Loudspeaker Driver in a Vented Enclosure

This example models the acoustic behavior of a loudspeaker driver mounted in a bass reflex enclosure.

Two of the most important design parameters for a loudspeaker driver is its sensitivity and the directivity of the system. The sensitivity is commonly defined as the on-axis sound pressure level, measured at a 1 m distance, as the driver is loaded by an AC voltage of 4 V. The directivity is assessed using the Directivity plot and represents the spatial sensitivity plotted against the frequency in a contour-like plot.

To isolate the driver’s performance from that of the environment it usually operates in, the driver is often set directly in an infinite baffle. This approach is used in another example Loudspeaker Driver — Frequency-Domain Analysis model in the Acoustics Module Application Library. The model described here borrows the electromagnetic results from that example and shows how the enclosure affects the sensitivity.

The model uses the Acoustic-Shell Interaction, Frequency Domain multiphysics interface. This interface provides automatic coupling between the Shell equation for the moving parts and the Pressure Acoustics equation in the surrounding air.

The model requires the Acoustics Module and the Structural Mechanics Module since it involves the use of the Shell interface.

Model Definition

Figure 1 shows the geometry of the considered driver in an infinite baffle, as modeled in the Loudspeaker Driver — Frequency-Domain Analysis example. In the model described here, the driver is set in a frame and placed in a bass reflex enclosure (Figure 2). The defining feature of this enclosure type is the vent, which in a properly designed enclosure acts to boost the sound at low frequencies.

Figure 3 shows the driver mounted in the enclosure. The infinite baffle is now flush with the front wall of the enclosure. The moving parts of the driver are drawn as surfaces rather than thin volumes. This lets you model them as shells, and vastly reduces the number of mesh elements required to resolve the model.

Figure 4 displays the complete model geometry, which includes a spherical domain for the air outside the enclosure. The concentric spherical mantle that envelopes the air domain is a perfectly matched layer (PML), acting to absorb the outgoing waves with a minimum of reflections.

Figure 1: The driver, here set in an infinite baffle as in the Loudspeaker Driver model.

Figure 2: The frame and the vented enclosure.

Figure 3: The geometry of the loudspeaker. The front of the box is set in an infinite baffle (not shown). The figure also shows the displacement of the moving parts at 3550 Hz.

Figure 4: The complete model geometry. Thanks to the symmetry with respect to the xz-plane, the model consists of only one half of the speaker and the outside air.

The model is driven by an electromagnetic force acting on the voice coil:

as derived in the documentation for the Loudspeaker Driver — Frequency-Domain Analysis model. BL is the so-called force factor of the voice coil and Zb is its blocked impedance; the electric impedance as measured when the coil is at stand-still. While BL is a constant 7.55 N/A, Zb is complex-valued function of the frequency. Both BL and Zb are taken directly from the Loudspeaker Driver — Frequency-Domain Analysis, the latter through interpolation from text files with the resistive and inductive contributions listed versus the frequency.

V0 is the applied driving voltage. The definition of sensitivity assumes a driving power that equals 1 W when the total impedance of the loudspeaker is at its nominal value. The modeled driver has a nominal impedance of 8 Ω, which translates to a driving voltage of V = V0eiωt with the amplitude V0 = 4 V. The second term in the expression for the driving force contains the velocity of the voice coil, v, which is unknown prior to the computation.

The electromagnetic force is applied as a homogeneous force density over the boundaries constituting the voice coil. As the cone and the suspension move along and deform, their local normal acceleration is automatically coupled over to become a sound source in the Pressure Acoustics equation. The computed acoustic pressure acts back as a load on the shell.

For a discussion of the material and damping parameters used in the moving parts of the driver, see the Loudspeaker Driver — Frequency-Domain Analysis model documentation.

Results and Discussion

The model produces the local stresses and strains in all the moving parts, as well as the sound pressure distribution inside and outside the enclosure at all frequencies solved for. As an example of how you can visualize the deformations, Figure 3 includes them as a deformed surface plot.

One way of looking at the sound distribution is as in Figure 5, which displays the pressure at 1000 Hz as an isosurface plot. An alternative option is shown in Figure 6, where the sound pressure level in dB is plotted on a slice near the symmetry plane. In Figure 7 and Figure 8, the same plot types illustrate the solution at the highest frequency, 3550 Hz. At this frequency, there is a complex standing wave pattern inside the enclosure.

