (\xi, \eta, \zeta)

where $ξ$ , $η$ , and $ζ$ are the displacements along the $x$ , $y$ , and $z$ directions, respectively (Fig).

Signal domain illustration — **Figure 1.** Microscopic view of a medium with sound showing particle displacement vectors.

Note that the sound pressure is a scalar. In acoustics the sound pressure is usually the most important quantity of interest, mainly because the human ear is sensitive to sound pressure. Hence calculations or measurements of sound pressure yield directly the input quantity of the human hearing system.

The particle velocity, which is the time derivative of displacement, is then defined as:

\vec{v} = \frac{\partial \vec{s}}{\partial t}

with components $v_x = \frac{\partial \xi}{\partial t}$ , $v_y = \frac{\partial \eta}{\partial t}$ , and $v_z = \frac{\partial \zeta}{\partial t}$ .

Because the displacements are neither homogeneous nor isotropic, the medium is subjected to compression and rarefaction, resulting in density fluctuations:

\rho = \rho_{tot} - \rho_0

where $\rho_{tot}$ is the instantaneous density and $\rho_0$ is the equilibrium density of the medium at rest. Similarly, the sound-induced pressure fluctuations are defined as:

p = p_{tot} - p_0

where $p_0$ is the static pressure of the medium. This quantity, $p$ , is commonly referred to as sound pressure.

Sound pressure is a scalar quantity, which is particularly important because the human ear responds directly to variations in pressure. This makes sound pressure a primary quantity for both theoretical analysis and experimental measurements in acoustics.

Acoustic Properties of Fluids

In fluids such as air, the elasticity of the medium is described by its compressibility, which characterizes how easily the medium can be compressed under an applied force. When sound waves propagate, particle displacements create local compressions and rarefactions. These, in turn, generate pressure fluctuations.
Temperature and heat transfer also influence the sound-induced pressure. For acoustic waves in air, however, we can often assume that heat diffusion between local regions is negligible. Under this assumption, the process is considered adiabatic, and we can apply the adiabatic Poisson equation:

\frac{p_{tot}}{p_0} = \left(\frac{\rho_{tot}}{\rho_0}\right)^{\kappa}

where $\kappa = \frac{C_p}{C_v}$ is the adiabatic exponent, or the ratio of specific heats at constant pressure and volume. For air, $\kappa \approx 1.4$ .
This relation links pressure fluctuations directly to density fluctuations, a fundamental concept in acoustics.

Derivation of the One-Dimensional Wave Equation

To build a mathematical description of sound propagation, we start with a small volume element of a one-dimensional fluid medium, such as a section of a tube. Let the element have thickness $\Delta x$ and cross-sectional area $S$ , and let it be subjected to a periodic acoustic source (e.g., a piston) that pushes and pulls on the medium with a volumetric flow $q S dx$ ([m³/s]).
The pressure and particle velocity at the left-hand side ( $x$ ) may differ from those at the right-hand side ( $x + \Delta x$ ), creating a net force on the volume element:

[(p_{tot})_x - (p_{tot})_{x + \Delta x}] S = \rho_{tot} S \Delta x \frac{dv}{dt}

Here, $v$ is the particle velocity and $\rho_{tot}$ is the local density. Using the chain rule for differentiation:

\frac{dv}{dt} = \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x}

This leads to the Euler equation for a one-dimensional fluid:

-\frac{\partial p_{tot}}{\partial x} = \rho_{tot}\left(\frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x}\right)

Conservation of Mass

The continuity equation describes conservation of mass. Any change in mass within the volume element must correspond to density variations:

-\frac{\partial (\rho_{tot} v)}{\partial x} = \frac{\partial \rho_{tot}}{\partial t} - \rho_{tot} q

Linearization for Small Sound Amplitudes

For most practical acoustics problems, sound-induced variations are small compared to equilibrium values: $p \ll p_0, \quad \rho \ll \rho_0$ . This allows linearization of the governing equations by neglecting higher-order terms in a Taylor expansion and replacing $\rho_{tot}$ with $\rho_0$ where appropriate. Using the adiabatic relation $p = c^2 \rho$ , we obtain the linearized sound field equations:

-\frac{\partial p}{\partial x} = \rho_0 \frac{\partial v}{\partial t}

-\rho_0 \frac{\partial v}{\partial x} = \frac{1}{c^2} \frac{\partial p}{\partial t} + \rho_0 q

where $c$ is the speed of sound.

The One-Dimensional Wave Equation

Eliminating the particle velocity $v$ yields the wave equation for sound pressure:

\frac{\partial^2 p}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} = -\rho_0 \frac{\partial q}{\partial t}

or, more compactly:

\Delta p - \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} = -\rho_0 \dot{q}

This classical wave equation can be generalized to three dimensions by introducing the Laplacian operator:

Cartesian coordinates $(x, y, z)$ :

\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}

Cylindrical coordinates $(r, \phi, z)$ :

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

Spherical coordinates $(r, \theta, \phi)$ :

\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}

The wave equation provides a universal framework for describing sound propagation in fluids, solids, and gases. It also serves as the starting point for studying solutions such as plane waves, spherical waves, and more complex modes.

