Piezoelectric Tonpilz Transducer

Transmitting Voltage Response (TVR)

The specific acoustic impedance Zaco is computed as the ratio of the impedance of the head mass surface exposed to water to the characteristic impedance Z0 of water as shown in Equation 1. The value of Z0 is computed from the product of density and speed of sound in water at a temperature of 293.15 K.

(1)

The impedance of the head mass surface is computed from the ratio of the area integral of the acoustic pressure p to the area integral of the normal component of the structural velocity vn of the surface. In this case the normal velocity is the same as the Z-component of the velocity vz which in frequency domain can be represented as the product of the Z-component of the structural displacement w, the variable j which is the imaginary number square root of -1 and the angular frequency of vibration ω. The area integral is computed by introducing a nonlocal integration coupling that is assigned to the surface of interest. To avoid division by zero, it is advised to add the built-in constant eps, which denotes the machine precision limit, to the variable w as shown in Table 2 in the definition of the variable Zaco.

The TVR represents the sensitivity of the transducer measured at a distance of 1 m and driven at 1 V RMS. This definition can be mathematically expressed using Equation 2.

(2)

The RMS pressure at 1 m from the transducer head mass surface can be computed from Equation 3.

(3)

In Equation 3, the pressure p at 1 m ahead of the transducer is obtained using the exterior field pressure variable pext by using an expression pext(0,0,-1). Note that the computational domain is much smaller than 1 m but by performing the exterior field calculation you are able to compute the pressure and phase at any point outside the domain. The variable p* is the complex conjugate of the pressure p and can be expressed in COMSOL using the expression conj(pext(0,0,-1)).

Directivity Index (DI)

The directivity index (DI) gives a measure of how directional the transmitted acoustic beam is (Ref. 3). It is a measure of the ratio of the intensity Ifront in front of the transducer at a certain distance in the far-field (beyond the Rayleigh radius) to the intensity Iave transmitted by an omnidirectional source of the same strength. The DI is evaluated at a distance of 10 m in front of the transducer. You can alter the value of the parameter Zeval to compute the DI at a different location. The DI can be computed from Equation 4.

(4)

The quantity Ifront can be computed from the ratio of the RMS pressure at the desired distance from the transducer to the characteristic impedance Z0. The quantity Iave can be computed from the ratio of total radiated power Ptot, to the surface area of a sphere having the same radius as the distance from the transducer at which the DI is computed.

The user-defined variables used to compute the transducer characteristics are shown in Table 2.

Table 2: List of Variables.
Name	Expression	Description
rho0	intop3(acpr.rho)	Density of water at room temperature
c0	intop3(acpr.c)	Speed of sound in water at room temperature
Zaco	intop2(p)/intop2(acpr.iomega(w+eps))/(rho0c0)	Specific acoustic impedance
pext_1	pext(0,0,-1)	Exterior field pressure at 1 m
prms	sqrt(0.5pext_1conj(pext_1))[Pa]	RMS pressure at 1 m
TVR	20*log10(prms/Vrms/1[uPa/V])	Transmitting Voltage Response (TVR)
pext_Zeval	pext(0,0,Zeval[1/m])	Exterior field pressure at Zeval
Ifront	0.5pext_Zevalconj(pext_Zeval)[Pa^2]/(rho0*c0)	On-axis intensity at Zeval
Ptot	intop1(down(acpr.Ix)acpr.nx+down(acpr.Iy)acpr.ny+down(acpr.Iz)*acpr.nz)	Total radiated power
Iave	Ptot/(4piZeval^2)	Average intensity of monopole source at Zeval
Di	Ifront/Iave	Intensity directivity
DI	10*log10(Di)	Directivity index of the tonpilz transducer
k0	2pifreq/c0	Wave number
DI_fl_pist	10log10((k0a)^2/(1-2besselj(1,2k0a)/(2k0*a)))	Directivity index of flanged piston
SPL_Zeval	intop3(subst(acpr.ffc1.Lp_pext,x,0,y,0,z,Zeval))	SPL at Zeval
SPL_rel	acpr.ffc1.Lp_pext-SPL_Zeval	SPL relative to 0 dB at Zeval

Results and Discussion

Figure 5: A multislice plot showing the total acoustic pressure variation in the water domain and PML at 40 kHz.

