Research of a Stepped Ultrasonic Radiator
- Open Access
- 01.06.2025
- CLASSICAL PROBLEMS OF LINEAR ACOUSTICS AND WAVE THEORY
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Abstract
INTRODUCTION
Research on sound radiators primarily focuses on sound field variables and parameters, such as axial sound pressure and directivity. It also involves analyzing the effect of the sound field on the vibration state of a sound radiator, namely radiation impedance, which is crucial for measuring the radiation efficiency of a radiator. The distribution of vibration velocity on a surface can be used to characterize sound radiators into piston radiators, if the vibration velocities are uniformly distributed, and flexural vibration radiators if the vibration velocity is nonuniformly distributed.
The sound field of a piston radiator in an infinite, planar, and rigid baffle has been extensively investigated using various methods [1‐4]. Moreover, calculating the radiation impedance of piston radiators is a classical problem. An equation for obtaining the radiation impedance of a circular piston has been derived using several approaches, including Green’s function, integral transform, and time domain techniques [5]. Similarly, several studies have investigated other piston radiator shapes, such as rectangular piston radiators with different aspect ratios. Stepanishen derived a formula for expressing the radiation impedance of a square piston radiator as a Bessel function series expansion of an impulse response function [5]. Swenson proposed that variable substitution and integration order exchange could be applied to express the quadruple integral of the radiation impedance of a rectangular piston radiator as an infinite series expansion [6]. Bank and Wright divided the surface of a rectangular radiator into eight triangular integration regions based on simple geometric relationships and then numerically calculated the self-radiation impedance of rectangular pistons with varying aspect ratios using Simpson’s formula [7]. Lee partitioned the piston surface into distinct integration regions based on geometric relationships, thereby establishing a quadruple formulation for the mutual-radiation impedance of two square pistons [8]. Yang employed the superposition principle to transform the quadruple integral expression of radiation impedance into a summation of a series of straightforward functions—an approach that led to the development of an efficient method for calculating the radiation impedance of pistons [9]. Considering other practical applications, the radiation impedance of a rectangular piston positioned at the wall of a rectangular duct and the radiation impedance of a piston in a rectangular wave guide were investigated [10, 11]. Min employed the transfer matrix method to investigate the radiation impedance of stepped acoustic resonators using air as the working medium at room temperature [12].
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Extensive research has been conducted on the sound fields generated by flexural vibration radiators. Based on the Rayleigh integral theory, the sound pressure distribution of a rectangular radiator with simply supported boundary conditions has been analytically obtained, and the sound pressure distribution was calculated [13]. The sound pressure of a circular radiator with simply supported boundary conditions can be derived via transformation coordinate system. Legendre polynomials and spherical Bessel functions have been used to numerically calculate sound pressure [14]. The principle of linear superposition has been applied to express sound pressure distribution using Bessel functions and calculate the near and far sound fields of cylindrical beams generated by strip-shaped acoustic radiators [15]. In addition, an ultrasonic radiation system with a flexural vibration radiator with free boundary conditions and a center driven by a transducer had a large radiation area and low impedance; this system has great application potential in the field of high-power gas medium ultrasonics [16]. In particular, it has improved the directionality of the planar radiator, turning it into a more directional stepped radiator by adding steps in the opposite vibration displacement on the surface of planar radiators [17‐19]. He et al. divided the stepped circular radiator into two components—a disk and a ring—and utilizing Kirchhoff’s and Rayleigh’s integral principles, derived radiation pressure expressions of the inner plate and outer ring, respectively. In addition, an expression was obtained for total radiation pressure through superposition, allowing for the derivation of directivity expressions for a stepped circular radiator [20]. However, this method is limited to stepped circular radiators with displacement analytical solutions, and there is no effective method for obtaining a solution for a rectangular or square stepped radiator without a displacement analytical solution.
Radiation impedance studies of flexural vibration radiators have primarily focused on simply supported boundary conditions. Approximate solutions for the radiation impedance of simply supported rectangular plates in various modes have been obtained through series expansion. The relationships between radiation impedance and frequency, as well as between radiation impedance and aspect ratio for a panel in different modes, have been analyzed, and the results have been compared with Maidanik’s approximate solution [21, 22]. Li simplified the quadruple integral to a double or single integral by using coordinate transformation and calculated the radiation resistance through numerical integration [23, 24]. The general expression of radiation impedance can be derived by using the spatial convolution approach, which simplifies the quadruple integral to a double integral, making numerical calculation more convenient [25]. The above methods divide the quadruple integration of the radiation impedance calculation into a double or single integral, or simplify the quadruple integration into series expansion to calculate the radiation impedance. These methods generally have a clear physical meaning; however, their mathematical derivations are complex. For circular radiators, an analytical displacement solution is available; however, for rectangular (including square) radiators, only approximate solutions exist. Calculating radiation impedance using either type of solution is challenging, as it requires a quadruple integration.