Figure 5: Isosurface plot of the sound pressure at 1000 Hz.

Figure 6: Slice plot of the sound pressure level at 1000 Hz.

Figure 7: Isosurface plot of the sound pressure at 3550 Hz.

Figure 8: Slice plot of the sound pressure level at 3550 Hz.

Thanks to the full integral exterior-field evaluation available in COMSOL, you can evaluate the pressure not only inside the computational domain but also anywhere outside the domain. This makes it possible to, for instance, plot the sound pressure level at a given distance versus the elevation angle, or evaluate the directivity. The step-by-step instructions for this model show you how to plot the sensitivity (Figure 9) and the directivity (Figure 11).

Figure 9: Loudspeaker sensitivity measured as the on-axis sound pressure level (dB) at a distance of 1 m from the unit. The pressure is evaluated using an RMS input signal of 2.83 V, corresponding to a power of 1 W at 8 Ω. Note the logarithmic scale for the frequency.

Compared to the sensitivity of the baffled driver alone (Figure 8 in the Loudspeaker Driver — Frequency-Domain Analysis model), adding the enclosure clearly results in a “boost” for the lower frequencies, roughly in the range between 30 Hz and 100 Hz. As in the other model, the dip at around 700 Hz is due to a Helmholtz resonance in the cavity beneath the voice coil. This resonance would be less pronounced if the model were to account for the thermoviscous losses in the thin air gap between the magnet and the pole piece.

When using the exterior field calculation feature it is also possible to use the dedicated radiation pattern plots to display the spatial response of the speaker. They exist as both 1D polar plots, as 2D plots, and as 3D polar plots. A slightly modified version of the default generated 3D radiation pattern plot is shown in Figure 10. The plot, evaluated for f = 1000 Hz, clearly shows that the speaker has a notch in the horizontal plane and a boost upward and downward. The sensitivity curve in Figure 9 is evaluated at a fixed point 1 m in front of the speaker for all frequencies and, thus, does not provide any spatial information about the speaker performance. This spatial information is visualized in the 3D radiation pattern plot (one frequency at the time) and is of course an important component in speaker design and optimization.

Figure 10: 3D radiation pattern plot of the speaker sensitivity at 1 m evaluated at 1000 Hz.

A plot that gives a mixed frequency and spatial information is the so-called directivity plot depicted in Figure 11. The response of the speaker is evaluated on a half circle 1 m in front of the speaker, the data is collected for all frequencies and plotted in this contour-like plot. The plot is predefined in the Acoustics Module and is simply called Directivity.

Figure 11: Directivity plot for the loudspeaker.

Notes About the COMSOL Implementation

This application requires the Acoustics Module and the Structural Mechanics Module.

The model instructions let you import the entire geometry, including the air and PML domains. In a similar real-life modeling situation it is usually straightforward to add air and PML domains to your own CAD drawing of the enclosure.

Acoustics_Module/Electroacoustic_Transducers/vented_loudspeaker_enclosure

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 .

Geometry 1

Import 1 (imp1)

In the toolbar, click .

In the window for , locate the section.

Click .

Browse to the model’s Application Libraries folder and double-click the file vented_loudspeaker_enclosure.mphbin.

Click .

Definitions

To make it easier to select some important sets of domains and boundaries in a complicated geometry such as the one you are looking at, it is good modeling practice to begin by defining selections. When working on your own model, these selections are most conveniently defined by clicking and selecting directly in the geometry. The instructions however refer to them by numbers. Whenever a selection is made, it is therefore recommended that you use the Paste Selection button. Enter the number or list of numbers in the text field that appears. Input such as 5, 5-8, 13, or 4, 7, and 9 is accepted.

Explicit 1

In the toolbar, click .

In the window for , type Model Domain in the text field.

Select Domains 1, 2, and 5 only.

Because the magnetic engine and the walls of the enclosure will be considered rigid structures, there is no need to include their interior in the simulation.

Explicit 2

In the toolbar, click .

In the window for , type Air Domain in the text field.

Select Domain 2 only.

