Probe Tube Microphone

It is often not possible to insert a normal microphone directly into the sound field being measured. The microphone may be too big to fit inside the measured system, such as for in-the-ear measurements for hearing aid fitting. The size of the microphone may also be too large compared to the wavelength, so that it disturbs the acoustic field. In these cases, a probe tube may be attached to the microphone case (see the sketch in Figure 1) in order to distance the microphone from the measurement point. In this model, the microphone sensitivity change due to the addition of this small probe is investigated, see Ref. 1 (Chapter 4, pp. 160–162) for further details.

This is a time-dependent model of a generic probe tube microphone setup consisting of an external acoustic domain, an elastic probe tube, and the cavity in front of the microphone diaphragm; see the sketch in Figure 1. The probe tube, modeled using the Pipe Acoustics, Transient interface, is connected to two separate 3D pressure acoustics domains, leading to a fully coupled acoustics simulation.

Figure 1: Sketch of the probe tube microphone setup, consisting of a probe tube of length L and outer diameter D0. The tube is connected to a cavity (with the indicated dimensions) in front of the microphone diaphragm. An external sound field hits the probe tube with wave vector k.

This application requires the Acoustics Module.

Model Definition

The probe tube microphone geometry and the computational domain considered in this model are shown schematically in Figure 1. From right to left, the model consists of the cavity in front of the microphone, which takes the form of a cylinder of radius R and height H connected to a cone of bottom radius R and top radius D0. D0 is also the outer diameter of the probe tube of length L. The probe is made of an elastic material with inner diameter 2dr = 2 mm and wall thickness dw = 0.3 mm. The tube material is set to have a Young’s modulus of 0.1 GPa and a Poisson’s ratio of 0.4.

Table 1 lists the model parameters,

Table 1: Model Parameters.
Variable	Value	Description
f	500 Hz	Incident wave frequency
T	1/f	Incident wave period
ω	2πf	Incident wave angular frequency
c0	343 m/s	Speed of sound at 20oC and 1 atm
λmin	c0/f	Wavelength of the incident wave
k	2π/λmin	Wave number of the incident wave
L	20 mm	Probe tube length
R	5 mm	Microphone casing radius
H	5 mm	Cavity height
dr	1 mm	Pipe inner radius
dw	0.3 mm	Pipe wall thickness
D0	2dr + 2dw = 2.6 mm	Probe tube outer diameter
kD0	kD0 = 0.02	Wave number times tube diameter
dvisc	0.1 mm	Viscous boundary layer thickness at 500 Hz

A sinusoidal wave moving in the positive x direction is incident on the probe tube microphone. The wave with an amplitude of 1 Pa is given by

(1)

where ω is the angular frequency, and k is the wave number (these are given in Table 1). Such a plane monochromatic wave exists as a built-in option for the feature in the interface. In the transient analysis, a ramp is automatically added to smoothly increase the background pressure field amplitude over the first period T = 1/f0. This is done to ensure numerical stability and can be deactivated under the with the turned on.

Modeled using the Pipe Acoustics, Transient interface, the long probe tube is treated as a1D structure. This assumption is valid as long as the interaction between the probe tube and the incoming sound field can be neglected. This, in turn, requires that k D0 << 1, which is true for the current choice of parameters because k D0 = 0.02; see Table 1.

Another important assumption is that there are no significant thermal and viscous boundary losses inside the probe tube. Also, this assumption holds for the current setup, where the incident field is a monochromatic wave. At a driving frequency of 500 Hz, the viscous boundary layer thickness inside the probe tube is of the order 0.1 mm, which is 10 times smaller than the internal radius dr. The thickness of the boundary layer is proportional to

Because the diaphragm is not a fully rigid structure, assume it to have a resistive loss given by the impedance Z = 100·106 Ns/m5. This value is of the order of magnitude observed in common condenser microphones.

Results and Discussion

An important parameter of a probe tube extension to a microphone is the relation between the pressure at the tip and the pressure at the diaphragm (as measured by the microphone). This transfer function is necessary to calibrate the measurement system. The pressure at the tip of the probe and the pressure at the diaphragm are shown in Figure 2. After an initial transient, the solution becomes periodic after about 4 ms. Hereafter, the system has a gain of about 1.4 and a phase shift equal to

. The gain and phase shift depend on the frequency content of the applied signal — in this case a pure harmonic tone of 500 Hz.