The above analysis indicates the various mathematical methods that have been employed to calculate and analyze the radiated sound field and radiation impedance of pistons and simply supported boundary radiators with different shapes. However, for a flexural vibration radiator with free boundary conditions, there remains a lack of an effective method for calculating the radiated sound field and radiation impedance, particularly for stepped radiators. This study proposes a method that combines the simplest Rayleigh method (a flat radiator in a flat infinite screen) with finite element simulation software to estimate the radiation field and impedance of stepped radiators—providing a theoretical basis for the design and application of radiators, and offers an evaluation method for assessing the acoustic performance of the stepped radiators.
METHODS
Consider a sound radiator in an infinite rigid baffle, as shown in Fig. 1. The plane of the radiator is the x and y planes of a Cartesian coordinate with the origin at the center of the radiator.
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Fig. 1.
Sound radiator in an infinite baffle.
Sound Field
Axial Sound Pressure
The radiator is divided into infinitesimal areas that can be considered as point sound sources, and \(d{{s}_{i}}\) is such a source. The coordinate, vibration displacement, and vibration velocity of \(d{{s}_{i}}\) are \(\left( {{{x}_{i}},{{y}_{i}}} \right)\), \({{\xi }_{i}} = {{\xi }_{{ai}}}{{e}^{{j\omega t}}}\), and \({{u}_{i}} = {{u}_{{ai}}}{{e}^{{j\omega t}}} = j\omega {{\xi }_{{ai}}}{{e}^{{j\omega t}}}\), where \({{\xi }_{{ai}}}\) and \({{u}_{{ai}}}\) are the vibration displacement and velocity amplitudes, respectively.
The sound pressure, \(d{{p}_{1}}\), generated by \(d{{s}_{i}}\) at point \({{Q}_{1}}\) iswhere \(k\) is the wave number, \(\rho \) is the density of air, \(c\) is the speed of sound in air, \({{\omega }}\) is the angular frequency, and \(H = \sqrt {x_{i}^{2} + y_{i}^{2} + {{z}^{{'2}}}} \) is the distance between \({{Q}_{1}}\) and \(d{{s}_{i}}\).
$$d{{p}_{1}} = j\frac{{k\rho c}}{{2\pi H}}{{u}_{{ai}}}{{e}^{{j\omega t}}}{{e}^{{ - jkH}}}d{{s}_{i}},$$
(1)
The sound pressure, \({{p}_{1}}\), generated by the entire radiator can be obtained using the Rayleigh integral:
$${{p}_{1}} = \iint {j\frac{{k\rho c}}{{2\pi H}}}{{u}_{{ai}}}{{e}^{{j\omega t}}}{{e}^{{ - jkH}}}d{{s}_{i}}.$$
(2)
Without considering the time factor in Eq. (2), substituting \({{u}_{{ai}}} = j\omega {{\xi }_{{ai}}}\) into Eq. (2) and using Euler expansion, the amplitude of the sound pressure \({{p}_{{a1}}}\) is
$${{p}_{{a1}}} = \frac{{k\rho c}}{{2\pi }}\sqrt {\left[ {\iint {\frac{{\omega {{\xi }_{{ai}}}}}{H}}\sin \left( {kH} \right)d{{s}_{i}}{{]}^{2}} + \,\,} \right[\iint {\frac{{\omega {{\xi }_{{ai}}}}}{H}}\cos \left( {kH} \right)d{{s}_{i}}{{]}^{2}}} .$$
(3)
Equation (3) can be further expressed as:
$${{p}_{{a1}}} = \frac{{k\rho c}}{{2\pi }}\sqrt {\left[ {\sum \frac{{\omega {{\xi }_{{ai}}}}}{H}\sin \left( {kH} \right)\Delta {{s}_{i}}{{]}^{2}} + \,\,} \right[\sum \frac{{\omega {{\xi }_{{ai}}}}}{H}\cos \left( {kH} \right)\Delta {{s}_{i}}{{]}^{2}}} .$$
(4)
Directivity
As shown in Fig. 1, \({{Q}_{2}}\) is located in the far sound field, and its Cartesian and spherical coordinates are \(\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}}} \right)\) and \(\left( {r,\theta ,\phi } \right)\) respectively.
Similarly, the sound pressure, \({{p}_{2}}\), produced by the radiator at point \({{Q}_{2}}\) can be calculated as follows:where \(R\) is the distance between \({{Q}_{2}}\) and \(d{{s}_{i}}\).