Explicit 3

In the toolbar, click .

In the window for , type PML Domain in the text field.

Select Domains 1 and 5 only.

Explicit 4

In the toolbar, click .

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

Locate the section. From the list, choose .

Click the button in the toolbar.

With wireframe rendering, you can see through boundaries and get a better view of which ones you are selecting.

Select Boundaries 50, 89, 97, and 108 only.

Explicit 5

In the toolbar, click .

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

Locate the section. From the list, choose .

Select Boundaries 49, 55, 56, 64, 74, 77, 80, 81, 85, 107, 112, 116, 117, 119, 120, 126, 130, and 131 only.

Explicit 6

In the toolbar, click .

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

Locate the section. From the list, choose .

Select Boundaries 35, 37, 133, and 135 only.

Explicit 7

In the toolbar, click .

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

Locate the section. From the list, choose .

Select Boundaries 87, 88, 105, and 106 only.

Union 1

In the toolbar, click .

In the window for , type Shell Boundaries in the text field.

Locate the section. From the list, choose .

Locate the section. Under , click .

In the dialog box, in the list, choose , , , and .

Click .

Next, define interpolation functions to bring in the blocked resistance and inductance from the model of the driver.

Interpolation 1 (int1)

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

Click .

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

Click .

In the text field, type Rb.

Locate the section. In the text field, type Hz.

In the text field, type ohm.

Locate the section. From the list, choose .

Set the extrapolation such that the results extend correctly just above the maximal frequency of 3500 Hz in the imported data. The simulation will run up to 3550 Hz.

Interpolation 2 (int2)

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

Click .

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

Click .

In the text field, type Lb.

Locate the section. In the text field, type Hz.

In the text field, type H.

Locate the section. From the list, choose .

To enable extraction of the velocity and application of the electric force on the apex, define average and integration operators.

Integration 1 (intop1)

In the toolbar, click and choose .

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

Locate the section. From the list, choose .

From the list, choose .

Average 1 (aveop1)

In the toolbar, click and choose .

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

Locate the section. From the list, choose .

From the list, choose .

Global Definitions

Define parameters for the driving voltage, the BL factor from the loudspeaker driver model, the frequency at which the material losses are specified, and the wavelength at 3500 Hz (used to set the mesh size).

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
V0	4[V]	4 V	Driving Voltage
BL	7.55[N/A]	7.55 Wb/m	Force factor from loudspeaker driver model
f_loss	40[Hz]	40 Hz	Frequency at which loss factor is given
omega_loss	2pif_loss	251.33 Hz	Angular frequency at which loss factor is given
lambda_min	343[m/s]/3500[Hz]	0.098 m	Wavelength at 3500 Hz (used in mesh)

Definitions

Next, create the expressions used in defining the electric driving force. The Description field is optional, but helps you keep track of what you are doing.

Variables 1

In the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
v0	av_apex(shell.u_tZ)	m/s	Cone velocity
vol_apex	2*int_apex(shell.d)	m³	Full coil volume
Zb	Rb(freq)+acpr.iomega*Lb(freq)	Ω	Blocked coil impedance
Fe	BLV0/Zb-v0BL^2/Zb	N·m/m	Electric driving force

Before creating the materials for use in this model, it is a good idea to specify where you want to solve the pressure acoustics equation and also which boundaries constitute moving parts and thus are to be modeled as shells. Using this information, the software can detect which material properties are needed.

Pressure Acoustics, Frequency Domain (acpr)

In the window, under click .

In the window for , locate the section.

From the list, choose .

The selection you just made removes the rigid structures from the volume where the pressure acoustics equation will be solved.

Shell (shell)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Add Material

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Material 2 (mat2)

In the window, under right-click and choose .

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

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	140[GPa]	Pa	Basic
Poisson’s ratio	nu	0.42	1	Basic
Density	rho	720	kg/m³	Basic

Material 3 (mat3)

Right-click and choose .

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

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	0.58[GPa]	Pa	Basic
Poisson’s ratio	nu	0.3	1	Basic
Density	rho	650	kg/m³	Basic

Material 4 (mat4)

Right-click and choose .

