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Lumped Loudspeaker Driver Using a Lumped Mechanical System
Introduction
This is a model of a moving-coil loudspeaker where a lumped parameter analogy represents the behavior of the electrical and mechanical speaker components. This lumped model is coupled to a 2D axisymmetric pressure acoustics model describing the surrounding air domain.
This example illustrates an alternative way of modeling mechanical components (mass, spring, and damper) using the Lumped Mechanical System interface in a lumped loudspeaker driver model. More information regarding the theory and a model setup can be found in the model Lumped Loudspeaker Driver.
Note: This model requires the Acoustics Module, the AC/DC Module, and the Multibody Dynamics Module.
Model Definition
A schematic representation of a moving coil loudspeaker is given in Figure 1. The figure shows a cross section of a loudspeaker. The speaker driver is placed in an infinite baffle with free space in front and on the back of the speaker. The speaker cone consists of the outer suspension, the diaphragm, and a dust cap (not marked in the figure).
Figure 1: Schematic representation of a moving coil speaker unit.
The mechanical and electrical components of the speaker that are lumped are shown inside the dotted box. On the electrical side it includes the voice-coil and magnetic system (permanent magnet and pole pieces), and on the mechanical side it includes the moving mass of the voice coil and speaker cone, the spring effect of the spider and outer suspension, as well as possible losses due to damping in these suspensions.
Lumped Representation of Electrical and mechanical components
The lumped or circuit representation for the electrical and mechanical parts of the system sketched in Figure 1 is shown in Figure 2. The upper figure represents the voice coil electrical system and the lower figure represents the lumped mechanical system of the speaker cone, suspensions, and mass of the voice coil. In both figures the node numbers are also shown — they are very useful when setting up the circuit model in COMSOL.
Figure 2: The lumped (circuit) representation for the electrical (top) and mechanical (bottom) components of the speaker driver.
In Figure 2 (top), the external voltage source is denoted by V0 and the generator output resistance is Rg, in this model Rg = 0 Ω. The voice coil resistance is RE, and the voice coil inductance is LE(ω), which is frequency dependent. The losses in the magnetic circuit are modeled through the frequency dependent resistance R'E(ω). The current controlled voltage source BL·uD, represents the back induced electromagnetic voltage generated when the voice coil (of length L) moves with velocity uD in the magnetic field B. Here, BL is the product of the magnetic field strength and the voice coil length L (see also Ref. 2 on how this can be modeled). In the electrical circuit, the current is denoted ic.
The mechanical system given in Figure 2 (bottom) has the following components:
•
Mass (MMD) representing the mass of voice coil and diaphragm assembly
•
Spring (1/CMS) representing the stiffness of speaker suspensions (both spider and outer suspension)
•
Damping (RMS) representing the possible losses in suspensions
The speaker diaphragm is subjected to the following forces acting in the axial direction:
•
Lorentz force: The Lorentz force is given by BL·ic for a voice coil of length L with current ic, where B is the magnetic flux density.
•
Pressure force: The acoustic pressure force is given by
(1)
where Δp is the pressure drop across the diaphragm and nz is the axial component of the surface normal n (see Figure 1). This expression gives the couplings from the acoustic finite element model to the lumped mechanical model. On the other hand, the coupling from the lumped mechanical model to the finite element model comes from specifying the acceleration on the surface of the diaphragm.
For more details about the model including small signal parameters, and finite element modeling refer to Ref. 3.
Results and Discussion
The generated pressure field is shown in Figure 3 for 1 kHz and 5 kHz. This plot shows the directive characteristic of the speaker cone at increasing frequencies; this nature is discussed more at the end of this section when discussing the directivity plot in Figure 7.
Figure 3: Acoustic pressure for a frequency of 1 kHz (top) and kHz (bottom).
Figure 4: The frequency dependent acoustic radiated power (upper left), electric input power (upper right), efficiency (lower left), and on-axis pressure field (lower right).
The first two graphs in Figure 4 (upper left and upper right) represent the acoustically radiated power PAR and the electric input power PE.
The third (lower-left) graph of Figure 4 represents the driver efficiency given in percent (%), that is, the ratio of the acoustic radiated power and the input electric power.
The pressure field along the z-axis is shown in the last graph of Figure 4 from z = Rair to z = 5 m, evaluated at a frequency of 1000 Hz. In the model, a Parametric Curve 2D is used to evaluate the exterior-field pressure outside the computational mesh.
Figure 5: The frequency dependent mechanical force amplitude (upper left), corresponding phase (upper right), diaphragm velocity amplitude (lower left), and corresponding phase (lower right).