$${{p}_{2}} = \iint {j\frac{{k\rho c}}{{2\pi R}}}{{u}_{{ai}}}{{e}^{{j\omega t}}}{{e}^{{ - jkR}}}d{{s}_{i}},$$
(5)
In the far sound field, \(R\) in the denominator of the fraction in Eq. (5) can be approximated as \(r\). Thus, \(R\) in the exponential term can be approximated as
$$R = r - \frac{{{{x}_{0}}{{x}_{i}} + {{y}_{0}}{{y}_{i}}}}{r}.$$
(6)
In addition, the Cartesian and spherical coordinates of point \({{Q}_{2}}\) satisfy the following relations:
$${{x}_{0}} = r\sin \theta \cos \varphi ,\,\,\,\,{{y}_{0}} = r\sin \theta \sin \varphi .$$
(7)
$$\begin{gathered} {{p}_{2}} = - \frac{{k\rho c\omega }}{{2\pi r}}\iint {{{\xi }_{{ai}}}{{e}^{{j\omega t}}}} \\ \times \,\,{{e}^{{ - jkr}}}{{e}^{{jk\left( {{{x}_{i}}\sin \theta \cos \varphi } \right)}}}{{e}^{{jk\left( {{{y}_{i}}\sin \theta \sin \varphi } \right)}}}d{{s}_{i}}. \\ \end{gathered} $$
(8)
Without considering the time factor in Eq. (8), the amplitude of the sound pressure, \({{p}_{{a2}}}\), can be obtained as:whereand
$${{p}_{{a2}}} = - \frac{{k\rho c\omega }}{{2\pi r}}{{e}^{{ - jkr}}}\sqrt {b_{1}^{2} + b_{2}^{2}} ,$$
(9)
$$\begin{gathered} {{b}_{1}} = \iint {{{\xi }_{{ai}}}}((\cos (k{{x}_{i}}\sin \theta \cos \varphi )\cos \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right) \\ - \,\,\sin \left( {k{{x}_{i}}\sin \theta \cos \varphi } \right)\sin \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right))d{{s}_{i}}, \\ \end{gathered} $$
$$\begin{gathered} {{b}_{2}} = \iint {{{\xi }_{{ai}}}}((\cos (k{{x}_{i}}\sin \theta \cos \varphi )\sin \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right) \\ + \,\,\sin \left( {k{{x}_{i}}\sin \theta \cos \varphi } \right)\cos \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right))d{{s}_{i}}. \\ \end{gathered} $$
Equation (9) can be further expressed as:whereand
$${{p}_{{a2}}} = - \frac{{k\rho c\omega }}{{2\pi r}}{{e}^{{ - jkr}}}\sqrt {B_{1}^{2} + B_{2}^{2}} ,$$
(10)
$$\begin{gathered} {{B}_{1}} = \sum {{\xi }_{{ai}}}((\cos (k{{x}_{i}}\sin \theta \cos \varphi )\cos \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right) \\ - \,\,\sin \left( {k{{x}_{i}}\sin \theta \cos \varphi } \right)\sin \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right))\Delta {{s}_{i}}, \\ \end{gathered} $$
$$\begin{gathered} {{B}_{2}} = \sum {{\xi }_{{ai}}}((\cos (k{{x}_{i}}\sin \theta \cos \varphi )\sin \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right) \\ + \,\,\sin \left( {k{{x}_{i}}\sin \theta \cos \varphi } \right)\cos \left( {k{{y}_{i}}\sin \theta \sin \varphi } \right))\Delta {{s}_{i}}. \\ \end{gathered} $$
According to the definition of directivity,where \({{({{p}_{{a2}}})}_{{\theta = 0}}} = - \frac{{k\rho c\omega }}{{2\pi r}}{{e}^{{ - jkr}}}\sum {{\xi }_{{ai}}}\Delta {{s}_{i}}.\)
$$D\left( {\theta ,\phi } \right) = \frac{{{{{({{p}_{{a2}}})}}_{\theta }}}}{{{{{({{p}_{{a2}}})}}_{{\theta = 0}}}}},$$
(11)
$$D\left( {\theta ,\phi } \right) = \frac{{\sqrt {B_{1}^{2} + B_{2}^{2}} }}{{\sum {{\xi }_{{ai}}}\Delta {{s}_{i}}}}.$$
(12)
Radiation Impedance
As shown in Fig. 1, \(d{{s}_{j}}\) is another infinitesimal area with the coordinates, vibration displacements, and vibration velocity of (\({{x}_{j}},{{y}_{j}}\)), \({{\xi }_{j}} = {{\xi }_{{aj}}}{{e}^{{j\omega t}}}\), and \({{u}_{j}} = {{u}_{{aj}}}{{e}^{{j\omega t}}} = j\omega {{\xi }_{{aj}}}{{e}^{{j\omega t}}}\), respectively. Thus, the sound pressure generated by \(d{{s}_{j}}\) near \(d{{s}_{i}}~\)can be obtained as:where \(h = \sqrt {{{{\left( {{{x}_{i}} - {{x}_{j}}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - {{y}_{j}}} \right)}}^{2}}} \) is the distance between \(d{{s}_{i}}\) and \(d{{s}_{j}}\).
$$d{{p}_{3}} = j\frac{{k\rho c}}{{2\pi h}}{{u}_{{aj}}}{{e}^{{j\omega t}}}{{e}^{{ - jkh}}}d{{s}_{j}},$$
(13)
Therefore, the sound pressure generated by the entire radiator near \(d{{s}_{i}}\) is
$${{p}_{3}} = \iint {j\frac{{k\rho c}}{{2\pi h}}}{{u}_{{aj}}}{{e}^{{j\omega t}}}{{e}^{{ - jkh}}}d{{s}_{j}}.$$
(14)
Similarly, the amplitude of sound pressure is
$${{p}_{{a3}}} = \iint {j\frac{{k\rho c}}{{2\pi h}}}{{u}_{{aj}}}{{e}^{{ - jkh}}}d{{s}_{j}}.$$
(15)
The sound power, \({\text{d}}W\), generated by \(d{{s}_{i}}\) iswhere \(u_{{ai}}^{*}\) is the conjugate complex number of \({{u}_{{ai}}}\).
