Acoustic waves

Linear acoustic waves

Strong formulation

This formulation of acoustic waves is derived under linear approximation assumption of Newtonian fluids. Such fluids follows the Navier–Stokes equations for compressible Newtonian fluids

$$\begin{align} \rho \Bigr(\frac{\partial \boldsymbol{v}}{\partial t} + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v} \Bigr) &= -\nabla p + \nabla \cdot (\mu (\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^T)) + \nabla (-\tfrac{2}{3} \mu \nabla \cdot \boldsymbol{v}) + \boldsymbol{f} \\[10pt] \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \boldsymbol{v}) &= 0. \end{align}$$

For small density and pressure variations the speed of sound in the fluid $c$ is given by

$$\begin{align} c = \sqrt{\frac{\partial p}{\partial \rho}}. \end{align}$$

We can neglect the viscosity terms in (1) since we are interesteed only in the region within a few acoustic wavelengths. Considering an inviscid fluid and no external forces, (1) rewrites as

$$\begin{align} \rho \Bigr(\frac{\partial \boldsymbol{v}}{\partial t} + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v}\Bigr) = -\nabla p \\[10pt] \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \boldsymbol{v}) = 0, \end{align}$$

which can, for tiny perturbations, be linearised around a mean value:

$$\begin{align} p = \overline{p} + \delta p,\\ \rho = \overline{\rho} + \delta \rho,\\ \boldsymbol{v} = \overline{\boldsymbol{v}} + \delta \boldsymbol{v} = \delta \boldsymbol{v}, \end{align}$$

where the overlined quantities are the mean values (constant in space and time) and the $\delta$ terms are tiny perturbations. Since the fluid is at rest, $\overline{\boldsymbol{v}} = 0$. Injecting equations (6) to (8) into equations (4) and (5) and neglecting nonlinear perturbations gives

$$\begin{align} \overline{\rho} \frac{\partial \delta \boldsymbol{v}}{\partial t} + \delta \rho \displaystyle\frac{\partial \delta \boldsymbol{v}}{\partial t} + \overline{\rho} (\delta \boldsymbol{v} \cdot \nabla) \delta \boldsymbol{v} = - \nabla \delta p\\[10pt] \frac{\partial \delta \rho}{\partial t} + \overline{\rho} \nabla \cdot \delta \boldsymbol{v} = 0. \end{align}$$

By algebraic manipulation (using the product rule and the continuity equation multiplied by $\delta \boldsymbol{v}$) and neglecting second‐order perturbations we obtain useful approximation

$$\begin{align} \overline{\rho}\,(\delta \boldsymbol{v}\!\cdot\nabla)\,\delta \boldsymbol{v} \approx - \delta \rho \frac{\partial \delta \boldsymbol{v}}{\partial t}. \end{align}$$

As a result, the two middle terms in the first relation of (9) cancel and we obtain obtains

$$\begin{align} \overline{\rho} \frac{\partial \delta \boldsymbol{v}}{\partial t} + \nabla\,\delta p &= 0 \\[10pt] \frac{\partial \delta \rho}{\partial t} + \overline{\rho} \nabla \cdot \delta \boldsymbol{v} &= 0. \end{align}$$

Taking the divergence of the first relation and the time derivative of the second we get

$$\begin{align} \overline{\rho} \nabla \cdot \frac{\partial \delta \boldsymbol{v}}{\partial t} + \Delta \delta p &= 0 \\[10pt] \frac{\partial^2 \delta \rho}{\partial t^2} + \overline{\rho} \nabla \cdot \frac{\partial \delta \boldsymbol{v}}{\partial t} &= 0, \end{align}$$

which can be combined into a single equation

$$\begin{align} \frac{\partial^2 \delta \rho}{\partial t^2} - \Delta \delta p = 0. \end{align}$$

With the isentropic approximation (3) the acoustic wave equation can be written as

$$\begin{align} \Delta \delta p - \frac{1}{c^2} \frac{\partial^2 \delta p}{\partial t^2} = 0, \end{align}$$

with $c$ the speed of sound in the fluid and $\delta p$ the pressure variation around the mean pressure.

Weak formulation

From now on we will write the $\delta p$ just as $p$. To form the weak formulation we will multiply both sides by test function $p^{\prime}$ and integrate over the whole domain $\Omega$.

$$\begin{align} \int_{\Omega}\ (\Delta p - \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2})\ p^{\prime}\ d\Omega = 0. \end{align}$$

We can use the Leibniz rule for a nabla operator to rewrite the Laplace term.

$$\begin{align} \int_{\Omega}\ \nabla \cdot (p^{\prime} \nabla p)\ d\Omega - \int_{\Omega}\ \nabla p \cdot \nabla p^{\prime}\ d\Omega - \int_{\Omega}\ \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2} \ p^{\prime}\ d\Omega = 0, \end{align}$$

and then apply the Divergence theorem on the divergence term to obtain the weak formulation

$$\begin{align} \int_{\Gamma}\ (\nabla p \cdot \boldsymbol{n})\ p^{\prime}\ d\Gamma - \int_{\Omega}\ \nabla p \cdot \nabla p^{\prime}\ d\Omega - \int_{\Omega}\ \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2} \ p^{\prime}\ d\Omega = 0. \end{align}$$

Available Interactions

Perfectly matched layer

Implements an absorbing boundary region to simulate infinite space and prevent reflections from truncated boundaries. Placed at the outer boundary of the acoustic domain, it absorbs outgoing sound waves without reflection.

Key properties:

• Used to simulate open boundaries for scattering acoustic structures.

Two types available:

• AML type: Suitable for smooth boundaries.
• Box PML: Suitable for rectangular or cornered boundaries.
• The number of PML layers can be tuned in the Shared PML settings.

Example:

Use case for Perfectly matched layer

Constraint

Applies a fixed acoustic pressure $p$ to a node or region, enforcing a Dirichlet boundary condition for the acoustic wave equation. Use this to define fixed-pressure boundaries, such as a driven pressure inlet.

How to use:

Provide a scalar pressure value in Pascals (Pa).

Example:

101325 applies standard atmospheric pressure to the selected node or region.

Unit: Acoustic pressure in Pascals (Pa)

Absorbing boundary

Periodicity

Imposes periodic boundary conditions on the acoustic pressure field $p$ between two boundaries. Reduces the computational domain size for geometrically symmetric problems, avoiding the need to model the full geometry.

Example:

Periodicity is similar in every physics section. This example is from $𝜑$-formulation, but workflows to setup Periodicity are identical:

Periodicity in electric motor

Acoustic damping

Normal acceleration

Continuity

Ensures continuity of the scalar acoustic pressure $p$ across an interface between two boundaries. Used to map the acoustic field continuously across mismatched or sliding mesh interfaces.

Example:

In a rotating machinery simulation, continuity maps the acoustic pressure between the rotor and stator meshes, which slide relative to each other during rotation. Continuity is similar in every physics section. This example is from $𝜑$-formulation, but workflows to setup Continuity are identical:

Continuity in electric motor

Couplings to Other Physics

This formulation supports the following couplings:

Acoustic structure (Solid mechanics)

Acoustic structure (Elastic waves)

Use cases

• Acoustic waves - Solid mechanics coupling in a loudspeaker
• Acoustic waves - Elastic waves coupling in a PMUT array