Figure 5 shows the total acoustic pressure in the water domain for 40 kHz excitation. The inner water domain in the model captures roughly half a wavelength at this frequency. As expected, the outer surface of the PML layer shows zero pressure which confirms that the PML layer effectively absorbs the outgoing waves. Setting the PML stretching type to rational allows us to effectively use the PML over a large range of wavelengths and angles of incidence of the pressure waves as encountered in this model.

Figure 6: A multislice plot showing the sound pressure level (SPL) in the water domain and PML at 40 kHz.

Figure 6 shows the sound pressure level (SPL) in the water domain for 40 kHz excitation. Note the 45 dB difference between the SPL near the transducer head mass and the outer surface of the PML layer. This once again confirms the effectiveness of the damping induced by the PML.

Figure 7: A 3D radiation pattern plot showing the sound pressure level (SPL) for 40 kHz excitation at a distance of 10 m from the head mass. The SPL is computed relative to 0 dB at an on-axis distance of 10 m from the head mass.

Figure 7 shows a 3D radiation pattern plot of the exterior field SPL computed at a radial distance of 10 m from the transducer head mass for 40 kHz excitation. The SPL shown here is computed relative to a value of 0 dB at an on-axis distance of 10 m directly ahead of the transducer head mass. This baseline value is computed by the user-defined variable SPL_Zeval and the SPL relative to this quantity is computed by the variable SPL_rel. Note that you can control the baseline value by changing the value of the parameter Zeval. You could also create similar plot for any of the other frequencies solved for and also at other distances outside the computational domain. The later can be controlled by manually specifying the radius of the sphere as the absolute value of the parameter Zeval in the Evaluation section of the Radiation Pattern plot settings.

Figure 8: A polar beam sensitivity plot showing the sound pressure level (SPL) for 1, 10 and 40 kHz excitation at a distance of 10 m from the head mass. The SPL is computed relative to 0 dB at an on-axis distance of 10 m from the head mass.

Figure 8 shows a polar beam sensitivity plot which is also known as the beam pattern. Here the relative exterior field SPL is computed at a radial distance of 10 m from the transducer head mass for 1, 10, and 40 kHz excitation. The polar plot here shows the radiation pattern in the XZ-plane. The right half of the circle from 270 o through 0 o up to 90 o correspond to the -Z hemisphere. The +Y direction is directed into the plane. The beam pattern shows that at a relatively lower frequency of 1 kHz, the sound emitted is fairly omnidirectional. This is because at low frequencies the transducer works in the piston mode.

Figure 9: Total structural displacement of the transducer at 40 kHz. An exaggerated deformation has been used for better visualization.

Figure 9 shows the total structural displacement of the tonpilz transducer at 40 kHz excitation. At this frequency, the head mass vibrates in a mode whose shape is somewhat toroidal. This produces the lobes in the sound radiation pattern as can be seen in Figure 7 and Figure 8. If you observe the vibration mode for relatively lower frequencies, say 1 kHz, observe that the head mass vibrates mainly along its axis similar to a flanged piston.

Figure 10: A multislice plot of the electric potential distribution within the four PZT-4 disks.

Figure 10 zooms on the piezo stacks. It shows the electric potential distribution through the thickness of the PZT-4 disks. The color bands indicate a successful implementation of the idea described in Figure 3.

Figure 11: A line plot of acoustic pressure variation directly ahead of the transducer head mass up to an on-axis distance of 500 mm.

Figure 11 shows a plot of acoustic pressure vs. z-coordinate along the axis of the transducer for an excitation frequency of 40 kHz. The blue curve shows the pressure variable p that is solved for vs. the z-coordinate of the geometric edge that passes through the radius of the inner water domain vertically downward from the head mass along the transducer axis. The green curve shows how the pressure outside the computational domain can be computed using the exterior-field variable pext by using an expression pext(x,y,z). The variable is evaluated using a parameterized curve dataset to evaluate the variable outside the computational mesh.

This approach shows how to use a relatively small computational domain for efficient modeling and still use the results from the exterior field calculation in postprocessing to visualize both the magnitude and phase of the pressure outside the computational domain.

