## Pressure Acoustics

Pressure acoustics is the most common use of the Acoustics Module. You can model pressure acoustics effects, such as the scattering, diffraction, emission, radiation, and transmission of sound. Simulations run in the frequency domain employ the Helmholtz equation, whereas in the time domain, the classical scalar wave equation is used. In the frequency domain, both FEM and BEM are available, as well as hybrid FEM–BEM. In the time domain, time implicit (FEM) as well as time explicit (dG-FEM) formulations are available.

There are many options to account for boundaries in acoustics models. For instance, you can add a boundary condition for a wall or an impedance condition for a porous layer. You can use ports to excite or absorb acoustic waves at the inlet and outlet of waveguides using multimode expansion. Sources like prescribed acceleration, velocity, displacement, or pressure can be applied on exterior or interior boundaries. Further, you are able to use radiation or Floquet periodic boundary conditions to model open or periodic boundaries.

The Acoustics Module can also be used to model pipe acoustics, computing the acoustic pressure and velocity in flexible pipe systems. Applications include HVAC systems, large piping systems, and musical instruments such as organ pipes.

## Electroacoustics: Speakers and Microphones

When modeling speakers and microphones, an essential part involves acoustic–structure interaction, where the fluid pressure causes a fluid load on the solid domain, and the structural acceleration affects the fluid domain as a normal acceleration across the fluid–solid boundary. The Acoustics Module includes a variety of acoustic–structure interaction capabilities.

For modeling transducers of all sorts, the capabilities included in the Acoustics Module are readily combined with functionality from the AC/DC Module, the MEMS Module, or the Structural Mechanics Module to create fully coupled multiphysics FEM models. This includes detailed modeling of magnets and voice coils in loudspeaker drivers or the electrostatic forces in condenser microphones. In electro-mechanical-acoustic transducer systems, it is easy to use lumped circuit models to simplify the electric and mechanical components. Both approaches are solved with a fully two-way coupling. In addition, the (linear) small signal behavior and nonlinear large signal dynamics can be modeled and analyzed. In miniature transducer systems, like mobile devices, condenser microphones, and hearing aid receivers, the important damping due to the thermoviscous boundary layer losses is included. There is also extensive functionality for modeling piezoelectric transducers of all kinds.

## Microacoustics

For an accurate microacoustic analysis of acoustic propagation in geometries with small dimensions, you need to account for losses associated with viscosity and thermal conduction; particularly, the losses in the viscous and thermal boundary layers. These effects are solved in full and automatically included when running a thermoviscous simulation using the Acoustics Module and are important for vibroacoustics modeling in miniature electroacoustic transducers like microphones, mobile devices, hearing aids, and MEMS devices. For detailed transducer modeling, you can use the built-in multiphysics couplings between structures and thermoviscous acoustic domains.

The software accounts for additional effects, including the full transitional behavior from adiabatic to isothermal at very low frequencies. Local nonlinear effects, such as vortex shedding in microspeaker ports or perforates, can be captured in the time domain with the addition of the nonlinear governing terms. There is also a dedicated feature for computing and identifying propagating and nonpropagating modes in narrow waveguides and ducts.

## Elastic Waves and Ultrasound in Solids

The propagation of sound in solids happens through small-amplitude elastic oscillations of the solid's shape and structure. These elastic waves are transmitted to surrounding fluids as ordinary sound waves.

You can use the Acoustics Module to model the propagation of elastic waves in solids and porous materials, for single-physics or multiphysics applications, such as vibration control, nondestructive testing (NDT), or mechanical feedback. Application areas range from micromechanical devices to seismic wave propagation. Elastic wave propagation over large domains containing many wavelengths is solved using a higher-order dG-FEM time-explicit method, and is multiphysics enabled for couplings with fluids as well as piezoelectric materials. The full structural dynamics formulation accounts for the effects of shear waves as well as pressure waves. You can model the coupled propagation of elastic and pressure waves in porous materials solving Biot's equations.

## Ultrasound in Fluids

Acoustic disturbances with frequencies that are not audible for humans are classified as ultrasound, which implies that ultrasonic waves have a short wavelength. For this, you can compute the transient propagation of acoustic waves in fluids over large distances in two ways: modeling wave propagation that includes a background flow or modeling the effects of high-power nonlinear acoustics.