In the top row of Figure 5, the frequency variation of the mechanical forces in mass, spring, and damper components are shown. The left graph shows the force amplitude whereas on the right graph, the phase is plotted. It can be seen that the spring force is dominant at low frequencies whereas the inertial force starts to dominate at higher frequencies. In the bottom row of Figure 5, the frequency variation of the amplitude and phase of the speaker cone axial velocity is shown.
The pressure and the sound pressure level evaluated at 1 m, using the dedicated radiation pattern plots, are shown in Figure 6, here evaluated at 10 kHz. In the figure the 0° mark corresponds to the axial z direction. Both figures show a very strong directive pattern as expected at this high frequency.
Figure 6: Exterior-field pressure and sound pressure level evaluated at a distance of 1 m the half sphere in front of the speaker and at 10 kHz.
In Figure 7 (top), the radiated intensity is illustrated at 100 Hz. The color plot represents the magnitude of the intensity vector I, the domain vector field represents the components of the intensity vector, and finally the vectors plotted on the edges represent the surface normals.
The final plot of this model is shown in Figure 7 (bottom); it is a so-called directivity plot of the speaker unit. The plot represents a contour plot of the sound pressure level Lp evaluated along a half circle in front of the speaker as function of the angle and the frequency, that is, Lp(θ, f). The x-axis represents the angle and runs from −90° to 90°. The y-axis is a logarithmic frequency axis running from 101 Hz to 104 Hz. The plot illustrates how the spatial response goes from a nearly omnidirectional constant value at the low frequencies, through a single lobe response at intermediate frequencies, and ends up as a complex directive pattern at high frequencies. This type of plot is very often used in industry, to characterize speakers and speaker units.
Figure 7: Top: intensity magnitude (color plot), intensity vector field (domain arrows), and surface normals (edge arrows) for 100 Hz. Bottom: directivity plot for the speaker. The x-axis is a scaled azimuthal angle that runs from -90° to 90° and the y-axis is a logarithmic frequency axis that runs from 10 Hz to 10 kHz.
References
1. W. Marshall Leach, Jr., Introduction to Electroacoustics and Audio Amplifier Design, Kendall Hunt, 2010.
2. Loudspeaker Driver Model Documentation, from the COMSOL Application Library.
3. Lumped Loudspeaker Driver Model Documentation, from the COMSOL Application Library.
Application Library path: Acoustics_Module/Electroacoustic_Transducers/lumped_loudspeaker_driver_mechanical
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Acoustics > Pressure Acoustics > Pressure Acoustics, Frequency Domain (acpr).
3
Click Add.
4
In the Select Physics tree, select AC/DC > Electrical Circuit (cir).
5
Click Add.
6
In the Select Physics tree, select Structural Mechanics > Lumped Mechanical System (lms).
7
Click Add.
8
Click  Study.
9
In the Select Study tree, select General Studies > Frequency Domain.
10
Click  Done.
Global Definitions
Load small signal and geometric parameters from a file.
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file lumped_loudspeaker_driver_mechanical_parameters.txt.
Import a simple 2D axisymmetric geometry of the speaker driver.
Geometry 1
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file lumped_loudspeaker_driver_mechanical.mphbin.
5
Click  Import.
6
In the Geometry toolbar, click  Build All.
Now, set up all the variables, selections, and component couplings under the Definitions node.
Definitions
Model variables
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, type Model variables in the Label text field.
3
Locate the Variables section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file lumped_loudspeaker_driver_mechanical_variables.txt.
Speaker
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Speaker in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 6, 11, and 14 only.
Internal wall
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Internal wall in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 7, 8, and 15 only.
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Speaker.
5
In the Operator name text field, type intop.
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
Select Domains 1 and 4 only.
3
In the Settings window for Perfectly Matched Layer, locate the Scaling section.
4
From the Coordinate stretching type list, choose Rational.
5
In the PML scaling factor text field, type 0.5.
6
In the PML scaling curvature parameter text field, type 5.
You have now changed the default settings for the perfectly matched layer (PML). The new settings will improve the performance of the PML at very low frequencies.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Air.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Now, set up the physics and the boundary conditions for the model.
Pressure Acoustics, Frequency Domain (acpr)
Interior Sound Hard Boundary (Wall) 1
1
In the Physics toolbar, click  Boundaries and choose Interior Sound Hard Boundary (Wall).
2
In the Settings window for Interior Sound Hard Boundary (Wall), locate the Boundary Selection section.
3
From the Selection list, choose Internal wall.
Interior Normal Acceleration 1
1
In the Physics toolbar, click  Boundaries and choose Interior Normal Acceleration.
