Comparison of Acoustic Radiation Characteristics in the Near and Far Fields Due to the Vibration of a Circular Disk Placed on an Infinite Rigid Baffle Using the Rayleigh Integral Method and Lumped Parameter Method

2.1 Acoustic radiation theory of circular disk

The far field sound pressure generated at a disk with a radius, a, vibrates on an infinite rigid baffle can be theoretically obtained as followed^(6,7). In Fig. 1, when the area element dS at point p' vibrates at a volume velocity dU and considering image source, the sound pressure at point p is followed as Eq. (1).

Fig. 1

Plane circular piston radiator in an infinite baffle

Where, dU = usS, dS = r'dϕdr'

In Fig. 1, the distance d between point p' on the disk and point p in space is given by Eq. (3).

If we consider the far field sound pressure in case of r > > r', and the far field sound pressure is followed as Eq. (4).

For θ ≈ 0, far field pressure p_{far field} is expressed like as Eq. (5)

When the observation point p is on-axis, the far field condition is such that when r' changes on the disk, the variation of d is induced a phase change according to the volume velocity dU position, resulting in acoustic cancellation at the observer's position. When using the approximation of the exponent component in Eq. (2), the condition for reducing the effect of acoustic cancellation due to phase change is that the variation of d should be less than λ/16⁽⁶⁾. When r' is at the origin, d becomes the minimum value d_min = r, and when r' is on the circumference, d becomes the maximum value $$ d_{\mathrm{m}\mathrm{a}\mathrm{x}}=\sqrt{r^2+a^2}\approx r+a^2/\left(2r\right)$$ and the path difference must satisfy d_max - d_min = a²/(2r) < λ/16.

In other words, when the far field on-axis condition is set to θ = 0, r → z like as Eq. (6),

If it is expressed in terms of frequency like as Eq. (7),

Under this condition, which is an exponent term of Eq. (2), becomes (2π/λ)(λ/16)=π/8=22.5°, Eq. (5) can be said to describe far fields on axis acoustic characteristics. Regardless of where r' is within the disk, if Eq. (6) or Eq. (7) is satisfied, the sound pressure arriving at point p on the on-axis is almost in phase, so sound pressure cancellation does not occur. Directivity, the direction characteristic of sound pressure, is calculated from Eq. (4) like as Eq. (8).

The direction index DI(directivity index) is as follows Eq. (9)^(6,7).

Since it is θ = 0 on-axis, it becomes D(ω,0)=1 and thus DI(ω,0)=0 (dB).

The near field sound pressure radiated by the source above the axis, the region close to the sound source where the far field approximation does not apply, is called Fresnel diffraction effects⁽⁷⁾. Although it is not possible to obtain closed form expressions for the near field pressure at arbitrary positions radiated by an acoustic source, it is possible to obtain the on-axis near field pressure radiated by a flat circular piston on an infinite baffle. The sound pressure caused by the disk on the baffle can be calculated using Eq. (1) is obtained by integrating with respect to the disk. Follow as Fig. 2, here, $$ d=\sqrt{z^2+{r\text{'}}^2}$$ and dS = 2πr'dr', and the exact on-axis solution p_exact can be expressed like as Eq. (10) ~ Eq. (11)⁽⁶⁾.

Fig. 2

Acoustic interference due to Fresnel effects

The condition for the sound pressure to be the maximum value in Eq. (11) is $$ \frac{k}{2}\left(\sqrt{z^2+a^2}-z\right)=\left(2m-1\right)\pi /2$$ is followed like as Eq. (12)⁽⁷⁾.

The conditions under which the sound pressure will be minimum is $$ \frac{k}{2}\left(\sqrt{z^2+a^2}-z\right)=m\pi $$ and given in Eq. (13).

The far field sound pressure is that if in Eq. (11), z/a > > 1 and are given $$ \sqrt{1+{\left(\frac{a}{z}\right)}^2}\approx a+\frac{1}{2}{\left(\frac{a}{z}\right)}^2$$ and sinθ ≈ θ becomes θ → ϵ and $$ \frac{z}{a}\gg \frac{ka}{4\epsilon }$$, then, the approximate sound pressure in the far field is Eq. (14).