Figure 12: Frequency response plot of the absolute value, real and imaginary components of the specific acoustic impedance at the interface between the head mass and water.

Figure 12 shows the frequency response of the specific acoustic impedance of the head mass surface that is exposed to water. A resonance and anti-resonance is observed within the frequency range of 20 kHz and 35 kHz.

Figure 13: Transmitting Voltage Response (TVR) as a function of frequency obtained at an on-axis distance of 1 m ahead of the head mass and computed relative to 1 μPa/V.

Figure 13 shows the variation in the TVR of the transducer as a function of operating frequency. The fairly flat region between 15 kHz and 25 kHz can be particularly useful for sensing applications.

Figure 14: Frequency response of the Directivity Index (DI) computed at an on-axis distance of 10 m from the head mass. The DI of the tonpilz transducer is compared to that of a flanged piston.

Figure 14 shows the Directivity Index (DI) of the tonpilz transducer (blue curve) and compares it with the DI of a flanged piston (green curve). The latter can be computed from analytical expression as shown in Table 2. It is defined by the variable DI_fl_pist. Note that when the tonpilz transducer operates like a piston at lower frequencies, its DI becomes very similar to that of a flanged piston. Another feature worth noting is that within the range of 15 kHz and 25 kHz, the DI of the tonpilz transducer changes from nearly 0 dB to 11 dB while its TVR remains nearly constant. This can make the transducer quite versatile within this operating range.

Figure 15: Total radiated power from the tonpilz transducer within the operating frequency range of 1 kHz to 40 kHz.

Figure 15 shows the total radiated power as a function of the operating frequency of the tonpilz transducer. Note that the acoustic radiated power should be always positive. In order to correctly compute the radiated power at the low frequency range you need to make some changes to the default settings of the Perfectly Matched Layer (PML).

The default settings produce negative radiated power at low frequencies. This happens because at low frequencies the evanescent waves extend into the PML layer. The interaction between the scaled coordinate system in the PML and these waves may create an erroneous energy contribution in the model (can be either positive or negative).

A good way to investigate the performance of the PML is to make a sensitivity analysis on some parameter (for example, the total radiated acoustic power) with respect to changes in the PML parameters. In this model such a rigorous sensitivity analysis is not performed. In general, increasing the curvature factor effectively shifts the resolving power of the PML toward the physical domain, which is necessary in this case since the evanescent components decay in only a fraction of a wavelength. However, if you increase it too much, you may lose resolution in the other end, that is, of the free space wavelength. Assuming that the PMLs work properly at high frequencies for a curvature parameter of 1, you can in principle perform a convergence study by increasing the value of the curvature parameter until the low-frequency (for 1 kHz) result converges while making sure that the high-frequency (for 40 kHz) result is not affected. It turns out that a value of 5 for the curvature parameter yields good results in this model. A scaling factor of 0.5 further improves the results, but only by a small amount. Decreasing the scaling factor corresponds to compressing the PML layer (shortening it), which in turn effectively increases the mesh resolution within the PML region.

Figure 16: Pressure evaluated outside the computational mesh using the exterior field operator pext(x,y,z).

As discussed above, the pressure (including phase) is evaluated outside the computational domain using the exterior field computation feature. The feature can evaluates the pressure (and phase) at any distance outside computational domain (as already seen in Figure 11) both in the near and in the far-field. The pressure outside the computational mesh is depicted in Figure 16 using the Grid 3D dataset. The sound pressure level is depicted in Figure 17.

Figure 17: Sound pressure level evaluated outside the computational domain using the exterior field feature as well as the sound pressure level inside the domain.

References

1. C. H. Sherman and J. L. Butler, Transducers and Arrays for Underwater Sound, Springer, New York, 2007.

2. M. Lasky, “Review of Undersea Acoustics to 1950,” J. Acoust. Soc. Am., vol. 61, pp. 283–297, 1976.

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

Acoustics_Module/Piezoelectric_Devices/tonpilz_transducer

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

In the tree, select .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Build a plane section of the geometry before revolving it.

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

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Work Plane 1 (wp1)

In the toolbar, click

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Start with the solid domain.

Work Plane 1 (wp1)>Rectangle 1 (r1)

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In the window for , locate the section.

In the text field, type 4.