2
In the Settings window for Interior Normal Acceleration, locate the Boundary Selection section.
3
From the Selection list, choose Speaker.
4
Locate the Interior Normal Acceleration section. Specify the a0 vector as
0
r
lms.M1.a
z
Exterior Field Calculation 1
1
In the Physics toolbar, click  Boundaries and choose Exterior Field Calculation.
2
Select Boundary 12 only.
3
In the Settings window for Exterior Field Calculation, locate the Exterior Field Calculation section.
4
From the Condition in the z = z0 plane list, choose Symmetric/Infinite sound hard boundary.
Proceed to set up the electrical circuit system and lumped mechanical system part of the model. When building this look at Figure 2 for the references to the node numbers used in the model.
Electrical Circuit (cir)
Voltage Source 1 (V1)
1
In the Electrical Circuit toolbar, click  Voltage Source.
2
In the Settings window for Voltage Source, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
1
n
0
4
Locate the Device Parameters section. In the vsrc text field, type V0.
Resistor 1 (R1)
1
In the Electrical Circuit toolbar, click  Resistor.
2
In the Settings window for Resistor, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
1
n
2
4
Locate the Device Parameters section. In the R text field, type R_g.
Resistor 2 (R2)
1
In the Electrical Circuit toolbar, click  Resistor.
2
In the Settings window for Resistor, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
2
n
3
4
Locate the Device Parameters section. In the R text field, type R_E.
Inductor 1 (L1)
1
In the Electrical Circuit toolbar, click  Inductor.
2
In the Settings window for Inductor, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
3
n
4
4
Locate the Device Parameters section. In the L text field, type L_E.
Resistor 3 (R3)
1
In the Electrical Circuit toolbar, click  Resistor.
2
In the Settings window for Resistor, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
3
n
4
4
Locate the Device Parameters section. In the R text field, type Rp_E.
Voltage Source 2 (V2)
1
In the Electrical Circuit toolbar, click  Voltage Source.
2
In the Settings window for Voltage Source, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p
4
n
0
4
Locate the Device Parameters section. In the vsrc text field, type BL*lms.M1.v.
Lumped Mechanical System (lms)
Spring 1 (K1)
1
In the Physics toolbar, click  Global and choose Spring.
2
In the Settings window for Spring, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p1
0
p2
1
4
Locate the Component Parameters section. In the k text field, type 1/C_MS.
5
Locate the Results section. Find the Add the following to default results subsection. Clear the Displacement checkbox.
Damper 1 (C1)
1
In the Physics toolbar, click  Global and choose Damper.
2
In the Settings window for Damper, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p1
0
p2
1
4
Locate the Component Parameters section. In the c text field, type R_MS.
5
Locate the Results section. Find the Add the following to default results subsection. Clear the Displacement checkbox.
Mass 1 (M1)
1
In the Physics toolbar, click  Global and choose Mass.
2
In the Settings window for Mass, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node names
p1
1
p2
2
4
Locate the Component Parameters section. In the m text field, type M_MD.
5
Locate the Results section. Find the Add the following to default results subsection. Clear the Displacement checkbox.
6
Select the Velocity checkbox.
Force Node 1 (frc1)
1
In the Physics toolbar, click  Global and choose Force Node.
2
In the Settings window for Force Node, locate the Node Connections section.
3
In the table, enter the following settings:
Label
Node name
p1
2
4
Locate the Terminal Parameters section. In the fp10 text field, type BL*cir.R2.i+F_D.
Mesh 1
Proceed and generate the mesh using the Physics-controlled mesh functionality. The frequency controlling the maximum element size is per default taken From study. Set the desired Frequencies in the study step. 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, use 8 elements per wavelength; the default Automatic is to have 5.
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
In the table, clear the Use checkboxes for Electrical Circuit (cir) and Lumped Mechanical System (lms).
4
Locate the Pressure Acoustics, Frequency Domain (acpr) section. From the Number of mesh elements per wavelength list, choose User defined.
5
In the text field, type 8.
Now, proceed to the study and set the frequencies, before building the mesh and solving.
Study 1
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
Click  Range.
4
In the Range dialog, choose ISO preferred frequencies from the Entry method list.
5
In the Start frequency text field, type 10.
6
In the Stop frequency text field, type 10000.
7
From the Interval list, choose 1/24 octave.
8
Click Replace.
Mesh 1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
Study 1
In the Study toolbar, click  Compute.
Results
Acoustic Pressure (acpr)
First, look at the default plots. Investigate the 2D Sound Pressure Level (acpr) plot to verify the performance of the perfectly matched layer (PML). After doing this you can disable plotting in the PML region, which is unphysical. Secondly, look at the default exterior-field plots and make a few changes, before setting up a range of plots to investigate the loudspeaker driver performance.