$${\text{d}}W = {{p}_{{a3}}}u_{{ai}}^{*}d{{s}_{i}},$$
(16)
By integrating Eq. (16), the radiated sound power of the radiator can be obtained as:
$$W = \iint {\iint {dW}} = \iint {\iint {j\frac{{k\rho c}}{{2\pi h}}}}{{u}_{{aj}}}{{e}^{{ - jkh}}}d{{s}_{j}}u_{{ai}}^{*}d{{s}_{i}}.$$
(17)
By substituting \({{u}_{{aj}}} = j\omega {{\xi }_{{aj}}}\) and \(u_{{ai}}^{*} = - j\omega {{\xi }_{{ai}}}\) into Eq. (17), the following equation is obtained:
$$\begin{gathered} W = j\frac{{k\rho c{{\omega }^{2}}}}{{2\pi }} \\ \times \,\,\iint {\iint {\frac{{{{\xi }_{{aj}}}{{\xi }_{{ai}}}}}{h}}}\left[ {\cos \left( {kh} \right) - j\sin \left( {kh} \right)} \right]d{{s}_{j}}d{{s}_{i}}. \\ \end{gathered} $$
(18)
Furthermore, Eq. (18) can be expressed as:
$$\begin{gathered} W = j\frac{{k\rho c{{\omega }^{2}}}}{{2\pi }}\sum \sum \frac{{{{\xi }_{{aj}}}{{\xi }_{{ai}}}}}{h} \\ \times \,\,\left[ {\cos \left( {kh} \right) - j\sin \left( {kh} \right)} \right]\Delta {{s}_{j}}\Delta {{s}_{i}}. \\ \end{gathered} $$
(19)
Finally, considering the average vibration velocity as the reference velocity, the radiation impedance iswhere \(\left\langle u \right\rangle \left\langle u \right\rangle {\text{*}} = \frac{{\sum \left( {j\omega {{\xi }_{{ai}}}} \right)\Delta {{s}_{i}}}}{S}\frac{{\sum \left( { - j\omega {{\xi }_{{ai}}}} \right)\Delta {{s}_{i}}}}{S}\), and S represents the radiator area.
$$Z = \frac{W}{{\left\langle u \right\rangle \left\langle u \right\rangle {\text{*}}}},$$
(20)
Substituting Eq. (19) into Eq. (20),where the radiation resistance function is given byand the radiation reactance function is
$$Z = \rho cS\left( {R + jX} \right),$$
(21)
$$R = \frac{{kS\sum \sum \frac{{{{\xi }_{{aj}}}{{\xi }_{{ai}}}}}{{\sqrt {{{{\left( {{{x}_{i}} - x_{j}^{'}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - y_{j}^{'}} \right)}}^{2}}} }}\sin \left[ {k\sqrt {{{{\left( {{{x}_{i}} - {{x}_{j}}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - {{y}_{j}}} \right)}}^{2}}} } \right]\Delta {{s}_{j}}\Delta {{s}_{i}}}}{{2\pi {{{(\sum {{\xi }_{{ai}}}\Delta {{s}_{i}})}}^{2}}}},$$
$$X = \frac{{kS\sum \sum \frac{{{{\xi }_{{aj}}}{{\xi }_{{ai}}}}}{{\sqrt {{{{\left( {{{x}_{i}} - x_{j}^{'}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - y_{j}^{'}} \right)}}^{2}}} }}\cos \left[ {k\sqrt {{{{\left( {{{x}_{i}} - {{x}_{j}}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - {{y}_{j}}} \right)}}^{2}}} } \right]\Delta {{s}_{j}}\Delta {{s}_{i}}}}{{2\pi {{{(\sum {{\xi }_{{ai}}}\Delta {{s}_{i}})}}^{2}}}}.$$
If the radiator is a piston, the amplitude and phase of the vibration velocity in each infinitesimal area are identical; therefore, \(\left\langle u \right\rangle \left\langle u \right\rangle * = {{\omega }^{2}}\xi _{{ai}}^{2}\) and Eq. (21) becomeswhere the radiation resistance function of a piston radiator is given byand the radiation reactance function is
$$Z{\kern 1pt} ' = \rho cS\left( {R{\kern 1pt} '\,\, + jX{\kern 1pt} '} \right),$$
(22)
$$R{\kern 1pt} ' = \frac{{k\sum \sum \frac{{\sin \left[ {k\sqrt {{{{\left( {{{x}_{i}} - {{x}_{j}}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - {{y}_{j}}} \right)}}^{2}}} } \right]\Delta {{s}_{j}}\Delta {{s}_{i}}}}{{\sqrt {{{{\left( {{{x}_{i}} - {{x}_{j}}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - {{y}_{j}}} \right)}}^{2}}} }}}}{{2\pi S}},$$
$$X{\kern 1pt} ' = \frac{{k\sum \sum \frac{{\cos \left[ {k\sqrt {{{{\left( {{{x}_{i}} - {{x}_{j}}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - {{y}_{j}}} \right)}}^{2}}} } \right]\Delta {{s}_{j}}\Delta {{s}_{i}}}}{{\sqrt {{{{\left( {{{x}_{i}} - {{x}_{j}}} \right)}}^{2}} + {{{\left( {{{y}_{i}} - {{y}_{j}}} \right)}}^{2}}} }}}}{{2\pi S}}.$$
The derivations above can be used to obtain discrete calculation formulas for the axial sound pressure, directivity, and radiation impedance. Consequently, software such as ANSYS or COMSOL can be used to perform the simulations. In this study, ANSYS 2021 was used for simulations as follows. Initially, by inputting the radiator’s material properties and boundary conditions, the radiator’s geometric model was established via the simulation software, the radiator was meshed into infinitesimal areas, and then the coordinates and vibration displacements of these areas were obtained through modal calculations. Subsequently, the coordinates and vibration displacements were substituted into Eqs. (4), (12), and (21) to calculate the axial sound pressure, directivity, and radiation impedance, respectively.