🧪 Interactive Examples

1D Acoustic Wave Simulation

Visualizes instantaneous pressure $p(x,t)=A\sin(kx-\omega t)$ , particle velocity (approx), and instantaneous intensity. Adjust parameters below.

Amplitude (Pa):

1.00 Pa

Frequency (Hz):

220 Hz

Speed of sound c (m/s):

343 m/s

Density ρ (kg/m³):

1.225 kg/m³

Show instantaneous intensityShow RMS / average

Notes: This simulation uses the plane-wave approximation where $v = p/(\rho c)$ and instantaneous intensity $I = p\cdot v$ . RMS and average intensity use $p_{rms}=A/\sqrt{2}$ and $I_{avg}=p_{rms}^2/(\rho c)$ .

📝 Key Takeaways

Particle velocity $\vec{v} = \frac{\partial \vec{s}}{\partial t}$ represents the rate of particle motion due to sound propagation.
Density fluctuations and sound pressure are given by $\rho = \rho_{tot} - \rho_0$ and $p = p_{tot} - p_0$ , describing deviations from equilibrium.
Under adiabatic conditions, pressure and density are related by
$\frac{p_{tot}}{p_0} = \left( \frac{\rho_{tot}}{\rho_0} \right)^{\kappa}$ ,
where $\kappa = \frac{C_p}{C_v} \approx 1.4$ for air.
For small-amplitude waves ( $p \ll p_0, \ \rho \ll \rho_0$ ), the linearized equations of acoustics apply: $-\frac{\partial p}{\partial x} = \rho_0 \frac{\partial v}{\partial t}$ ,
$-\rho_0 \frac{\partial v}{\partial x} = \frac{1}{c^2} \frac{\partial p}{\partial t} + \rho_0 q$ .
Combining these yields the wave equation:
$\Delta p - \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} = -\rho_0 \dot{q}$ ,
the foundation of linear acoustics.
Sound intensity expresses the rate of sound energy flow per unit area:
$I = \frac{P}{A}$ or $I = \frac{P}{4 \pi r^2}$ for a point source.
Intensity is a directional power quantity, while sound pressure and sound power are non-directional scalar quantities.

🧠 Quick Quiz

Test your understanding - select and submit an answer.

1) What physical quantity does sound intensity represent?

A) Sound pressure
B) Flow of sound energy per unit area
C) Sound power
D) Particle displacement

2) Which of the following quantities is a vector (directional) quantity?

A) Sound pressure
B) Sound power
C) Sound intensity
D) Sound energy

3) The particle velocity is mathematically defined as:

A) The derivative of pressure with respect to time
B) The derivative of displacement with respect to time
C) The ratio of density to pressure
D) The inverse of acceleration

4) The relation between total pressure and total density in adiabatic conditions is given by:

A) $p_{tot} = \rho_{tot} c$
B) $\frac{p_{tot}}{p_0} = \left(\frac{\rho_{tot}}{\rho_0}\right)^{\kappa}$
C) $p = \rho c^2$
D) $p_0 = \frac{\rho_{tot}}{\rho_0}$

5) For small-amplitude waves, which assumption is valid?

A) $p \approx p_0$ and $\rho \approx \rho_0$
B) $p \ll p_0$ and $\rho \ll \rho_0$
C) $p \gg p_0$ and $\rho \gg \rho_0$
D) $p = \rho c^2$ always

6) The one-dimensional linearized wave equation for sound pressure is:

A) $\frac{\partial p}{\partial x} = \rho_0 \frac{\partial v}{\partial t}$
B) $\Delta p - \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} = -\rho_0 \dot{q}$
C) $\rho_0 \frac{\partial v}{\partial x} = \frac{\partial p}{\partial t}$
D) $p = \rho_0 c^2$

7) The Laplacian operator (Δ) in Cartesian coordinates is expressed as:

A) $\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$
B) $\frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial}{\partial r}\right)$
C) $\frac{1}{r^2} \frac{\partial^2}{\partial \phi^2} + \frac{\partial^2}{\partial z^2}$
D) $\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial}{\partial r}\right)$

8) A sound source emits 2 W of acoustic power uniformly in all directions. What is the sound intensity at a distance of 4 m?

A) 0.02 W/m²
B) 0.01 W/m²
C) 0.005 W/m²
D) 0.002 W/m²

9) The speed of sound in air (c) can be expressed in terms of adiabatic properties as:

A) $c = \sqrt{\frac{p_0}{\rho_0}}$
B) $c = \sqrt{\frac{\kappa p_0}{\rho_0}}$
C) $c = \kappa p_0 \rho_0$
D) $c = \left(\frac{\rho_0}{p_0}\right)^2$

10) If a sound wave travels in air with an intensity of 0.1 W/m² and the characteristic impedance of air is 415 Pa·s/m, what is the RMS sound pressure?

A) $20.4\,\text{Pa}$
B) $6.4\,\text{Pa}$
C) $0.64\,\text{Pa}$
D) $0.204\,\text{Pa}$

Chapter - 10

Key Words: Sound, Waves, Sound Field Quantities"