Sound Pressure Level (acpr)
Look at the sound pressure level (SPL) plots at the frequencies of 10 kHz, 1 kHz, and 10 Hz. Note that the SPL decreases nearly 100 dB over the width of the thickness of the PML. This means that the outgoing waves are extremely damped.
1
In the Model Builder window, click Sound Pressure Level (acpr).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 1000.
4
In the Sound Pressure Level (acpr) toolbar, click  Plot.
5
From the Parameter value (freq (Hz)) list, choose 10.
6
In the Sound Pressure Level (acpr) toolbar, click  Plot.
Study 1/Solution 1 (sol1)
In the Model Builder window, expand the Results > Datasets node, then click Study 1/Solution 1 (sol1).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 2 and 3 only.
Acoustic Pressure (acpr)
1
In the Model Builder window, under Results click Acoustic Pressure (acpr).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 1000.
4
In the Acoustic Pressure (acpr) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
From the Parameter value (freq (Hz)) list, choose 5000.
7
In the Acoustic Pressure (acpr) toolbar, click  Plot.
These two plots should reproduce Figure 3.
Radiation Pattern 1
1
In the Model Builder window, expand the Exterior-Field Sound Pressure Level (acpr) node, then click Radiation Pattern 1.
2
In the Settings window for Radiation Pattern, locate the Evaluation section.
3
Find the Angles subsection. From the Restriction list, choose Manual.
4
In the ϕ start text field, type -90.
5
In the ϕ range text field, type 180.
6
In the Exterior-Field Sound Pressure Level (acpr) toolbar, click  Plot.
Radiation Pattern 1
1
In the Model Builder window, expand the Exterior-Field Pressure (acpr) node, then click Radiation Pattern 1.
2
In the Settings window for Radiation Pattern, locate the Evaluation section.
3
Find the Angles subsection. From the Restriction list, choose Manual.
4
In the ϕ start text field, type -90.
5
In the ϕ range text field, type 180.
6
In the Exterior-Field Pressure (acpr) toolbar, click  Plot.
These two polar plots should reproduce Figure 6.
Now modify the default plots of the Lumped Mechanical System interface to reproduce Figure 5.
Mechanical Force, Amplitude
1
In the Model Builder window, under Results click Force, Amplitude (lms) 1.
2
In the Settings window for 1D Plot Group, type Mechanical Force, Amplitude in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section. In the y-axis label text field, type Force (N).
5
Locate the Legend section. From the Position list, choose Upper left.
6
Click the  x-Axis Log Scale button in the Graphics toolbar.
7
In the Mechanical Force, Amplitude toolbar, click  Plot.
Mechanical Force, Phase
1
In the Model Builder window, under Results click Force, Phase (lms) 1.
2
In the Settings window for 1D Plot Group, type Mechanical Force, Phase in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section. In the y-axis label text field, type Phase (rad).
5
Locate the Legend section. From the Position list, choose Lower right.
6
Click the  x-Axis Log Scale button in the Graphics toolbar.
7
In the Mechanical Force, Phase toolbar, click  Plot.
Diaphragm Velocity, Amplitude
1
In the Model Builder window, under Results click Velocity, Amplitude (M1).
2
In the Settings window for 1D Plot Group, type Diaphragm Velocity, Amplitude in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section. In the y-axis label text field, type Velocity (m/s).
5
Click the  x-Axis Log Scale button in the Graphics toolbar.
6
In the Diaphragm Velocity, Amplitude toolbar, click  Plot.
Diaphragm Velocity, Phase
1
In the Model Builder window, under Results click Velocity, Phase (M1).
2
In the Settings window for 1D Plot Group, type Diaphragm Velocity, Phase in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section. In the y-axis label text field, type Phase (rad).
5
Click the  x-Axis Log Scale button in the Graphics toolbar.
6
In the Diaphragm Velocity, Phase toolbar, click  Plot.
Acoustic Radiated Power
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Acoustic Radiated Power in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type f (Hz).
6
Select the y-axis label checkbox. In the associated text field, type Power (W).
7
Locate the Legend section. From the Position list, choose Lower middle.
Global 1
1
Right-click Acoustic Radiated Power and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Definitions > Variables > P_AR - Radiated power - W.
3
In the Acoustic Radiated Power toolbar, click  Plot.
4
Click the  x-Axis Log Scale button in the Graphics toolbar.
This plot should reproduce Figure 4 (top left).