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

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	1.6[MPa]	Pa	Basic
Poisson’s ratio	nu	0.4	1	Basic
Density	rho	67	kg/m³	Basic

Add Material

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click to close the window.

Materials

Copper (mat5)

In the window for , locate the section.

From the list, choose .

NOTE: In the 2D model of the driver, the apex and the voice coil that is wound around it are two separate domains. In the model at hand, they are lumped together and treated as Copper. Because the apex is not expected to deform considerably, this is fine as long as the shell thickness is tuned so that the total mass becomes the same as in the 2D model.

Definitions

Perfectly Matched Layer 1 (pml1)

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Find the subsection. In the table, enter the following settings:


Xm (m)	Ym (m)	Zm (m)
-0.07	0	0

The default stretching is kept as it ensures better convergence of the iterative solver used in the study.

Pressure Acoustics, Frequency Domain (acpr)

With the materials defined, it is now time to set up the remaining physics of the model. Begin by specifying the symmetry condition in the acoustics domain.

Symmetry 1

In the window, under right-click and choose .

Select Boundaries 1, 4, and 44 only.

The way the geometry is set up, there are two boundaries just outside the surround, separating the air inside and outside the enclosure. To avoid the air leaking through these boundaries, turn them into (interior) hard walls.

Interior Sound Hard Boundary (Wall) 1

In the toolbar, click and choose .

Select Boundaries 33 and 136 only.

The acoustics physics is now ready, add an exterior-field calculation.

Exterior Field Calculation 1

In the toolbar, click and choose .

Select Boundaries 6 and 45 only.

In the window for , locate the section.

From the list, choose .

You have now supplied a source boundary encompassing all local sound sources and applied symmetry planes to account for the infinite baffle and the geometric symmetry. The Full integral setting lets you compute the pressure field (including phase) at any finite distance from speaker.

Shell (shell)

Now set up the shell physics. Begin by specifying the individual thicknesses and damping properties of the moving parts of the driver.

Damping 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the ηs list, choose . In the associated text field, type 0.04.

Linear Elastic Material 1

In the window, click .

Damping 2

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the βdK text field, type 0.14/omega_loss.

Linear Elastic Material 1

In the window, click .

Damping 3

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the βdK text field, type 0.46/omega_loss.

Linear Elastic Material 1

In the window, click .

Damping 4

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the ηs list, choose . In the associated text field, type 0.04.

Thickness and Offset 1

Change the thickness in the default node to 1[mm], this will be applied to the Cone of the speaker. Set the thickness of the other speaker parts.

In the window, click .

In the window for , locate the section.

In the d text field, type 1[mm].

Thickness and Offset 2

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the d text field, type 0.35[mm].

Thickness and Offset 3

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the d text field, type 1.4[mm].

Thickness and Offset 4

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the d text field, type 0.166[mm].

Next, apply the electric load onto the apex. Because the load is entered as a volume force density, you need to divide the force by the volume of the apex.

Body Load 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the FV vector as


0	x
0	y
Fe/vol_apex	z

Fix the outer rim of the spider and the surround.

Fixed Constraint 1

In the toolbar, click and choose .

Select Edges 64, 94, 264, and 275 only.

Keep in mind that you can use the button for all selections. This is also good idea for the following edge setting, which serves to impose the mirror symmetry.

Symmetry 1

In the toolbar, click and choose .

Select Edges 63, 68, 93, 95, 107, 109, 124, 139, 147, 155, 157, 164, 169, 171, 173, 196, 311–314, 318, 322, 323, 332, 336, 345, 357, 358, 368, and 376 only.

In the window for , locate the section.

From the list, choose .

Note that the to use as symmetry plane normal is set to 2. This means that the solution will be symmetric with respect to the xz-plane.

Add Physics

In the toolbar, click to open the window.

You have now set up all the relevant physics. This may be a good time to have a second look at all the features of the model, to verify that all settings make sense and apply to the right domains or boundaries. In particular, you can check the node, which has been set automatically.

Mesh 1

The mesh needs to resolve the fine details of the geometry as well as the waves at all frequencies. Furthermore, the PML should have at least 5 elements across its thickness (for the rational scaling). To improve the exterior-field evaluation add a thin boundary layer, of thickness lambda_min/5/10, adjacent to the PML domain. Turn off the smooth transition option - the single layer is used to get a good normal gradient evaluation. These requirements are all met by the following settings.