The ratio [%] and error of sound pressure to far field are calculated from Eq. (15) and Eq. (16).

Assuming that a disk with a radius of 0.1 m vibrates as a rigid body on an infinite baffle, the sound waves radiating from this disk below 545.9 Hz can be viewed as spherical waves. If we apply arbitrarily ϵ = 5° = π/36 in the far field sound pressure condition z/a > > ka/(4ϵ), we get z/a > > 2.85. Meanwhile, using Eq. (16), it can be seen that error between the exact axis response p_exact and far field sound pressure p_{far field} is within 5 % is corresponding to z/a ≥ 2.2. Assume three cases in which a rigid disk with a radius of 0.1 m vibrates at 1715 Hz, 5145 Hz, and 8575 Hz, respectively. At this time, it corresponds to a = λ/2, 3λ/2, 5λ/2 for each frequency, Fresnel diffraction effects. The distance that can be considered as a far field when the disk vibrates is calculated based on Eq. (6) and r₁ = 2λ = 0.4 m, r₂ = 18λ = 1.2 m, r₃ = 50λ = 2.0 m can be considered as the far field sound field region⁽⁶⁾. However, if the region where the difference between the exact sound pres-sure and the far field approximate sound pressure is within 5 % is called the far field region(or area), it represents 0.26 m, 0.48 m, and 0.75 m, respectively. Meanwhile, the peak value positions can be calculated by using Eq. (12) when the exact sound pressure fluctuates according to the excitation frequency of the disk. This is shown in Fig. 3 and is summarized in Table 1. In Table 1, the meanings of 0, 0.53, and 2.4 in the 3rd column in case 3 are shown in Fig. 3. This means that the peak of the solid line (8575 Hz), which is the exact sound pressure p_exact of case 3, is located at z/a = 0, 0.53 and 2.4. Looking at the 4th column of the 4th row of Table 1, when a circular disk vibrates on the baffle at a frequency of 8575 Hz and radiates sound, the distance vertically from the center of the baffle can be called the start position of the far field sound region. When predicting sound pressure in the far field sound field, it was shown that, based on Eq. (16), within 5 % error, it is possible to use the far field sound pressure approximation p_{far field} instead of p_exact to predict sound pressure level in case of z/a > = 7.5 from farther away from the circular panel. In other words, at a location 7.5 times larger than the radius of the circular disk, the error in sound pressure is within 5 % even if the sound pressure is predicted using the far field sound pressure approximation p_{far field}.

Fig. 3

Comparison of on-axis pressure curves between pexact and pfar field for circular disk(radius = 0.1 m) on rigid baffle, vibrating frequency; pexact : 1715 Hz(dash-dot), 5145 Hz(dashed) and 8575 Hz(solid), pfar field : 1715 Hz(triangle), 5145 Hz(diamond) and 8575 Hz(x)

Table 1

Comparison of far field region start position between Eq. (6) and Eq. (16) for circular disk (radius a = 0.1 m) on infinite rigid baffle

