Impedance
The Impedance node adds an impedance boundary condition with the option to select between several built-in impedance models and engineering relations. The impedance condition is a generalization of the sound-hard and sound-soft boundary conditions:
Here, Zn is the (normal) specific input impedance of the external domain and it has the SI unit Pa·s/m — a pressure divided by a velocity. From a physical point of view, the acoustic input impedance is the ratio between the local pressure and local normal particle velocity. The Impedance boundary condition is a good approximation of a locally reacting surface — a surface for which the normal velocity at any point depends only on the pressure at that exact point.
For plane waves, the specific impedance Zn is related to the acoustic impedance Zac (ratio of average surface pressure and flow rate) and the mechanical impedance Zmech (ratio of force and velocity) via the area A of the boundary, according to
This relation only defines the relation between average or lumped impedance values at a surface.
All built-in impedance models only exist in the frequency domain. The only exception is the User defined impedance, which can be applied also in the time domain. In the frequency domain, the impedance can be any complex-valued number and thus models a surface that is both resistive and reactive. In the time domain, only resistive (real-valued impedance) effects can be included.
In the two opposite limits Zi → ∞ and Zi → 0, this boundary condition is identical to the Sound Hard Boundary (Wall) condition and the Sound Soft Boundary condition, respectively.
COMSOL Multiphysics supports the use of the rayl unit when specifying a value of the impedance (1 rayl = 1 Pa·s/m). Two variants exist: [rayl] and [rayl_cgs]. The latter is the definition of the unit in the cgs unit system. Notice that inconsistent definitions give: 1[rayl_cgs] = 10[rayl] = 10[Pa*s/m].
On an impedance boundary, postprocessing variables exist to evaluate the equivalent absorption coefficient. They are the absorption coefficient for normal incidence acpr.imp1.alpha_n and for random incidence acpr.imp1.alpha_ran (here given for the impedance feature imp1).
Notice that the Impedance boundary condition cannot directly be combined with a source like Normal Acceleration. In cases where such a behavior is desired, modeling a source impedance, this can be achieved by coupling the boundary to an Electrical Circuit model. See, for example, the Lumped Loudspeaker Driver model under Electroacoustic Transducers in the Application Library.
Impedance
A number of different types of impedance boundary conditions are included to address standard situations in many typical applications of pressure acoustics:
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Simple RCL models define lumped-parameter circuits describing, for example, the mechanical response of a microphone diaphragm. This model is only fully valid for plane waves at normal incidence.
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A set of physiological models enable easy modeling of the surface impedance of human skin and the acoustic loads of the human ear and eardrum. The ear and eardrum models are based on lumped models and only really valid for plane waves at normal incidence. They can be used as engineering approximations in other cases.
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The waveguide end impedances define simplified models of the acoustic losses at the end of pipes and ducts. Like for the RCL impedance, this model is only valid for plane waves at normal incidence at the end of waveguides. It is not valid when higher order modes start to propagate.
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A porous layer backed by a sound-hard wall model can be defined. This condition is of particular to room acoustic simulations as it has some options to handle random incidence with an effective formulation.
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Simple wave-type specific characteristic impedance models exist to address open boundaries in a simplified way.
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An option to define the impedance through the absorption coefficient of the surface.
The ear impedance, skin impedance, and RCL models provide tools for engineers to add realistic acoustic loads when, for example, developing and simulating headphones, hearing aids, headsets, and other mobile devices.
The theory for the impedance models is discussed in detail in Theory for the Boundary Impedance Models.
Choose an Impedance model — User defined (the default), RCL, Physiological, Waveguide end impedance, Porous layer, Specific characteristic impedance, or Absorption Coefficient.
User Defined
Allows the user to enter any expression and is the only impedance model that applies to time-dependent models. It is advantageous to enter complicated user-defined models as a variable under the Definitions node or use an interpolation function for measured data.
Enter the value of the Specific impedance Zn (SI unit: Pa·s/m). The default value is set to the characteristic specific acoustic impedance of air: 1.2 kg/m3·343 m/s.
RCL
The RCL model includes all possible circuits involving a source of damping (a resistor Rac), an acoustic mass or inertance (an inductor Lac), and a source of acoustic compliance (a capacitor Cac). The circuit elements are entered in acoustic units. These can be used as a simple model of, for example, the input impedance of a microphone, a loudspeaker cone, or other electromechanical applications. Other applications include general transmission line/circuit models with applications in materials with exotic acoustic properties. More advanced circuit models may be entered manually in the User defined option or by coupling to an Electric Circuit model (this requires the AC/DC Module) to a Lumped Port.
Choose an option from the list: Serial coupling RCL, Parallel coupling RCL, Parallel LC in series with R, Parallel RC in series with L, Parallel RL in series with C, Serial RC in parallel with L, Serial LC in parallel with R, or Serial RL in parallel with C.
Notice the matching diagram and Equation section information for each choice. Then enter the following:
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Equivalent acoustic resistance Rac (SI unit: kg/(m4·s)).
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Equivalent acoustic compliance Cac (SI unit: m4·s2/kg).
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Equivalent acoustic inertance Lac (SI unit: kg/m4).
