Acoustic waves

This page derives the weak formulation for linear acoustic wave propagation and describes the available boundary conditions and couplings in Allsolve.

Linear acoustic waves

Strong formulation

This formulation is derived under the linear approximation for Newtonian fluids. The governing equations are the Navier–Stokes equations for compressible flow:

\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} \tag{1} \\[10pt] \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \boldsymbol{v}) &= 0. \tag{2} \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}}. \tag{3} \end{align*}

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

\begin{align*} \rho \Bigr(\frac{\partial \boldsymbol{v}}{\partial t} + (\boldsymbol{v} \cdot \nabla) \boldsymbol{v}\Bigr) = -\nabla p \tag{4} \\[10pt] \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \boldsymbol{v}) = 0, \tag{5} \end{align*}

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

\begin{align*} p = \overline{p} + \delta p, \tag{6}\\ \rho = \overline{\rho} + \delta \rho, \tag{7}\\ \boldsymbol{v} = \overline{\boldsymbol{v}} + \delta \boldsymbol{v} = \delta \boldsymbol{v}, \tag{8} \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 $\text{(6)}$ to $\text{(8)}$ into equations $\text{(4)}$ and $\text{(5)}$ 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 \tag{9} \\[10pt] \frac{\partial \delta \rho}{\partial t} + \overline{\rho} \nabla \cdot \delta \boldsymbol{v} = 0. \tag{10} \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}. \tag{11} \end{align*}

As a result, the two middle terms in the first relation of $\text{(9)}$ cancel and we obtain

\begin{align*} \overline{\rho} \frac{\partial \delta \boldsymbol{v}}{\partial t} + \nabla\,\delta p &= 0 \tag{12} \\[10pt] \frac{\partial \delta \rho}{\partial t} + \overline{\rho} \nabla \cdot \delta \boldsymbol{v} &= 0. \tag{13} \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 \tag{14} \\[10pt] \frac{\partial^2 \delta \rho}{\partial t^2} + \overline{\rho} \nabla \cdot \frac{\partial \delta \boldsymbol{v}}{\partial t} &= 0, \tag{15} \end{align*}

which can be combined into a single equation

\begin{align*} \frac{\partial^2 \delta \rho}{\partial t^2} - \Delta \delta p = 0. \tag{16} \end{align*}

With the isentropic approximation $\text{(3)}$ 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. \tag{17} \end{align*}

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

Weak formulation

From this point, $\delta p$ is written simply as $p$ . To form the weak formulation, multiply both sides by the test function $p^{\prime}$ and integrate over the domain $\Omega$ :

\begin{align*} \int_{\Omega}\ (\Delta p - \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2})\ p^{\prime}\ d\Omega = 0. \tag{18} \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, \tag{19} \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. \tag{20} \end{align*}

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

Applies a first-order absorbing boundary condition to truncate the computational domain, allowing outgoing acoustic waves to pass through with minimal reflection. Lighter-weight than a Perfectly matched layer — suitable when a simpler setup is acceptable and some small residual reflection is tolerable.


How to use	Select the outer boundary of the acoustic domain where waves should exit. No additional parameters are needed — the condition automatically matches the acoustic impedance of the medium.

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 follows the same principles, regardless of which physics module it belongs to. See how periodicity is used in magnetism as a reference: Periodicity in electric motors

Acoustic damping

Applies acoustic energy dissipation within a region, modelling losses due to porous materials, viscous absorption, or other damping mechanisms. Reduces the acoustic pressure amplitude as waves propagate through the damped region.


How to use	Select the target region and set the damping value.

Normal acceleration

Applies a prescribed normal acceleration $a_n$ to a boundary, acting as an acoustic source. Use this to model vibrating surfaces that radiate sound into the acoustic domain — a loudspeaker cone or a vibrating plate for example.


How to use	Select the target boundary and set the normal acceleration value.
Example	`1.0` applies an inward normal acceleration of 1 m/s² to the boundary.
Unit	Acceleration in meters per second squared (m/s²)

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	Continuity follows the same principles, regardless of which physics module it belongs to. See how continuity is used in magnetism as a reference: Continuity in electric motors

Couplings to Other Physics

This formulation supports the following couplings:

Acoustic structure
- Solid mechanics
- Elastic waves