You can solve for transient linear acoustics in a simulation that contains many wavelengths in a stationary background flow by modeling the convected wave equation. Applications include flowmeters; exhaust systems; and biomedical applications, for instance, ultrasonic imaging and high-intensity focused ultrasound (HIFU).

For high-power nonlinear acoustics applications, you can capture progressive wave propagation phenomena where the cumulative nonlinear effects surpass the local nonlinear effects. This includes modeling the formation and propagation of shocks.

For both options, there are multiphysics capabilities available for fully coupling your model with elastic waves in structures and/or with piezoelectric materials.

## Aeroacoustics

You can efficiently run computational aeroacoustics (CAA) simulations with a decoupled two-step approach in the Acoustics Module. First, you find the background mean flow using tools from the CFD Module or a user-defined flow profile; then, you solve for the acoustic propagation.

For convected acoustics simulations, there are finite element formulations including linearized Navier–Stokes, linearized Euler, and linearized potential flow aeroacoustics simulations. You can compute acoustic variations in pressure, density, velocity, and temperature in the presence of any stationary isothermal or nonisothermal background mean flow. The formulations readily account for convection, damping, reflection, and diffraction of acoustic waves by the flow. There is also functionality for FSI analyses in the frequency domain with predefined couplings to elastic structures.

Flow-induced noise can be included in a pressure acoustics analysis by the addition of aeroacoustic flow sources using Lighthill's acoustic analogy with input from a transient large eddy simulation (LES) or detached eddy simulation (DES) with the CFD model.

## Geometrical Acoustics

The geometrical acoustics capabilities of the Acoustics Module can be used to evaluate high-frequency systems where the acoustic wavelength is smaller than the characteristic geometric features. There are two methods available: ray acoustics and acoustic diffusion.

For ray acoustics, you can compute the trajectories, phase, and intensity of acoustic rays. Additionally, you can calculate impulse responses, energy and level decay curves, as well as the classical objective room acoustic metrics. The rays can propagate in graded media, which is necessary in underwater acoustics applications. For simulating ray acoustics in both air and water, specialized atmosphere and ocean attenuation material models are available that are important for wave propagation over large distances and at high frequencies.

For acoustic diffusion, you can determine the sound pressure level distribution in coupled rooms and the reverberation times at different locations. The acoustics are modeled in a simplified way using a diffusion equation for the acoustic energy density. This method is well suited for quick analyses inside buildings and other large structures.

## Acoustic Streaming

With the Acoustics Module, it is possible to simulate acoustic streaming that describes the physical process where an acoustic field can induce movement in a fluid. The module contains multiphysics capabilities for coupling acoustics and fluid flow with model acoustic streaming phenomena for pressure and thermoviscous acoustics.

Acoustic streaming is a nonlinear phenomenon that occurs due to the nonlinearity of the Navier–Stokes equations. The Acoustics Module computes the forces, stresses, and boundary slip velocities that the acoustic field induces in a fluid in order to generate the streaming flow field. This phenomena is used widely in biotech and semiconductor processing and is important in microfluidics and lab-on-a-chip systems for applications such as particle handling, the mixing of fluids, and microfluidic pumps.

## Features and Functionality in the Acoustics Module

Explore the features and functionality of the Acoustics Module in more detail in the sections below.

### Built-In User Interfaces

The Acoustics Module provides built-in user interfaces covering all of the application areas listed above. These interfaces define sets of domain equations, boundary conditions, initial conditions, predefined meshes, predefined studies with solver settings, as well as predefined plots and derived values. All of these steps are accessed within the COMSOLMultiphysics^{®} environment. Meshing and solver settings are handled automatically by the software, with options for manual editing.

The COMSOLMultiphysics^{®} workflow for building acoustics models is the same as for building a model with any other physics interface. In this way, it is easy to incorporate multiple physics into one acoustics model, and there are several multiphysics interfaces built into the Acoustics Module and accessible when combining with other add-on modules from the COMSOL product suite.

### Pressure Acoustics Interfaces

For modeling pressure acoustics, there are multiple user interfaces where the sound field is represented by a scalar pressure variable. The general-purpose interfaces, based on FEM, include the capability of solving in both the frequency and time domain. For the transient case, nonlinear effects can be included and are based on the Westervelt equation.

To efficiently solve large radiation and scattering problems, frequency-domain BEM is available that couples seamlessly with the finite-element-based interfaces, both acoustic and structural.

To efficiently solve large transient models, a specialized user interface based on the discontinuous Galerkin finite element method and a time-explicit solver is available. This interface can be coupled to the corresponding time-explicit interface for elastic and piezoelectric waves.