RESULTS AND DISCUSSION
This section presents the analyses of several examples to verify the sound field and radiation impedance calculated using the proposed method.
Sound Field
The free boundary rectangular stepped radiator depicted in Fig. 2 was analyzed. It was enhanced by incorporating steps in regions exhibiting opposite vibration displacement compared with the (0,4) stripe mode of the flat rectangular radiator. The radiator was assumed to be steel with a Poisson ratio of ν = 0.28, Young’s modulus of E = 1.96 × 1011 N/m2, and density of ρ = 7.91 × 103 kg/m3. Figure 3 illustrates the selected working mode, and the frequency of this mode is f = 31893.9 Hz. According to the method discussed, the vibration displacement and coordinates of each microelement on the surface of the radiator were calculated by the finite element simulation software, and the vibration displacement and coordinates were substituted into Eqs. (4) and (12) to obtain the axial sound pressure amplitude and directionality of the radiator. The results of the simulation are shown in Figs. 4 and 5.
Fig. 2.
Rectangular stepped radiator: (a) dimensions, (b) fabricated radiator.
Fig. 3.
Mode of vibration for the rectangular stepped radiator: (a) top view, (b) side view.
Fig. 4.
Axial sound pressure amplitude of the rectangular stepped radiator, f = 31893.9 Hz.
Fig. 5.
Directivity of the rectangular stepped radiator, f = 31893.9 Hz.
To verify the sound field calculations, the radiator shown in Fig. 2 was constructed, and its axial sound pressure and directionality were tested in an unobstructed environment. The schematic of the experimental setup is shown in Fig. 6. The apparatus comprises an ultrasonic generator, digital power meter, transducer, radiator, sound level measurement amplifier (model NA-42, manufactured by Japan Liyin company, with a measurement range from 1 Hz to 100 kHz), and an acoustic pressure probe (a type UC-29 microphone, manufactured by Japan Liyin company, with a diameter of \(\frac{1}{4}\) inch and a working frequency range from 20 Hz to 100 kHz).
Fig. 6.
Experimental setup for the measurement of a sound field.
Axial Sound Pressure
At the beginning of the axial sound pressure experimental test, a microphone was placed 1 mm away from the radiator. It was then moved toward the far end of the radiator at step lengths of 10 mm along the central perpendicular line. Consequently, a test point was produced, and the sound intensity pSPL at different test positions was obtained. After the experiment, according to the relationship between the effective sound pressure pe and sound intensity \({{p}_{{{\text{SPL}}}}} = 20\log \frac{{{{p}_{{\text{e}}}}}}{{{{p}_{{{\text{ref}}}}}}}\) (pref is the reference sound pressure and was considered to be 2 × 10–5 Pa), the sound pressure along the axial direction of the radiator was obtained as shown by the solid line in Fig. 4.
As shown in Fig. 4, the axial sound pressure amplitude in the near field fluctuated considerably, with the maximum and minimum values of the sound pressure amplitude alternating. In the far sound field, the amplitude of sound pressure gradually decreased with the axial distance. Moreover, the critical point between the near and far sound fields was d2/λ = 1.7 m away from the radiator’s center, where d and λ represent, respectively, half the radiator’s diagonal length and the wavelength of the radiated sound.
Both the theoretical and experimental results exhibit oscillations in the near field and a monotonic decay trend in the far field, consistent with the fundamental principles of acoustic propagation. However, there are some differences between the experimental and theoretical results, primarily in terms of the sound pressure amplitudes and particularly in the far field. Furthermore, the theoretical calculations show fluctuations in axial sound pressure amplitudes within the near field, with clear and numerous peaks and valleys, whereas the experimental measurements reveal more dispersed and less numerous near field peaks and valleys that exhibit shifts in the coordinate position.