When changing the radius of the circular disk from 0.1 m to 0.05 m, it can be calculated that the sound waves radiated when the disk rigidly vibrates at frequencies of 1715 Hz, 5145 Hz, and 8575 Hz show at the far field region start of 0.1 m, 0.3 m, and 0.5 m based on Eq. (6), respectively. According to Eq. (16), it can be calculated that the radiated sound waves start at the far field region 0.11 m, 0.15 m, and 0.21 m. In terms of z/a, it is 2.2, 3.0, and 4.2, respectively. In summary, as the frequency of the circular disk vibrating as a rigid body increases, the starting position of the far field region moves away from the center of the disk. In addition, it can be seen that as the diameter of the circular disk becomes smaller, the starting position of the far field region at the same frequency becomes closer to the disk. When measuring the sound pressure in a free sound field, the total sound power can be accurately measured by surrounding the radiating object and measuring the intensity at a location where the far field sound field characteristic, where the sound pressure decreases in inverse proportion to the distance, appears. However, in the near field region, which is inside the far field region, an oscillate phenomenon of sound pressure occurs due to sound interference, Fresnel diffraction effects, making it impossible to properly measure the actual sound power radiated by the object. In LEAP^® software⁽⁸⁾, as shown in Fig. 4, the methodology of small source arrays is used to model circular and rectangular loudspeaker as small source arrays to analyze the characteristics in the near field. For example, when analyzing a woofer with diameter 15 cm, 6 small circular disks are used in the diameter direction, so the radius of the small disk is a = 0.5 inch = 0.0127 m. When the upper frequency limit of interest for the woofer’s radiated sound pressure is 3 kHz according to Eq. (16). The closest distance at which the far field approximation p_{far field} can be used is 0.03 m, which indicates that reliable results can be obtained even if the near field sound pressure at 3 cm or more is obtained by the far field sound pressure approximation p_{far field}.

Fig. 4

Schematic diagram of circular and rectangular shape loudspeaker modeling by LEAP software

2.2 Brief description of RIM

Kirchhoff-Helmholtz Integral Theorem is as followed Eq. (17) and Eq. (18)⁽²⁾.

Applying the Rayleigh Integral Method on the baffle surface follow as Eq. (19),

where, $$ R_{\mathrm{1,2}}=\left[{\left(x_s-x\right)}^2+{\left(y_s-y\right)}^2+{\left(z_s\mp z\right)}^2\right]$$

The sound pressure is as follows Eq. (20).

When calculating integral equations, there are simple source method and collocation method⁽⁹⁾. Direct acoustic BEM can analyze the interior or exterior of a closed space. In the case of an object that is open on one side, the interior acoustic analysis is directly analyzed using acoustic internal BEM, and physical quantities are calculated at the boundary between the interior space and the open space. Using this data, we can interpret the acoustic radiation from an open space interface wall as the sound radiating from the boundary surface. In this paper, acoustic analysis was performed by using the collocation method proposed by Kirkup and developing a modified program for user convenience by linking RIM3⁽²⁾, a program provided by Kirkup, with the GiD^® program^(1~3,9,10).

2.3 LPM

Koopmann and Fahnline presented LPM, a method to easily analyze acoustic radiation in open spaces. Koopmann et al. selected acoustic power, a scalar quantity, as a physical quantity that can characterize acoustic radiation as a design parameter for noise reduction design of structures⁽⁴⁾. While BEM uses the element velocity boundary condition, LPM adopts the surface volume velocity as the velocity boundary condition in the Kirchhoff-Helmholtz integral equation and converts it to an average form. In this method, mathematical difficulties such as non-uniqueness that arise in acoustic BEM are eliminated by analyzing the external sound field by creating equivalent volume velocity fields using a combination of simple, dipole, and tripole sources according to boundary conditions⁽⁴⁾. In other words, BEM uses velocity boundary conditions at elements, but LPM uses volume velocities at elements, suggesting a method to avoid non-uniqueness and non-existence problems that occur in BEM⁽⁴⁾. In addition, the analysis time is very short compared to the BEM method, so it can be used for external acoustic radiation analysis instead of BEM^(9,12). LPM is briefly described as follows. In one element, the sound pressure caused by the simple source and the dipole source is expressed as Eq. (21)⁽⁴⁾.

Where, $$ \widehat{g}\left(\mathbf{x},\mathbf{q}\right)=\frac{1}{R}{e}^{jkR}$$

Using Euler’s equation, the velocity in the direction at the field point x is given by Eq. (22).

The volume velocity across elements of the boundary surface is calculated, that is, Eq. (22) is integrated over the element surface to be analyzed. We can obtain an expression for the volume velocity as in Eq. (23).

After obtaining the coefficients s_υ using the volume velocity boundary conditions using Eq. (23), the sound pressure at any field point can be calculated using Eq. (24).