### High-Frequency Pressure Acoustics

Two highly specialized interfaces are available for quick high-frequency acoustics analysis in the frequency domain. These interfaces are based on computing the Kirchhoff–Helmholtz integral and include one interface for scattering analysis and another interface for radiation analysis. This type of analysis can be used as a first step before moving on to a more computationally demanding analysis based on FEM or BEM.

### Elastic Waves Interfaces

The Acoustics Module includes user interfaces for modeling the propagation of linear elastic waves in solids, porous, and piezoelectric materials. These interfaces readily couple to fluid domains using a set of built-in multiphysics couplings.

The solid mechanics interfaces have the capability of representing full elastodynamics and can be used for modeling elastic waves in solids in both the frequency and time domain. A port boundary condition is specifically implemented to model and handle various propagating modes in elastic waveguide structures.

The poroelastic interfaces are used for modeling poroelastic waves in porous materials. These waves result from the complex two-way interaction between acoustic pressure variations in the saturating fluid and the elastic deformation of the solid porous matrix. The poroelastic interfaces solve Biot’s equations in the frequency domain and include loss mechanisms from viscous losses (Biot), for modeling rocks and soils, as well as thermal and viscous losses (Biot–Allard), for sound-absorbing materials in air.

Two interfaces, based on a time-explicit discontinuous Galerkin formulation, can be used for modeling linear elastic waves in solid and piezoelectric domains. These interfaces can be coupled and are suited for modeling domains with several wavelengths efficiently. A dedicated *Fracture* boundary condition can be used to model two solids with nonideal bonding, for example, if the goal is to simulate the acoustic response of a defect or a delamination zone. In addition, these interfaces can be coupled with the time-explicit interfaces for pressure acoustics and the convected wave equation.

### Aeroacoustics Interfaces

For modeling detailed convected acoustics, or flow-borne noise, a number of aeroacoustics interfaces are available in both the frequency and time domain. These interfaces are used for simulating one-way interaction of a background fluid flow with an acoustic field. There are different physics interfaces that solve the governing equations under various physical approximations.

The linearized Navier–Stokes interfaces are used for solving for the acoustic variations in pressure, velocity, and temperature.

The linearized Euler interfaces are used for computing the acoustic variations in density, velocity, and pressure in the presence of a stationary background mean flow that is well approximated by an ideal gas flow.

Special boundary mode interfaces are available for computing propagating and nonpropagating modes in waveguides and ducts in the presence of a background flow.

For simplified analysis, interfaces for linearized potential flow can be used in both the time and frequency domains.

### Open Domains and Radiation

To model an unbounded computational domain, you can truncate it using so-called perfectly matched layers (PMLs) in both time and frequency domains. Alternative methods include using radiation boundary conditions or an exterior domain modeled using a boundary element method interface.

For finite-element-based interfaces, an exterior field calculation feature can be used to determine the pressure in any point outside the computational domain. Dedicated results and analysis capabilities exist for visualizing the radiation pattern of the exterior field (near and far field) in polar, 2D, and 3D plots.

### Flow-Induced Noise

By combining the Acoustics Module with the CFD Module, you get access to a hybrid aeroacoustic (CAA) method for modeling flow-induced noise.

The computational method is based on the FEM discretization of Lighthill's acoustic analogy (wave equation). This formulation of the equations ensures that any solid (fixed or vibrating) boundaries are implicitly taken into account.

The functionality relies on coupling an LES fluid flow simulation, using the CFD Module, to an aeroacoustic flow source for pressure acoustics, available in the Acoustics Module.

### Finite Element and Boundary Element Methods

Most user interfaces in the Acoustics Module are based on different versions of FEM. User interfaces based on BEM are available as well, and can be seamlessly combined with FEM-based interfaces. Hybrid FEM–BEM is very efficient for modeling acoustic–structure interaction involving vibrating structures.

Applications for hybrid FEM–BEM include transducers and radiation simulations with complex geometries where you model the transducer (piezo or electromagnetic) with FEM and the exterior acoustics with BEM.

A BEM-based interface can be used to replace an FEM-based radiation condition or PML, as well as the FEM-based exterior-field calculations.