Boundary Layers 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Click to expand the section. Clear the check box.

Boundary Layer Properties

In the window, click .

In the window for , locate the section.

In the text field, type 1.

Select Boundaries 6 and 45 only.

From the list, choose .

In the text field, type lambda_min/5/10.

Size

In the window, click .

In the window for , click to expand the section.

In the text field, type lambda_min/5.

This gives you a minimum of 5 elements per wavelength at the highest frequency, 3500 Hz. In practice, remembar that a mesh convergence analysis is always recommended.

In the text field, type 1[mm].

The shortest edges of the geometry have length 1 mm, which you will resolve with this setting. You can increase the setting to get a coarser mesh in the narrow region. This can trigger a warning as COMSOL checks if the minimum size is larger than a gometric entity. The model will solve when a warning is present. It is up to the user to decide if the warning is relevant for the quality of the simulation results.

Use a swept mesh for the PML (with the size settings you will get 6 elements in the thickness). For more details on meshing PMLs see the modeling sections in the Acoustics Module User’s Guide.

Swept 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Distribution 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 8.

Finally, mesh the loudspeaker box. The mesh here is only for postprocessing purposes to be able to plot the box as a solid structure. It is not used in the computation.

Free Tetrahedral 1

In the window, right-click and choose .

Click .

Study 1

In the window, click .

In the window for , locate the section.

Clear the check box.

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

Click , the icon at the right of the edit-filed.

In the dialog box, choose from the list.

In the text field, type 10.

In the text field, type 3550.

From the list, choose .

Click .

This gives you frequencies with a 1/6 octave resolution from 10 to 3550 Hz with values specificed by the ISO standard. The model takes about 5 GB of RAM to solve and will solve in around 40 min (depending on your hardware). If you are short on time you can, for example, select 1/3 octave or simply octave spacing.

Generate the default solver sequence, then expand and enable the suggested iterative solve for this acoustic-shell interaction problem. The iterative solver is both faster and more memory efficient than the default direct solver.

Solution 1 (sol1)

In the toolbar, click .

In the window, expand the node.

Right-click and choose .

In the toolbar, click .

Results

3D Plot Group 1

In the toolbar, click and choose .

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

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

Select the check box.

In the text area, type Total displacement (m), f=eval(freq,Hz) Hz.

Clear the text field.

Surface 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Deformation 1

Right-click and choose .

Surface 1

In the toolbar, click .

Click the button in the toolbar.

In the window, click .

In the window for , locate the section.

From the list, choose .

This first plot shows how the shell displaces. The colors represent the total displacement and the deformation shows the position of the membrane at zero phase. Try looking at a couple of different solutions to see how the displacement amplitude and phase change with the frequency.

In the toolbar, click .

Displacement

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click .

From the list, choose .

In the toolbar, click .

Datasets

In order to display the box and reproduce Figure 3, create a new data set defined only in the solid parts of the model.

Study 1/Solution 1 (2) (sol1)

In the window, expand the node.

Right-click and choose .

In the window for , type Study 1/Solution 1 - Speaker box in the text field.

Selection

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domains 3, 4, 6, and 7 only.

Volume 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

Clear the check box.

Select the check box.

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

By plotting the domain number in grayscale, you have now added a black and white visualization of the enclosure and the solid parts of the driver. Lowering the resolution makes the visualization faster. If you rotate the plot a little, it should now resemble Figure 3.

Next, create a plot of the sound pressure level in a centered slice of the geometry. It is convenient to copy and paste the plot group from the first plot.

Displacement 1

Right-click and choose .

In the window for , type Sound Pressure Level in the text field.

Locate the section. In the text area, type Sound Pressure Level (dB), f=eval(freq,Hz) Hz.

Surface 1

In the window, expand the node, then click .

In the window for , locate the section.

Clear the check box.

Slice 1

In the window, right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

From the list, choose .

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

The smoothing algorithm will attempt to make the pressure continuous across the membrane. You get a sharper plot if you remove it.

In the toolbar, click .

The plot range is dominated by the large pressure drop in the PML. You can hide the PML domain by letting the solution data set be defined only in the air.