Here, the constants α_υ and β_υ are known constant values used to represent simple, dipole, or tripole sources of acoustic radiation from each element of the structure. Table 2 shows how the source shape is determined according to the characteristics of each surface element and the constants α_υ and β_υ are determined accordingly⁽⁴⁾. If the element is on the baffle, it is considered a simple source. If the element surrounds a finite volume, it is assumed to be a tripole source. In other cases, if the element is neither on the baffle nor surrounding a finite volume, it is assumed to be a dipole source^(3~5,11,12).

Table 2

Constants αυ and βυ according to surface elements

3.1 Acoustic radiation analysis using RIM

The program presented by Kirkup was slightly modified⁽²⁾, and a program linked to the GiD^® software was programmed and used to make it easier for users to use. A program system consisting of batch files was used to enable analysis of rigid body vibration using RIM.

(2) Acoustic radiation analysis of flexible body vibration

The acoustic radiation analysis process according to flexible body vibration was carried out in the following order. Step 1: After selecting the flat plate to be analyzed on an infinite rigid baffle, velocity boundary conditions and analysis frequency range were specified. The modeling process was performed and an input file for FEM analysis was created and saved. Step 2: The previously created input file was read using a commercial FEM program. Vibration analysis was performed by inputting boundary conditions and material properties for vibration analysis in FEM. It was analyzed in the time domain using a boundary condition where the edges of the circular disk were fixed and a condition when a random force of 1 N was applied to the center of the circular disk. The random waveform was filtered using a 10 kHz low-pass filter to create a random waveform with an upper limit frequency of 10 kHz, and this was used as the 1 N random force. The acoustic elements used in the acoustic analysis were three-node triangular elements, and for vibration, six-node triangular elements were used for vibration analysis. Fig. 5 shows the vibration pattern of the instantaneous velocity magnitude at which a circular disk vibrates depending on the applied force.

Fig. 5

Vibration analysis on circular disk(radius = 0.1 m), fixed boundary condition, 1 N random force on center point, Δt = 9.765625E-5 s, T = 0.1 s

After vibration analysis, file written the velocity data at each node and file written the node coordinates were saved. Step 3: The time domain velocity data at each node was converted to frequency domain data using fast Fourier transform. The analysis time step was Δt = 97.65625 μs and the analysis time was 0.1 s. Therefore, the number of cases that can be analyzed ranges from 10 Hz to 5120 Hz, and vibration analysis results for 512 cases of harmonic excitation were obtained at 10 Hz intervals. It was programmed that leakage error was reduced by adopting a Hanning window when performing fft in case more data than 0.1 sec was used. Step 4: Using complex velocity data in the frequency domain at each node, the average velocity in normal direction at the centroid of the three-node triangular element, which is an acoustic element, was calculated and saved as a file. Step 5: Using the RIM program presented by Kirkup, which has been modified to have GUI (graphic user interface) characteristics so that users can use it conveniently, use velocity boundary conditions on the centroids of each element at each frequency specified in step 1. Then, RIM analysis was performed. As a result of the analysis, the sound pressure pattern on the surface to be analyzed and the sound pressure at the field points(not meshes) were calculated. In addition, in order to calculate sound pressure patterns for subsequent field point meshes, data related to sound pressure on the surface to be analyzed were saved in a file in advance. To calculate the sound pressure pattern in the new field point meshes, the field point meshes were modeled. In order to analyze acoustic characteristics in the new field point mesh, a program was written and used for post analysis. By executing this batch file, we analyzed the acoustic pattern in the field point mesh and plotted the results in GiD^®.

3.2 Acoustic radiation analysis using LPM

The program used in this section was slightly modified from the program presented by Koopmann⁽⁴⁾, and was programmed in conjunction with GiD^®, a commercial program, to make it easier for users to use using a GUI. A batch file was created and used to enable analysis using LPM for rigid body vibration in a process similar to the RIM analysis process. For the flexible body vibration analysis, another batch file was created, and acoustic analysis was performed using LPM using these batch files⁽⁵⁾. We programmed a post program, batch file that can calculate and plot the LPM analysis results on field point meshes, and using this, we were able to obtain sound pressure pattern results on field point meshes.