These discrepancies can be explained as follows. First, in terms of discrepancies caused by the attenuation of acoustic waves during propagation through air. The attenuation, which is primarily caused by classical absorption and molecular relaxation absorption, is not considered in the theoretical model. When considering the attenuation, it is necessary to revise the Eq. (2), and the revised formula is shown as follows:
$${{p}_{1}} = \iint {j\frac{{k\rho c}}{{2\pi H}}}{{u}_{{ai}}}{{e}^{{j\omega t}}}{{e}^{{ - jkH}}}{{e}^{{ - \alpha H}}}d{{s}_{i}}.$$
(23)
And the sound pressure amplitude \({{p}_{{a1}}}\), after accounting for attenuation, is given bywhere \({{e}^{{ - \alpha {\text{H}}}}}\) is the attenuation term, \({{\alpha }}\) is the total attenuation coefficient, \(\alpha = {{\alpha }_{{{\text{class}}}}} + {{\alpha }_{{{\text{relax}}}}}\). \({{\alpha }_{{{\text{class}}}}}\) is the classic attenuation coefficient, \({{\alpha }_{{{\text{relax}}}}}\) is relaxation attenuation coefficient. All experiments involved in this paper were conducted under a temperate atmosphere environment, therefore the attenuation coefficient was calculated using the formula specified in ISO 9613-1:1993 [26, 27]. Taking an air temperature of 20°C, pressure of 1 atm, and relative humidity of 50% as an example, \(\alpha = 0.083\,\,~{\text{Np}}\,/\,{\text{m}}\). The axial sound pressure amplitude, accounting for acoustic waves attenuation, is shown by the dotted line in the Fig. 4. It indicates that the deviation between theory and the experiment is related to attenuation.
$${{p}_{{a1}}} = \frac{{k\rho c}}{{2\pi }}\sqrt {\left[ {\sum {{e}^{{ - \alpha H}}}\frac{{\omega {{\xi }_{{ai}}}}}{H}\sin \left( {kH} \right)\Delta {{s}_{i}}{{]}^{2}} + \,\,} \right[\sum {{e}^{{ - \alpha H}}}\frac{{\omega {{\xi }_{{ai}}}}}{H}\cos \left( {kH} \right)\Delta {{s}_{i}}{{]}^{2}}} ,$$
(24)
Second, in terms of discrepancies caused by the experimental operation. A step interval of 10 mm for microphone movement may not be fine enough to capture all pressure peaks and valleys, potentially missing some peak and valley points. In addition, during the manual movement of the microphone, ensuring consistent spacing of exactly 10 mm was not possible, leading to deviations in the measured peak and valley positions.
Third, in terms of discrepancies caused by microphone. A UC-29 model microphone, manufactured by the Japanese company Liyin with a diameter of 0.25 inch, was used in the experiment. However, due to the large size of the microphone’s diaphragm, its high inertia may make it prone to phase deviation when measuring the sound pressure of high-frequency acoustic waves. Finally, in terms of discrepancies caused by deviations in material and geometric parameters. The material parameters and geometric dimensions of the rectangular stepped radiator used in theoretical calculations cannot be completely consistent with those used in experiments, introducing discrepancies in vibration mode prediction and sound field distribution, which in turn led to discrepancies between the theoretical and experimental results.
Directivity
In directivity experiments, the microphone was first placed in the far sound field directly facing the radiator’s center. Then, the turntable was rotated counterclockwise every 5° (up to 90°) to record data. Finally, mirror tests were performed to complete the directivity test, and its results are shown by the dashed line in Fig. 5.
As shown in Fig. 5, the solid and dashed lines represent, respectively, the theoretical calculation and experimental measurement results. The theoretical and experimental results of the directivity are consistent, a sharp major lobe and some side lobes; however, the side lobes are much smaller than the main lobe, demonstrating the concentration of the sound energy and the sharp directionality. Although the theoretical calculation results consistent well with the experimental results, there are still some discrepancies, primarily exhibited in the width of the main lobe and side lobes measured in the experiment are wider than that calculated theoretically, besides, the position of the side lobe is offset.
In part, these discrepancies arise from the experimental operational conditions. As manually adjusting the turntable’s angle cannot ensure a consistent accuracy of exactly 5°, with the accumulation of multiple measurement discrepancies of this type leading to a shifting of the side lobe’s position. Another factor is the discrepancies caused by the microphone. Due to the large size of the microphone’s diaphragm, its high inertia may make it prone to deviation when measuring the directionality of high-frequency acoustic waves. Finally, there are discrepancies caused by deviations between the theoretical and experimental material and geometric parameters.
Despite the discrepancies between the theoretical and experimental results, the experimental outcomes are largely consistent with the calculated results, which also align with the fundamental laws of acoustics. Therefore, the method provided in this paper can be justifiably considered to provide a theoretical basis for the design and application of radiators and offers an evaluation method for assessing the acoustic performance of sound radiators, particularly the performance of stepped radiators.
In addition, we have conducted detailed studies on the directionality of square flat, square-stepped, square piston, and axisymmetric circular stepped radiators [20, 28, 30, 31]. These studies indicated the feasibility of using simulations to calculate the sound field.
RADIATION IMPEDANCE
Simulations were used to calculate the radiation impedance of radiators with different boundary conditions and shapes. Simultaneously, the results of the proposed simulation were compared with legacy calculations to verify its effectiveness.