Electric Input Power
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Electric Input Power in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type f (Hz).
6
Select the y-axis label checkbox. In the associated text field, type Power (W).
7
Locate the Legend section. From the Position list, choose Lower right.
Global 1
1
Right-click Electric Input Power and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Definitions > Variables > P_E - Electric input power (rms) - W.
3
In the Electric Input Power toolbar, click  Plot.
4
Click the  x-Axis Log Scale button in the Graphics toolbar.
This plot should reproduce Figure 4 (top right).
Driver Efficiency
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Driver Efficiency in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type f (Hz).
6
Select the y-axis label checkbox. In the associated text field, type Efficiency (%).
Global 1
1
Right-click Driver Efficiency and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
eta*100
1
Driver efficiency
4
In the Driver Efficiency toolbar, click  Plot.
5
Click the  x-Axis Log Scale button in the Graphics toolbar.
This plot should reproduce Figure 4 (bottom left).
Set up a parametric curve used to evaluate the exterior-field outside of the computational mesh.
Parametric Curve 2D 1
1
In the Results toolbar, click  More Datasets and choose Parametric Curve 2D.
2
In the Settings window for Parametric Curve 2D, locate the Expressions section.
3
In the z text field, type s*5[m]+(1-s)*Rair.
4
Select the Only evaluate globally defined expressions checkbox.
On-axis Pressure Field
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type On-axis Pressure Field in the Label text field.
3
Locate the Data section. From the Dataset list, choose Parametric Curve 2D 1.
4
Locate the Title section. From the Title type list, choose Label.
5
Locate the Plot Settings section.
6
Select the x-axis label checkbox. In the associated text field, type z (m).
7
Select the y-axis label checkbox. In the associated text field, type Pressure (Pa).
8
Locate the Data section. From the Parameter selection (freq) list, choose From list.
9
In the Parameter values (freq (Hz)) list box, select 1000.
10
Locate the Legend section. Clear the Show legends checkbox.
Line Graph 1
1
Right-click On-axis Pressure Field and choose Line Graph.
2
Select Boundary 4 only.
3
In the Settings window for Line Graph, locate the y-Axis Data section.
4
In the Expression text field, type pext(r,z).
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type z.
7
Click to expand the Quality section. From the Evaluation settings list, choose Manual.
8
From the Resolution list, choose Extra fine.
9
In the On-axis Pressure Field toolbar, click  Plot.
This plot should reproduce Figure 4 (bottom right).
Create a 2D intensity plot that includes the magnitude of the intensity vector acpr.I_rms as well as an arrow surface (vector field plot) of the intensity vector, with the components (acpr.Ir,acpr.Iz).
Intensity
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Intensity in the Label text field.
Surface 1
1
Right-click Intensity and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type acpr.I_rms.
Arrow Surface 1
1
In the Model Builder window, right-click Intensity and choose Arrow Surface.
2
In the Settings window for Arrow Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Pressure Acoustics, Frequency Domain > Intensity > acpr.Ir,acpr.Iz - Intensity.
3
Locate the Coloring and Style section. From the Color list, choose Black.
Arrow Line 1
1
Right-click Intensity and choose Arrow Line.
2
In the Settings window for Arrow Line, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Pressure Acoustics, Frequency Domain > Geometry > acpr.nr,acpr.nz - Normal vector.
3
Locate the Coloring and Style section. From the Color list, choose Black.
4
In the Intensity toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Now change the evaluation frequency to 5000 Hz, 1000 Hz, and 100 Hz in order to plot and reproduce Figure 7.
Intensity
1
In the Model Builder window, click Intensity.
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 5000.
4
In the Intensity toolbar, click  Plot.
5
From the Parameter value (freq (Hz)) list, choose 1000.
6
In the Intensity toolbar, click  Plot.
7
From the Parameter value (freq (Hz)) list, choose 100.
8
In the Intensity toolbar, click  Plot.
Next create the directivity plot of the speaker.
Directivity
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Directivity in the Label text field.
Directivity 1
1
In the Directivity toolbar, click  More Plots and choose Directivity.
2
In the Settings window for Directivity, locate the Evaluation section.
3
Find the Angles subsection. From the Restriction list, choose Manual.
4
In the ϕ start text field, type -90.
5
In the ϕ range text field, type 180.
6
Click to expand the Coloring and Style section. From the Layout list, choose Frequency on y-axis.
7
In the Directivity toolbar, click  Plot.
8
Click the  y-Axis Log Scale button in the Graphics toolbar.
This should reproduce the directivity plot depicted in Figure 7. You can tailor the plot to your needs using the normalization options or defining the specific levels to use in the contour plot.