4.1 Comparison of acoustic radiation according to rigid body vibration of a disk

It was assumed that a circular disk with a diameter of 0.1 m was placed on an infinite baffle. The sound pressure radiated when a rigid body vibrates at 1 m/s was compared with the near field sound pressure, the far field sound pressure, and the results of the RIM. The sound pressure was calculated at 100 field points locations at 0.01 m intervals at z = 0.01 m in the normal direction from the center of the circular disk. The analyzed frequencies were 1715 Hz and 8575 Hz. Analysis was performed for four cases depending on the size of the elements used. Looking at Fig. 6, the first column showed the sound pressure when a disk with radius a = 0.1 m was vibrating at 1715 Hz, 1 m/s, and moving away from the center of the disk in the normal direction. Comparing the results analyzed by and RIM in all four cases described in Table 3 (Fig. 6(a)), it was shown that the error was within 5 % in all four cases compared to p_exact. Fig. 6(b), the second column of Fig. 6, compares the radiated sound pressure calculated by RIM, p_exact and p_{far field} when the disk vibrates at a frequency of 8575 Hz. The first figure showed case 1 in Table 3, which was the result of modeling a circular panel with 16 elements and analyzing it with RIM. The RIM results showed an error of up to 114.14 % when compared to p_exact. A dip phenomenon in sound pressure occurred near z/a = 1 (z = 0.1 m), and the maximum error occurred at this point. However, overall, the RIM analysis results were in good agreements with the exact pressure p_exact. Case 1 represents a desirable frequency limit of up to 583 Hz if the 1/6 rule of the wavelength is followed, and if the 1/3 rule is followed, the upper frequency limit that can be interpreted is up to 1166 Hz. However, even when the on-axis sound pressure was 8575 Hz, the RIM analysis results showed good agreement with the exact solution when z/a is greater than 1. As the error decreased as the element size decreased, RIM passed the convergence test.

Fig. 6

Comparison of pexact, pfar field and Pressure by RIM, circular disk(radius a = 0.1 m) on infinite baffle, max(Err%) = max(|(RIM-pexact)/pexact|) × 100, rigid body velocity(1 m/s) B.C., zrange = 0.01 m ~ 1 m, interval = 0.01 m

Table 3

Comparison of max(Err%) of beam pattern between RIM and LPM in case of circular disk(radius = 0.1 m) on rigid baffle corresponding to three frequencies, 1715 Hz, 3430 Hz, 8575 Hz

Figure 7 showed the results calculated by LPM. The first column in Fig. 7 is shown in Fig. 7(a), which is the 1715 Hz case. Cases 1 and 2 in Table 3 correspond to the first and second figures above, and the maximum error between the LPM analysis result and the exact solution was about 6 %. The location where the maximum error occurred was 0.01 m in front of the circular disk. The second column, Fig. 7(b) is the result of comparing the sound pressure radiated from a disk vibrating at 8575 Hz. Unlike the RIM analysis results, the LPM analysis showed large errors. It was shown that errors in the near field can be greatly reduced only when a circular disk is modeled with an element size that satisfies the 1/3 rule. As the error decreased as the element size decreased, LPM passed the convergence test. Table 3 showed the errors of LPM compared to RIM in four modeling cases. It was found that at least the elements must be modeled using the 1/3 rule to ensure that the error in the near field is within about 5 %.