In the text field, type 2.

Locate the section. In the text field, type 25.

Work Plane 1 (wp1)>Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.

In the text field, type 20.

Locate the section. In the text field, type 5.

Work Plane 1 (wp1)>Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 6.

In the text field, type 10.

Locate the section. In the text field, type 2.

In the text field, type 15.

Work Plane 1 (wp1)>Rectangle 4 (r4)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.

In the text field, type 8.

Locate the section. In the text field, type 4.

In the text field, type 7.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	2
Layer 2	2
Layer 3	2

Work Plane 1 (wp1)>Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

Then build the water domain.


xw (mm)	yw (mm)
0	0
0	5
2	5
2	7
20	7
a	0

Work Plane 1 (wp1)>Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rwater+Rpml.

In the text field, type 90.

Locate the section. In the text field, type -90.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	Rpml

In the toolbar, click

button in the toolbar.

Revolve 1 (rev1)

In the window, right-click and choose .

In the window for , locate the section.

Click the button.

In the text field, type 90.

Rotate 1 (rot1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type range(90,90,270).

Locate the section. Select the check box.

button in the toolbar.

Create domain selections that will be used in the modeling.

button in the toolbar.

Definitions

Water domain - Inner

In the toolbar, click

In the window for , type Water domain - Inner in the text field.

Select Domains 3, 4, 22, and 32 only.

Water domain - PML

In the toolbar, click

In the window for , type Water domain - PML in the text field.

Select Domains 1, 2, 21, and 31 only.

Aluminum

In the toolbar, click

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

Select Domains 5, 6, 23, and 33 only.

Steel

In the toolbar, click

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

Select Domains 7, 8, 17–20, 24, 29, 30, and 34–36 only.

+Z poled Piezo

In the toolbar, click

In the window for , type +Z poled Piezo in the text field.

Select Domains 9, 10, 13, 14, 25, 27, 37, and 39 only.

-Z poled Piezo

In the toolbar, click

In the window for , type -Z poled Piezo in the text field.

Select Domains 11, 12, 15, 16, 26, 28, 38, and 40 only.

Water domains

In the toolbar, click

In the window for , type Water domains in the text field.

Locate the section. Under , click

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

Piezo domains

In the toolbar, click

In the window for , type Piezo domains in the text field.

Locate the section. Under , click

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

Solid domains

In the toolbar, click

In the window for , type Solid domains in the text field.

Locate the section. Under , click

In the dialog box, select in the list.

Non-PML domains

In the toolbar, click

In the window for , type Non-PML domains in the text field.

Locate the section. Under , click

In the dialog box, select in the list.

Create boundary selections that will be used further in the modeling.

Ground boundaries

In the toolbar, click

In the window for , type Ground boundaries in the text field.

Locate the section. From the list, choose .

Select Boundaries 28, 29, 38, 39, 45, 46, 93, 99, 103, 145, 151, and 155 only.

Voltage boundaries

In the toolbar, click

In the window for , type Voltage boundaries in the text field.

Locate the section. From the list, choose .

Select Boundaries 33, 34, 43, 44, 96, 102, 148, and 154 only.

Fixed boundaries

In the toolbar, click

In the window for , type Fixed boundaries in the text field.

Locate the section. From the list, choose .

Select Boundaries 19, 20, 88, and 164 only.

Exterior Field boundaries

In the toolbar, click

In the window for , type Exterior Field boundaries in the text field.

Locate the section. From the list, choose .

Select Boundaries 9, 10, 82, and 125 only.

Define a nonlocal integration coupling on the exterior-field boundary.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Define a nonlocal integration coupling on the acoustic-structure interface.

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 14, 15, 85, and 128 only.

Define a nonlocal integration coupling to retrieve values at the summit point of the water domain.

Integration 3 (intop3)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Point 46 only.

Define a rotated system that will be used for the poling of the -Z poled piezoelectric disks.

Rotated System 2 (sys2)

In the toolbar, click

and choose .

In the window for , locate the section.

Find the subsection. In the β text field, type pi.

Insert a Perfectly Matched Layer to model the absorption of acoustic wave far away from the source.

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 .

In the text field, type 0.5.

In the text field, type 5.

Materials

Add water, aluminum, steel, and peizoelectric materials from the Material Library.