A full transfer function could be obtained, for example, by using a Gaussian pulse as the incident signal and then performing a Fourier transform of the signal. This requires resolving the frequency content with an appropriate mesh that would become very dense. Moreover, the model assumptions might not hold for the full range of frequencies.

Figure 3 depicts the pressure distribution in the xz-plane at the end of the time interval shown in Figure 2.

Figure 2: Pressure as function of time at the probes tip (green line) and at the microphone diaphragm (blue line).

Figure 3: Pressure distribution in the xz-plane at time t = 10 ms.

Notes About the COMSOL Implementation

Coupling between 1D and 3D domains

The Pipe Acoustics domain (probe domain) is connected to the first Pressure Acoustics domain (exterior domain) through a pressure condition at the pipe end. This manual coupling ensures continuity of the pressure between the pipe domain and the incident pressure field.

The Pipe Acoustics domain is connected to the second Pressure Acoustics domain (diaphragm cavity domain) through the Acoustics-Pipe Acoustics multiphysics coupling. This coupling guarantees continuity of pressure and velocity between both domains through the equations

(2)

(3)

where ppipe is the pipe pressure at the connection point, A is the total area of the connected acoustic boundary, pt is the total pressure in the acoustic domain, and S is the connected acoustic boundary. Furthermore, n is the vector normal to the boundary, qd is a dipole source, and upipe is the pipe velocity at the connection point.

Further examples of coupling between 1D pipes and 3D domains are found in the tutorial model Acoustics of a Pipe System with 3D Bend and Junction, also available in the Application Gallery: www.comsol.com/model/acoustics-of-a-pipe-system-with-3d-bend-and-junction-40831.

Solver SETUP

The only adjustment needed to be done to the solver is to change it from using a segregated approach to using a fully coupled approach. The default logic in the study is to use segregated groups for physics that do not have a built-in multiphysics coupling. Here, the coupling is set up manually in one of the ends of the pipe. The settings for the transient solver are handled automatically by setting the in the section (as done in the instructions below.)

Reference

1. D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley & Sons, 2000.

Acoustics_Module/Tutorials,_Pipe_Acoustics/probe_tube_microphone

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

In the tree, select .

Click .

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file probe_tube_microphone_parameters.txt.

Geometry 1

Next, define the geometry of the model. Either build the geometry by following the instructions at the end of the document or import the geometry from a sequence, and then jump to the section. To import the geometry sequence, click in the toolbar. Browse to the model’s Application Libraries folder and open the file probe_tube_microphone_geom_sequence.mph.

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file probe_tube_microphone_geom_sequence.mph.

In the toolbar, click

Use the wireframe rendering to more easily see and access faces and lines inside the geometry.

Click the

button in the toolbar.

Click the

button in the toolbar.

The geometry should look like the figure below.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Air 1 (mat2)

In the window for , locate the section.

From the list, choose .

Select Edge 25 only.

Definitions

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , type aveop_mic in the text field.

Locate the section. From the list, choose .

Select Boundary 12 only.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , type intop_tip in the text field.

Locate the section. From the list, choose .

Select Point 14 only.

Pipe Acoustics, Transient (patd)

Proceed to set up the physics of the problem. Select the domains where the different physics are applied and add the appropriate boundary conditions to couple these.

In the window, under click .

In the window for , locate the section.

Click

Select Edge 25 only.

Pipe Properties 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the di text field, type 2*dr.

Locate the section. From the list, choose .

From the E list, choose . In the associated text field, type 0.1[GPa].

From the ν list, choose . In the associated text field, type 0.4.

In the Δw text field, type dw.

In the window, click .

In the window for , locate the section.

In the fmax,sol text field, type f.

Pressure 1

In the toolbar, click

and choose .

Select Point 14 only.

In the window for , locate the section.

In the pin text field, type p2.

This manual coupling applies the pressure of the first Pressure Acoustics Domain into the Pipe Acoustics Domain.

Pressure Acoustics, Transient (actd)

In the window, under click .