Study 1/Solution 1 (3) (sol1)

In the toolbar, click and choose .

In the window for , type Study 1/Solution 1 - Air domain in the text field.

Selection

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Sound Pressure Level

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click .

Click the button in the toolbar.

The plot should now like Figure 8. Switch to 1000 Hz to reproduce Figure 6.

From the list, choose .

In the toolbar, click .

Create a third plot group to show the 3D exterior-field polar plot. It is the spatial sensitivity characteristic evaluated at a distance of 1 m and at 1000 Hz.

3D Plot Group 3

In the toolbar, click and choose .

In the window for , type Exterior-Field Sound Pressure Level in the text field.

Locate the section. From the list, choose .

Radiation Pattern 1

In the toolbar, click and choose .

In the window for , locate the section.

In the text field, type (acpr.efc1.Lp_pext-77)/20.

Clear the check box.

Locate the section. Find the subsection. In the text field, type 30.

In the text field, type 60.

From the list, choose .

In the θ text field, type 90.

Locate the section. From the list, choose .

The first expression defines the shape of the 3D polar plot. In this case it is modified to set the reference at 77 dB and then scaled by a factor of 20 in order to visualize it relative to the model geometry. Changing the reference to 77 dB enhances the visualization of the notches. The surface color (second expression, enabled with the check box) gives the true SPL at 1 m from the speaker. Change the evaluation frequency to see how the directivity of the speaker changes with frequency. Set the expression back to acpr.efc1.Lp_pext and use the tool.

Volume 1

In the window, right-click and choose .

Volume 1

In the window, right-click and choose .

In the toolbar, click .

Click the button in the toolbar.

Rotate the plot in order to reproduce Figure 10.

Now, plot the sensitivity versus the frequency and reproduce Figure 9 as follows.

1D Plot Group 4

In the toolbar, click and choose .

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

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

In the text area, type Sensitivity.

Locate the section. Select the check box.

In the associated text field, type Frequency (Hz).

Select the check box.

In the associated text field, type SPL (dB).

Octave Band 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type pext(0,0,1).

Locate the section. From the list, choose .

This will plot the sound pressure level using the reference pressure, which defaults to the commonly used value 20 μPa. The pext(x,y,z) operator extracts the sound pressure in (x,y,z) = (0,0,1[m]).

You can change the style to 1/3 octave or octave to plot the sensitivity in bands if needed.

In the toolbar, click .

The next set of instructions show you how to create an isosurface plot of the pressure distribution in and outside of the speaker. The last plot is the directivity plot for the speaker.

Displacement 1

In the window, right-click and choose .

In the window for , type Acoustic Pressure in the text field.

Locate the section. In the text area, type Acoustic pressure (Pa), f=eval(freq,Hz) Hz.

Locate the section. From the list, choose .

Surface 1

In the window, expand the node, then click .

In the window for , locate the section.

Clear the check box.

Isosurface 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type -5 -2.5 0 2.5 5.

Locate the section. From the list, choose .

In the toolbar, click .

Click the button in the toolbar.

The plot should resemble Figure 5. To reproduce Figure 7, change the frequency and the levels for the isosurface plot.

Acoustic Pressure

In the window, click .

In the window for , locate the section.

From the list, choose .

Isosurface 1

In the window, click .

In the window for , locate the section.

In the text field, type -20 -10 -5 5 10 20.

In the toolbar, click .

Click the button in the toolbar.

Finally, create the directivity plot depicted in Figure 11 for the loudspeaker.

1D Plot Group 6

In the toolbar, click and choose .

In the window for , type Directivity Plot in the text field.

Directivity 1

In the toolbar, click and choose .

In the window for , locate the section.

Find the subsection. From the list, choose .

In the φ text field, type -90.

In the φ text field, type 180.

In the text field, type 90.

Find the subsection. In the text field, type 0.

In the text field, type 1.

Find the subsection. In the text field, type 1.

In the text field, type 0.

These settings give a directivity plot taken in the yz-plane where 0 deg. correspond to the z-axis direction. The evaluation plane can easily be visualized using the feature. The radius of the red evaluation circle displayed is normalized relative to the geometry.

Click .

In the toolbar, click .

Click the button in the toolbar.