Add Material

In the toolbar, click

to open the window.

Go to the window.

Click in the window toolbar.

Click in the window toolbar.

Click in the window toolbar.

Lead Zirconate Titanate (PZT-4) (mat4)

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Water, liquid (mat1)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Aluminum (mat2)

In the window, click .

In the window for , locate the section.

From the list, choose .

Steel AISI 4340 (mat3)

In the window, click .

In the window for , locate the section.

From the list, choose .

In the window, click .

In the window for , locate the section.

From the list, choose .

Pressure Acoustics, Frequency Domain (acpr)

Define physics settings and boundary conditions.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the cref text field, type 1500[m/s].

Exterior Field Calculation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Piezoelectric Material 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Piezoelectric Material 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Fixed Constraint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Electrostatics (es)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Ground 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Electric Potential 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the V0 text field, type V0.

Mesh the geometry; create a tetrahedral mesh in the solid and the water-inner domains and create a swept mesh in the PML.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Free Tetrahedral 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Define a mesh size in the water domain to ensure that the smallest wavelength is resolved by at least 5 elements (here we select lambda0/6).

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type 1500[m/s]/f0max/6.

Size 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type 2.

This setting ensures at least two mesh elements in the narrow layers of the piezo disks and bolt.

Swept 1

In the toolbar, click

Distribution 1

Right-click and choose .

Create a boundary layer at the external boundaries of the water domain. This will ensure numerically well defined normal gradients used in the exterior-field calculation feature. Turn off the smoothing option.

Boundary Layers 1

In the toolbar, click

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.

From the list, choose .

Locate the section. In the text field, type 1.

From the list, choose .

In the text field, type 1500[m/s]/f0max/6/20.

In the window, right-click and choose .

Import the file containing the variable definitions. These variables will mainly be used for postprocessing calculations.

button in the toolbar.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

Delete the default surface plot and replace it by a multislice plot, which shows the value of the acoustic pressure inside the water domain.

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

Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type range(f0min,f0step,f0max).

In the toolbar, click

Results

Acoustic Pressure (acpr)

In the window for , locate the section.

From the list, choose .

In the window, expand the node.

Right-click and choose .

Acoustic Pressure (acpr)

In the window, click .

Multislice 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select the check box.

In the toolbar, click

This plot should look like Figure 5.

button in the toolbar.

Sound Pressure Level (acpr)

In the window, click .

In the window for , locate the section.

From the list, choose .

Do the same as for the acoustic pressure plot.

Sound Pressure Level (acpr)

In the window, expand the node.

Right-click and choose .

In the window, click .

Multislice 1

In the toolbar, click

and choose .

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

In the toolbar, click

This plot should look like Figure 6.

Exterior-Field Sound Pressure Level (acpr)

Plot the sound pressure level at 10 meters relative to the value on the Z-axis

Radiation Pattern 1

In the window, click .

In the window for , locate the section.

In the text field, type SPL_rel.

Locate the section. Find the subsection. From the list, choose .

In the θ text field, type 90.

Find the subsection. From the list, choose .

In the text field, type abs(Zeval).

In the toolbar, click

Change the view in the graphics window to obtain a plot similar to Figure 7.

Plot the Exterior-Field Sound Pressure Level in XZ-plane for several frequencies.

Relative polar beam sensitivity

In the window, expand the node, then click .

In the window for , type Relative polar beam sensitivity in the text field.

Locate the section. From the list, choose .

In the list, choose , , and .

Radiation Pattern 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type SPL_rel.

Locate the section. Find the subsection. From the list, choose .

In the φ text field, type -90.

In the φ text field, type 180.

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

In the text field, type -1.

The reference direction defines that 0 deg. in the polar plot corresponds to the negative z-axis direction.

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

In the text field, type 0.

Find the subsection. In the text field, type abs(Zeval).

Relative polar beam sensitivity

In the window, click .

In the toolbar, click

This plot should look like Figure 8.

Replace the stress plot by displacement in solid.

Displacement

In the window, under click .

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

Locate the section. From the list, choose .

This plot should look like Figure 9.

In the window, expand the node, then click .

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

Displacement

In the window, click .