Piston Radiator
First, a circular piston radiator with a radius of a was considered. As shown in Fig. 7, according to the Rayleigh integral, the sound pressure generated by the entire radiator at \(ds{\kern 1pt} '\) iswhere \({{u}_{a}}\) represents the vibration velocity amplitude of \(ds\), and \(h{\kern 1pt} '\) represents the distance between \(ds\) and \(ds{\kern 1pt} '\).
$$p = j\frac{{k\rho c}}{{2\pi }}{{u}_{a}}{{e}^{{j\omega t}}}\iint {\frac{{{{e}^{{ - jkh{\kern 1pt} '}}}}}{{h'}}}ds,$$
(25)
Fig. 7.
Circular piston radiator in an infinite baffle.
Therefore, the reaction force of the sound field on the radiator is
$${{F}_{r}} = j\frac{{k\rho c}}{{2\pi }}{{u}_{a}}{{e}^{{j\omega t}}}\iint {ds{\kern 1pt} '}\iint {\frac{{{{e}^{{ - jkh{\kern 1pt} '}}}}}{{h{\kern 1pt} '}}ds}.$$
(26)
From Fig. 7,where \(\rho {\kern 1pt} '\) is the distance from the center of the circle to \(ds{\kern 1pt} '\), and \({{\theta }}\) is the angle between \(h{\kern 1pt} '\) and the diameter. Then, Eq. (24) can be simplified asIntegrating Eq. (26):where J1 is the first-order Bessel function, and H1 is the first-order Struve function.
$$ds{\kern 1pt} ' = \rho {\kern 1pt} 'd\rho {\kern 1pt} 'd\phi ,\,\,\,\,ds = h{\kern 1pt} 'd\theta dh{\kern 1pt} ',$$
(27)
$${{F}_{{\text{r}}}} = j\frac{{k\rho c}}{\pi }{{u}_{a}}{{e}^{{j\omega t}}}\mathop \smallint \limits_0^a \rho {\kern 1pt} 'd\rho {\kern 1pt} '\mathop \smallint \limits_0^{2\pi } d\phi \mathop \smallint \limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} d\theta \mathop \smallint \limits_0^{2\rho {\kern 1pt} '\cos \theta } {{e}^{{ - jkh{\kern 1pt} '}}}dh{\kern 1pt} '.$$
(28)
$${{F}_{{\text{r}}}} = \rho c\pi {{a}^{2}}{{u}_{a}}\left[ {1 - \frac{{2{{J}_{1}}\left( {2ka} \right)}}{{2ka}} + j\frac{{2{{H}_{1}}\left( {2ka} \right)}}{{2ka}}} \right]{{e}^{{j\omega t}}},$$
(29)
According to the definition, the radiation impedance of a circular piston radiator iswhere the radiation resistance is \({{R}_{{\text{r}}}} = 1 - \frac{{2{{J}_{1}}\left( {2ka} \right)}}{{2ka}},\) and the radiation reactance is \(~{{X}_{{\text{r}}}} = \frac{{2{{H}_{1}}\left( {2ka} \right)}}{{2ka}}.\)
$${{Z}_{{\text{r}}}} = \frac{{{{F}_{{\text{r}}}}}}{U} = \rho c\pi {{a}^{2}}\left( {{{R}_{{\text{r}}}} + j{{X}_{{\text{r}}}}} \right),$$
(30)
Equation (30) can be used to analytically calculate the radiation impedance of a circular piston radiator. In the following section, we further calculate the radiation impedance of the circular piston radiator using the simulations. Similarly, the radiator model was established, and the coordinates of all infinitesimal areas on the surface of the radiator were extracted and substituted into Eq. (22). Figure 8 shows that the results obtained by the two calculation methods are in good agreement. Moreover, Table 1 shows that the relative errors of radiation resistance between the proposed simulation and the analytical results were not greater than 4.4810%, and the relative radiation reactance errors were not greater than 9.9368%.
Fig. 8.
Radiation resistance and reactance of a circular piston radiator.
Table 1.
Radiation resistance function and reactance function for a circular piston radiator
2ka | Analytic calculation | Our results | (R2 − R1)/R1, % | (X2 − X1)/X1, % | ||
|---|---|---|---|---|---|---|
R1 | X1 | R2 | X2 | |||
0.1 | 0.0012495 | 0.0424130 | 0.0012494 | 0.0416164 | –0.0080 | –1.8782 |
0.2 | 0.0049917 | 0.0846565 | 0.0049913 | 0.0830605 | –0.0080 | –1.8853 |
0.3 | 0.0112079 | 0.1265620 | 0.0112063 | 0.1241609 | –0.0143 | –1.8972 |
0.4 | 0.0198671 | 0.1679627 | 0.0198624 | 0.1647485 | –0.0237 | –1.9136 |
0.5 | 0.0309262 | 0.2086950 | 0.0309150 | 0.2046569 | –0.0362 | –1.9349 |
1.0 | 0.1198988 | 0.3969147 | 0.1197322 | 0.3885241 | –0.1390 | –2.1140 |
2.0 | 0.4232752 | 0.6467637 | 0.4209409 | 0.6282364 | –0.5515 | –2.8646 |
3.0 | 0.7739607 | 0.6800731 | 0.7643694 | 0.6511633 | –1.2392 | –4.2510 |
4.0 | 1.0330217 | 0.5348633 | 1.0105241 | 0.5000181 | –2.1778 | –6.5148 |
5.0 | 1.1310317 | 0.3231248 | 1.0941336 | 0.2910166 | –3.2623 | –9.9368 |
7.0 | 1.0013379 | 0.0989445 | 0.9564681 | 0.0910749 | –4.4810 | –7.9535 |
8.0 | 0.9413409 | 0.1220290 | 0.9061372 | 0.1218675 | –3.7397 | –0.1323 |
9.0 | 0.9454863 | 0.1663428 | 0.9216113 | 0.1642187 | –2.5252 | –1.2769 |
10.0 | 0.9913055 | 0.1783665 | 0.9728556 | 0.1672581 | –1.8612 | –6.2279 |
Similarly, using a square piston radiator and a rectangular piston radiator with a length–width ratio of 4 : 1 as examples, the simulation results are compared with those of Burnett [29]. The solid lines in Figs. 9 and 10 represent the calculation results from [29], and the dashed lines are the results of our simulations. The simulation results are consistent with Burnett’s results. Moreover, Table 2 indicates that the maximum relative error of the radiation resistance of a square piston is 1.1577%, and the maximum relative error of the radiation reactance is 5.7976%. Similarly, Table 3 shows that the maximum relative errors for the radiation resistance and reactance of a rectangular piston radiator with a length–width ratio of 4 : 1 are 0.3257 and 4.6194%, respectively.