Fig. 7

Comparison of pexact, pfar field and Pressure by LPM, circular disk(radius a = 0.1 m) on infinite baffle, max(Err%) = max(|(RIM-pexact)/pexact|) × 100, rigid body velocity(1 m/s) boundary condition, zrange = 0.01 m~ 1 m, interval = 0.01 m

Figure 8 compares the direction index DI, which is the beam pattern of far field sound pressure p_farfield, LPM and RIM, for case 2 in Table 3. In case 2, according to the 1/3 wavelength rule, up to 1718 Hz is the upper frequency limit for reliable analysis. In the case of RIM, it was shown to match well with the beam pattern of far field sound pressure even at 3430 Hz and 8575 Hz, which are frequencies exceeding the upper analysis limit frequency. However, in the case of LPM analysis, in the case of 3430 Hz (Fig. 8(a)), a large error occurred above 40° based on the axis, and in the case of 8575 Hz, a large error occurred above 30°. When the size of the element was larger than the 1/3 wavelength rule, the beam pattern analyzed by LPM was inaccurate. On the other hand, the RIM method showed that the beam pattern matched well with the theoretical value even though it was modeled with elements larger than 1/3 wavelength rule.

Fig. 8

Comparion of beam pattern between RIM and LPM and pfar field in case of circular disk for case 2, 0° means z-axis, rigid body vibration, at radius = 1 m

Figure 9 shows the beam pattern for case 3 in Table 3. According to the 1/3 wavelength rule, the upper frequency limit for analysis was up to 3924 Hz. In Fig. 9(a), both LPM and RIM showed that the beam pattern matched well with the theoretical value. In the case of 8575 Hz, Fig. 9(b), which exceeds the upper analysis limit frequency, the RIM method showed better agreement with the theoretical beam pattern than that obtained by the LPM method.

Fig. 9

Comparion of beam pattern between RIM and LPM and pfar field in case of circular disk for case 3, 0° means z-axis, rigid body vibration, estimated at radius = 1 m from circular disk center

4.2 Comparison of acoustic radiation according to the flexible vibration of the disk

The sound pressure radiated when a circular disk placed on an infinite rigid baffle vibrates under a random force of 1 N at the center with the edge fixed was analyzed using RIM and LPM methods. Fig. 10 is for case 4 in Table 3 According to the 1/3 wavelength rule, the upper analysis limit frequency was 6258 Hz. When excited at 3430 Hz, the maximum error in sound pressure as LPM compared to RIM was 2.7 %. Since almost identical results were obtained with the LPM and RIM analysis methods, both analysis methods could be said to be useful in the case of a circular disk placed on a baffle surface.

Fig. 10

Comparison of Pressure between RIM and LPM, circular disk(radius a = 0.1 m) on infinite baffle, max(Err%) = max(|(LPM-RIM)/RIM|) × 100 = 2.7 %, case 4, at 3430 Hz, flexible random vibration excited on circular disk center by 1 N force, zrange = 0.01 m ~ 1 m, interval = 0.01 m

Figure 11 shows the result of calculating the beam pattern in a circular disk vibrating at 3430 Hz using RIM and LPM, and the maximum error of LPM compared to RIM was 15.7 % around 60° offset on the axis. The reason why the error appeared larger than it appeared was around the dip, where the level of sound pressure was very small, so even a small difference in sound pressure level resulted in a large error.

Fig. 11

Comparison of Pressure beam pattern between RIM and LPM, circular disk(radius a = 0.1 m) on infinite baffle, max(Err%) = max(|(LPM-RIM)/RIM|) × 100 = 15.7 % at 60°, case 4, at 3430 Hz, flexible random vibration excited by 1 N random force at center of circular disk, fixed boundary condition.

Figure 12 compares the sound pressure pattern on the surface of a hemisphere (radius 1 m, center, 0,0,0) due to the radiated sound generated from the circular disk modeled as case 4 in Table 3. The maximum sound pressure difference between RIM and LPM was 0.19 dB. It was shown that almost the same results can be obtained using RIM and LPM, which used volume velocity boundary conditions.

Fig. 12

Comparion of field mesh surface pressure(hemisphere, center: 0, 0, 0, radius = 1 m) in case of circular disk radiation, case 4, 3430 Hz, flexible vibration excited at circular disk(radius = 0.1 m) center by 1 N random force.