In the toolbar, click

Definitions

Create a new view to plot the electric potential in the piezoelectric disks.

In the window, under right-click and choose .

Results

Electric Potential (es)

In the window, expand the node, then click .

In the toolbar, click

Use the in the graphics window to obtain a plot similar to Figure 10.

Insert a 1D plot group to obtain the on-axis pressure graph shown in Figure 11.

First, create a parameterized curve dataset to enable plotting of the exterior-field variable outside the computational mesh: from the edge of the exterior-field boundary to 500 mm in front of the transducer.

Parameterized Curve 3D 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type (1-s)*(-1.02*Rwater)-s*500[mm].

Select the check box.

On-axis Pressure

In the toolbar, click

In the window for , type On-axis Pressure in the text field.

Locate the section. From the list, choose .

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

Locate the section. Select the check box.

In the associated text field, type Distance from the transducer (mm).

Select the check box.

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

Line Graph 1

Right-click and choose .

Select Edge 136 only.

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

Locate the section. From the list, choose .

In the text field, type -z.

Click to expand the section. Select the check box.

From the list, choose .


Legends
Domain evaluation

Line Graph 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type pext(x,y,z).

Locate the section. From the list, choose .

In the text field, type -z.

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

Find the subsection. From the list, choose .

Locate the section. Select the check box.

From the list, choose .

Insert a 1D plot to obtain the calculated specific acoustic impedance shown in Figure 12.


Legends
Exterior field evaluation

In the toolbar, click

In the toolbar, click

and choose .

In the window for , type Specific Acoustic Impedance in the text field.

Locate the section. From the list, choose .

Locate the section. Select the check box.

In the associated text field, type Z/(\rho c).

Locate the section. From the list, choose .

Right-click and choose .

In the window for , locate the section.


Expression	Unit	Description
abs(Zaco)	1
real(Zaco)	1
imag(Zaco)	1

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


Legends
Absolute value
Real part
Imaginary part

In the toolbar, click

Duplicate the last plot group and replace it by the transmitting voltage response to obtain the graph shown in Figure 13.

Transmitting Voltage Response

In the window, right-click and choose .

In the window for , type Transmitting Voltage Response in the text field.

Locate the section. In the text field, type TVR (dB rel. 1\mu Pa/V).

Locate the section. From the list, choose .

Ctrl-click to select table rows 2 and 3.

In the window, expand the node, then click .

In the window for , locate the section.

An alternative approach to create the above plot is to use the Octave Band Plot. Set the geometric-entity level to global, set the expression to pext(0,0,-1), reference to 1[uPa/V]/Vrms, and the style to continous.


Expression	Unit	Description
TVR		Transmitting Voltage Response (TVR)

Locate the section. Clear the check box.

In the toolbar, click

Duplicate the previous plot group and modify it to plot the directivity index like in Figure 14.

Directivity Index (DI)

In the window, right-click and choose .

In the window for , type Directivity Index (DI) in the text field.

Locate the section. In the text field, type DI (dB).

Locate the section. From the list, choose .

In the window, expand the node, then click .

In the window for , locate the section.

Click to select row number 3 in the table.


Expression	Unit	Description
DI		Directivity index of Tonpilz transducer
DI_fl_pist		Directivity index of flanged piston

Transmitting Voltage Response

Locate the section. From the list, choose .

In the toolbar, click

Duplicate the previous plot group and modify it to total radiated power like in Figure 15.

Total Radiated Power

In the window, right-click and choose .

In the window for , type Total Radiated Power in the text field.

Locate the section. Clear the check box.

Locate the section. From the list, choose .

In the window, expand the node, then click .

In the window for , locate the section.


Expression	Unit	Description
Ptot	mW	Total radiated power

In the toolbar, click

Create a grid dataset in order to plot the exterior-field pressure pext(x,y,z) outside of the computational mesh. This will reproduce Figure 16.

Grid 3D 1

In the toolbar, click

and choose .

In the window for , locate the section.

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

Find the subsection. In the text field, type -200.

In the text field, type 200.

Find the subsection. In the text field, type -200.

Click to expand the section. In the text field, type 2.

In the text field, type 150.

Exterior Pressure Slice

In the toolbar, click

In the window for , type Exterior Pressure Slice in the text field.