In the window for , locate the section.

Click

Select Domain 1 only.

Locate the section. In the fmax,sol text field, type f.

Cylindrical Wave Radiation 1

In the toolbar, click

and choose .

Select Boundaries 1, 2, 13, and 18 only.

In the window for , locate the section.

Specify the raxis vector as


0	x
0	y
1	z

Add an incident plane wave as defined in Equation 1 using the built-in option. The wave is automatically multiplied with a smoothing ramp function over the first period. This option can be seen by enabling the view.

Incident Pressure Field 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the p0 text field, type 1.

From the c list, choose .

From the list, choose .

Specify the ek vector as


1	x
0	y
0	z

In the f0 text field, type f.

Pressure Acoustics, Transient 2 (actd2)

In the window, under click .

In the window for , locate the section.

Click

Select Domains 2 and 3 only.

Locate the section. In the fmax,sol text field, type f.

Impedance 1

In the toolbar, click

and choose .

Select Boundary 7 only.

In the window for , locate the section.

In the Zi text field, type 100e6[N*s/m^5]*R^2*pi.

Multiphysics

Acoustic-Pipe Acoustic Connection 1 (apc1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Selecting this second domain is important, as otherwise the probe will have both ends connected to the same domain.

Mesh 1

Proceed and generate the mesh based on the suggestion. The frequency controlling the maximum element size is per default taken , that is, from the . In general, 5 to 6 second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. In this model, we use 10 elements per wavelength; the default is to have 5.

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type 10.

Locate the section. From the list, choose .

In the text field, type 10.

Locate the section. From the list, choose .

Size 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Edge 25 only.

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type L/10.

Size 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 12 only.

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type dr/2.

Click

Study 1

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,T/25,5*T).

The data will be recorded for 5 periods (5*T) with a sampling of 25 points per period (increase the number of points to get better resolution in the stored solution).

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node.

Right-click and choose .

In this model, the coupling between the first Pressure Acoustics domain and the Pipe Acoustics Domain is set up manually. The default behavior, in the study, is for COMSOL to generate solver suggestions for the individual constituting physics (notice the tags) and sets up a segregated solver suggestion. This model is best solved using the fully coupled approach.

Now, solve the model and proceed to look at the results. First, look at the default plots and then create two custom plot groups. The model only takes a few seconds to solve.

In the window, under click .

In the window for , click

Results

Acoustic Pressure (patd)

Acoustic Pressure (actd)

Acoustic Pressure, Isosurfaces (actd)

Acoustic Pressure (actd2)

Acoustic Pressure, Isosurfaces (actd2)

Follow the steps below to reproduce the plot in Figure 3.

3D Plot Group 7

In the toolbar, click

and choose .

Slice 1

Right-click and choose .

In the window for , locate the section.

In the text field, type p2.

Locate the section. From the list, choose .

In the text field, type 1.

Locate the section. From the list, choose .

From the list, choose .

Slice 2

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type p3.

Locate the section. From the list, choose .

In the text field, type 1.

Click to expand the section. From the list, choose .

Line 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 0.5*patd.dh.

Select the check box.

Click to expand the section. From the list, choose .

In the toolbar, click

Cross Sections

In the window, under click .

In the window for , type Cross Sections in the text field.

The figure should look like the one depicted in Figure 3.

1D Plot Group 8

In the toolbar, click

and choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
aveop_mic(p3)	Pa	Diaphragm Pressure
intop_tip(p)	Pa	Probe Tip Pressure

Pressure Profiles

In the window, click .

In the window for , locate the section.

Select the check box.

In the associated text field, type Pressure (Pa).

In the toolbar, click

In the text field, type Pressure Profiles.

The figure should look like the one in Figure 2.

Appendix: Geometry Sequence Instructions

If you want to create the geometry yourself, follow these steps.

From the menu, choose .

In the window, click

In the window, click

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
L	20[mm]	0.02 m	Probe tube length
R	5[mm]	0.005 m	Microphone casing radius
H	5[mm]	0.005 m	Probe tube volume height
dr	1[mm]	0.001 m	Pipe inner radius
dw	0.3[mm]	3E-4 m	Pipe wall thickness