### Boundary Conditions and Sources for Pressure Acoustics

There is a large variety of boundary conditions available for pressure acoustics, including hard walls and conditions for applying sources. There are radiation, symmetry, periodic, and port conditions for modeling open boundaries. Impedance conditions include models for different parts of the human ear, human skin, simple RCL circuit models, and more. By using the interface for boundary mode analysis, you can study propagating modes in the cross sections of waveguides and ducts. The options for modeling idealized sources include built-in options for monopole, dipole, and quadrupole point sources.

### Acoustic–Structure Interaction Interfaces

The interfaces for acoustic–structure interaction apply to phenomena where the fluid pressure causes a load on the solid domain and the structural acceleration affects the fluid domain across the fluid–solid boundary. This is also known as vibroacoustics.

The interfaces include the capability of solving in either the frequency or the time domain. The solids included in the simulations can be isotropic, anisotropic, porous, or piezoelectric.

By combining with the Structural Mechanics Module, the structural side of the coupling can additionally include structural shells or membranes.

By combining with the Multibody Dynamics Module, you can include the effects of multiple moving rigid or flexible parts connected through various types of joints.

For more advanced options, by combining with the AC/DC Module or MEMS Module, you can analyze fluid–structure interaction involving electric or magnetic forces, including solids having electrostrictive or magnetostrictive material properties.

### Thermoviscous Acoustics Interfaces

In order to accurately model acoustics in geometries with small dimensions, it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations. Near walls, there are viscous and thermal boundary layers. Here, viscous losses due to shear and thermal conduction become important because of large gradients.

The interfaces for thermoviscous acoustics include the capability to simultaneously model the effects of pressure, particle velocity, and acoustic temperature oscillations. Thermoviscous acoustics is, for example, used when modeling the response of small transducers like microphones and receivers, also known as microacoustics. A multiphysics coupling with thermoelasticity physics allows for detailed modeling of damping in MEMS applications, including detailed thin-film damping.

The interfaces are available for solving in both the frequency and time domains. In the time domain, nonlinear effects can also be modeled.

Lumped acoustic and electroacoustic representations can readily be extracted from and/or coupled with the computational domain using ports, lumped ports, or the *Lumped Speaker Boundary* feature. This is useful for system simulation using, for example, the Thiele–Small representation of a microtransducer in a mobile phone.

### Ultrasound and Convected Wave Equation Interfaces

For analyzing transient linear ultrasound devices and processes, you can use the convected wave equation user interface. This interface can be used to efficiently solve large transient linear acoustic models containing many wavelengths in a stationary background flow.

For simulating the propagation of high-amplitude nonlinear acoustic waves, you can use the nonlinear pressure acoustics user interface. This interface includes special functionality for capturing shocks.

Both interfaces include absorbing layers that are used to set up effective nonreflecting-like boundary conditions. The interfaces are based on the discontinuous Galerkin method and use a computationally efficient time-explicit solver.

### Ray Acoustics and Acoustic Diffusion Interfaces

For running simulations in the high-frequency limit, where the acoustic wavelength is much smaller than the characteristic geometric features, you can use the user interfaces for ray acoustics. In addition, for quick analyses, there is a user interface for solving the acoustic diffusion equation, also known as energy finite elements.

Both user interfaces are suited for modeling acoustics in rooms and concert halls. The ray acoustics interface can also be used in outdoor or underwater scenarios, for example.

The ray acoustics interface is used to compute the trajectories, phase, and intensity of acoustic rays. It includes the capabilities of impulse response analysis, showing the level decay curves and computed objective room acoustic metrics, such as EDT, T_{60} values, etc.

### Acoustic Losses and Porous Materials

A more approximate way of introducing losses is to use the equivalent fluid models available in the pressure acoustics interfaces. In a homogenized way, this introduces attenuation properties to the bulk fluid that mimic different loss mechanisms. The fluid models include losses due to bulk thermal conduction, viscosity and relaxation in the atmosphere (air) and the ocean (seawater), and models for simulating the damping in porous materials.

In addition to the *Thermoviscous Acoustics* interface that simultaneously models the effects of pressure, particle velocity, and acoustic temperature oscillations, the *Pressure Acoustics* interface can also account for thermoviscous boundary layer losses. Narrow-region acoustics can be used in narrow ducts and waveguides of constant cross sections, while the thermoviscous boundary layer impedance (BLI) condition is applicable for geometries larger than the boundary layer.

When applicable, the equivalent fluid and homogenized models are computationally very efficient. However, for representing losses in porous materials with higher fidelity, you can combine pressure acoustics with the effects of poroelastic wave propagation.