Generic 711 Coupler — An Occluded Ear-Canal Simulator: Application Library path Acoustics_Module/Tutorials,_Thermoviscous_Acoustics/generic_711_coupler
Physiological
This is a set of simple models to address applications involving interactions of acoustics with the human body. The models comprise human skin, the impedance of the entire human ear including or excluding the pinna, the outward radiation impedance caused by the pinna, and the inward impedance experienced at the ear drum comprising the drum and the entire inner ear. For the two models of the human ear (with/without pinna), the pressure at the ear drum is automatically calculated. The variable has the form acpr.imp1.p_ear_drum and is available for postprocessing.
Note that the model for the ear drum impedance (see Ref. 28-30), also known as the model of Hudde and Engel, is defined in the references up to a maximum frequency of 16 kHz. The whole-ear models are based on the geometry of the ear canal and pinna of a specific ear (see Ref. 31 and appendix in Ref. 30), but person-to-person variations are to be expected. The ear canal geometry used here has an ear canal entrance corresponding to a circle of radius 4.25 mm, that is an area of 56.7 mm2. For applications where a specific ear canal geometry can be obtained, better results are expected by explicitly modeling this and applying the eardrum impedance at the end. For more details see the theory sections for the Human skin model and Models related to the human ear.
Choose an option from the list: Human skin, Outward human ear radiation, Human ear drum, Human ear without pinna, or Human ear, full. Then select material properties either From material (the default) or User defined, for the properties of air (the fluid). These are used to define the losses in the fluid. Select as required:
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Ratio of specific heats γ (SI unit: 1).
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Heat capacity at constant pressure Cp (SI unit: J/(kg·K)).
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Thermal conductivity k (SI unit: W/(m·K)).
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Dynamic viscosity μ (SI unit: Pa·s).
When the From material option is selected, remember to add a material under the Materials node and assign it to the specific boundary. The boundary will not automatically assume the physical properties of the domain.
Waveguide End Impedance
This is a set of idealized models for the acoustic losses at the end of pipes opening into vast domains. The models consider both square and circular cross sections, as well as flanged and unflanged pipe ends. These models are based on a plane wave assumption (propagation below the cutoff frequency).
Choose an option from the list: Flanged pipe, circular (the default), Flanged pipe, rectangular, Unflanged pipe, circular (low ka limit), or Unflanged pipe, circular. Then enter the following as required:
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Inner radius a (SI unit: m) or
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Inner width wi (SI unit: m) and Inner height hi (SI unit: m).
Open Pipe: Application Library path Acoustics_Module/Verification_Examples/open_pipe
Porous Layer
This choice models the acoustic losses of an incident field on a porous layer of user-defined thickness d backed by a sound-hard wall. The angle of incidence can be controlled to be normal to the surface or to use a specific angle or direction. An automatic option assigns an effective angle of incidence useful for room acoustics simulation. Use this boundary condition as an alternative to modeling the porous layer explicitly using the Poroacoustics feature. All material models from Poroacoustics are implemented in this feature.
Note that the Porous Layer condition is not compatible with the Anisotropic Acoustics domain condition. To model a porous layer, next to an Anisotropic Acoustics domain, it has to be modeled as a domain using the Poroacoustics feature.
Enter the Thickness of porous layer d (Si unit: m), select the Direction of incident wave, and select a Poroacoustic model. The rest of the settings are the same as for Poroacoustics. For the Direction of incident wave select Normal, Automatic, User defined, or From angle of incidence.
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For Normal, the normal incidence impedance value is used (the angle of incidence is set to 0o).
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For Automatic, the angle of incidence is set to 50o behind the scenes. This angle gives an on average value, valid for random incidence, and is useful when modeling closed spaces like rooms.
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For User defined, enter the Wave direction ek, the default is the surface normal. If a Background Pressure Field feature is present you can, for example, use the wave direction components: acpr.bpf1.kdirx, acpr.bpf1.kdiry, and acpr.bpf1.kdirz (here in 3D, use the appropriate tag).
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For From angle of incidence, enter the Incidence angle θ (default value: 50[deg]).
Specific Characteristic Impedance
This is a set of models describing the specific characteristic impedance associated with three basic wave types (ratio of local particle velocity to pressure): plane wave, cylindrical wave, and spherical wave. Although mostly of academic interest, these serve as good first-order and wave-type specific boundary-condition models of infinite domains (open boundaries). They can be applied to all cases where the tangential components of the acoustic field at the boundary may be ignored, that is when the angle of incidence is well defined and the wave direction is well known. Use the radiation conditions (Plane Wave Radiation, Spherical Wave Radiation, or Cylindrical Wave Radiation) if a nonreflecting open boundary is modeled.
Select a Wave type: Plane wave (the default), Cylindrical wave, or Spherical wave. Then enter the Wave direction ek for the plane wave (default is normal to the surface); the Radiating field source location r0 and Radiating field source axis rac for the cylindrical wave; or the Radiating field source location r0 for the spherical wave.
Absorption Coefficient
With this option the specific impedance of a boundary is defined through the normal incidence absorption coefficient αn (SI unit: 1) of the boundary. Since the absorption coefficient carries no phase information, it is also possible to define the phase of the associated reflection coefficient. If no phase is entered the impedance will be purely resistive (no reactive component is defined). This is typically an acceptable approximation at higher frequencies and it is also the assumption in ray tracing models. Sometimes surfaces are only specified through an absorption coefficient and in such cases using the Absorption coefficient option can be a first good approximation.
Enter the Normal incidence absorption coefficient αn (SI unit: 1) and the Phase (SI unit: rad).