Postprocessing Variables

Several specialized variables specific to acoustics are predefined in the Acoustics Module and can be used when analyzing the results of an acoustic simulation. The variables are available from the expression selection menus when plotting.

In this section:

•

Intensity Variables

•

Power Dissipation Variables

•

Pressure Acoustics, Boundary Mode Variables

In the COMSOL Multiphysics Reference Manual:

•

Results Analysis and Plots

•

Operators, Functions, and Constants

Intensity Variables

The propagation of an acoustic wave is associated with a flow of energy in the direction of the wave motion, the intensity vector I. The sound intensity in a specific direction (through a specific boundary) is defined as the time average of (sound) power per unit area in the direction of the normal to that area.

Knowledge of the intensity is important when characterizing the strength of a sound source — that is, the power emitted by the source. The power is given by the integral of n·I on a surface surrounding the source, where n is the surface normal. The intensity is also important when characterizing transmission phenomena, for example, when determining transmission loss, insertion loss curves, or absorption.

The acoustic intensity vector I (SI unit: W/m2) is defined as the time average of the instantaneous rate of energy transfer per unit area (sound power) pu, such that

(2-4)

where p is the pressure and u the particle velocity.

Frequency Domain

In the frequency domain (harmonic time dependence), the time averaging integral Equation 2-4 reduces to

(2-5)

where

denotes complex conjugation. In the frequency domain the velocity is readily expressed in terms of the pressure as

. Using the specific characteristic acoustic impedance for plane waves, the intensity can also be expressed in terms of the root mean square (RMS) pressure as

In the Pressure Acoustics, Frequency Domain interface the general formulation, valid for all types of waves given in Equation 2-5, is used to define the intensity.

When using the scattered field formulation, by adding a Background Pressure Field feature, the intensity variables for the total, background, and scattered fields are available in postprocessing if the option is selected.

Time Domain

For time-dependent problems, the equivalent quantity is the instantaneous value of the intensity, defined as

This expression is difficult to recover in pressure acoustics and would require the solution of an additional PDE to calculate the velocity from the pressure dependent variable. Only the intensity I (time averaged) is available as postprocessing variable in the frequency domain and can be selected from the expressions menus when plotting. The instantaneous intensity does exists as a postprocessing variable in transient interfaces such as Linearized Euler, Linearized Navier–Stokes, or Thermoviscous Acoustics where the velocity is solved for explicitly, see Modeling with the Aeroacoustics Branch or Modeling with the Pressure Acoustics Branch (FEM-Based Interfaces) for details.

Postprocessing Variables

The variables are defined in Table 2-3, Table 2-4, and Table 2-5. In the variable names, phys_id represents the interface name, for example, acpr for a Pressure Acoustics, Frequency Domain interface.

Table 2-3: Intensity Variables in 3D
Variable	Description
phys_id.I_mag	Magnitude of the intensity vector (frequency domain only)
phys_id.Ix	x-component of the intensity vector (frequency domain only)
phys_id.Iy	y-component of the intensity vector (frequency domain only)
phys_id.Iz	z-component of the intensity vector (frequency domain only)

Table 2-4: Intensity Variables in 2D Axisymmetric
Variable	Description
phys_id.I_mag	Magnitude of the intensity vector (frequency domain only)
phys_id.Ir	r-component of the intensity vector (frequency domain only)
phys_id.Iz	z-component of the intensity vector (frequency domain only)

Table 2-5: Intensity Variables in 2D
Variable	Description
phys_id.I_mag	Magnitude of the intensity vector (frequency domain only)
phys_id.Ix	x-component of the intensity vector (frequency domain only)
phys_id.Iy	y-component of the intensity vector (frequency domain only)

In the COMSOL Multiphysics Reference Manual:

•

Results Analysis and Plots

•

Expressions and Predefined Quantities

Power Dissipation Variables

Common to the Pressure Acoustics fluid models (also porous materials) and The Thermoviscous Acoustics, Frequency Domain Interface is that all the interfaces model some energy dissipation process, which stem from viscous and thermal dissipation processes. The amount of dissipated energy can be of interest as a results analysis variable or as a source term for a multiphysics problem. An example could be to determine the amount of heating in the human tissue when using ultrasound. In the Acoustics Module special variables exist for the dissipation.

•

For the case of a plane wave propagating in the bulk of a fluid (the general thermal and viscous fluid models described in Thermally Conducting and Viscous Fluid Model) the dissipation is

and in the frequency domain after averaging over one period

(2-6)

where * in Equation 2-6 is the complex conjugate operator.

•

In addition, an approximate expression for the dissipated energy density from a propagating plane wave exists for the Narrow Region Acoustics, the Poroacoustics models and attenuation in Pressure Acoustics. This total dissipated power density Qpw is defined by

where

is the magnitude of the intensity vector I, and k is the wave number. This expression is an approximation and is only valid for traveling plane waves (or waves that are close to plane); however, it has many uses as a first estimate of the dissipation since it is easy to calculate in many different situations. The expression is, for example, not valid for standing waves in resonant systems. When the above expression is not valid, the dissipated energy should be calculated using an energy balance approach.

The power dissipation variables are defined in Table 2-6. In the variable names, phys_id represents the name (acpr, for example, for a pressure acoustics interface).

Table 2-6: Power dissipation Variables
Variable	Description
phys_id.diss_therm	Thermal power dissipation density
phys_id.diss_visc	Viscous power dissipation density
phys_id.diss_tot	Total thermal and viscous power dissipation density
phys_id.Q_pw	Plane-wave total dissipated power density

Pressure Acoustics, Boundary Mode Variables

A series of special variables exist for postprocessing after solving a boundary mode acoustics problem. They include in-plane and out-of-plane components of the velocity v and acceleration a.

The in-plane (ip) and out-of-plane (op) components to the acceleration and velocity are defined as

where n is the normal to the surface being modeled. The velocity and acceleration are defined in terms of the gradient of the pressure p as follows

and

where kn is the out-of-plane wave number solved for, m is a possible radial wave mode number, and

is the tangential derivative along the boundary.

The boundary mode acoustics variables are defined in Table 2-7. In the variable names, phys_id represents the name (acbm, for example, for a Boundary Mode Acoustics interface).

Table 2-7: Boundary Mode Acoustics variables in 3D
Variable	Description
phys_id.vipx	In-plane velocity, x-component
phys_id.vipy	In-plane velocity, y-component
phys_id.vipz	In-plane velocity, z-component
phys_id.vip_rms	In-plane velocity RMS value
phys_id.aipx	In-plane acceleration, x-component
phys_id.aipy	In-plane acceleration, y-component
phys_id.aipz	In-plane acceleration, z-component
phys_id.aip_rms	In-plane acceleration RMS value
phys_id.vopx	Out-of-plane velocity, x-component
phys_id.vopy	Out-of-plane velocity, y-component
phys_id.vopz	Out-of-plane velocity, z-component
phys_id.vop_rms	Out-of-plane velocity RMS value
phys_id.aopx	Out-of-plane acceleration, x-component
phys_id.aopy	Out-of-plane acceleration, y-component
phys_id.aopz	Out-of-plane acceleration, z-component
phys_id.aop_rms	Out-of-plane acceleration RMS value