Fig. 9.
Radiation resistance and reactance of a square piston radiator.
Fig. 10.
Radiation resistance and reactance of a rectangular piston radiator with a length–width ratio of 4 : 1.
Table 2.
Radiation resistance function and reactance function for a square piston radiator
\(k\sqrt s \) | Burnett’s results [26] | Our results | (R2 − R1)/R1, % | (X2 − X1)/X1, % | ||
|---|---|---|---|---|---|---|
R1 | X1 | R2 | X2 | |||
0.1 | 0.0015907 | 0.0472786 | 0.0015906 | 0.0467634 | −0.0049 | −1.0897 |
0.2 | 0.0063521 | 0.0943088 | 0.0063518 | 0.0932772 | −0.0048 | −1.0939 |
0.3 | 0.0142525 | 0.1408439 | 0.0142515 | 0.1392935 | −0.0069 | −1.1008 |
0.4 | 0.0252395 | 0.1866412 | 0.0252368 | 0.1845686 | −0.0108 | −1.1105 |
0.5 | 0.0392400 | 0.2314636 | 0.0392340 | 0.2288642 | −0.0154 | −1.1230 |
1.0 | 0.1505592 | 0.4332708 | 0.1504762 | 0.4279421 | −0.0552 | −1.2299 |
2.0 | 0.5101172 | 0.6613308 | 0.5090247 | 0.6500517 | −0.2142 | −1.7055 |
3.0 | 0.8730617 | 0.6199537 | 0.8689103 | 0.6030836 | −0.4755 | −2.7212 |
4.0 | 1.0709715 | 0.4124478 | 1.0622283 | 0.3923452 | −0.8164 | −4.8740 |
5.0 | 1.0769733 | 0.2104150 | 1.0645047 | 0.1982161 | −1.1577 | −5.7976 |
Table 3.
Radiation resistance function and reactance function for a rectangular piston radiator with a length–width ratio of 4 : 1
\(k\sqrt s \) | Burnett’s results [26] | Our results | (R2 − R1)/R1, % | (X2 – X1)/X1, % | ||
|---|---|---|---|---|---|---|
R1 | X1 | R2 | X2 | |||
0.1 | 0.0015897 | 0.0422813 | 0.0015896 | 0.0418793 | −0.0038 | −0.9507 |
0.2 | 0.0063362 | 0.0842235 | 0.0063361 | 0.0834196 | −0.0013 | −0.9545 |
0.3 | 0.0141728 | 0.1254942 | 0.0141727 | 0.1242883 | −0.0011 | −0.9609 |
0.4 | 0.0249899 | 0.1657740 | 0.0249897 | 0.1641662 | −0.0008 | −0.9699 |
0.5 | 0.0386375 | 0.2047630 | 0.0386375 | 0.2027533 | 0.0000 | −0.9815 |
1.0 | 0.1417857 | 0.3718460 | 0.1417952 | 0.3678225 | 0.0067 | −1.0820 |
2.0 | 0.4144608 | 0.5329509 | 0.4145626 | 0.5247543 | 0.0246 | −1.5380 |
3.0 | 0.6241440 | 0.5489272 | 0.6241580 | 0.5360232 | 0.0022 | −2.3508 |
4.0 | 0.7832267 | 0.5364557 | 0.7821945 | 0.5185282 | −0.1318 | −3.3418 |
5.0 | 0.9282847 | 0.4807726 | 0.9252609 | 0.4585637 | −0.3257 | −4.6194 |
Simply Supported Rectangular Radiators
A simply supported rectangular radiator (255 × 85 × 5 mm) was considered as an example. It was made of steel with a Poisson ratio of ν = 0.28, Young’s modulus of E = 1.96 × 1011 N/m2, and a density of ρ = 7.91 × 103 kg/m3. The radiation resistance and reactance of the radiator in (1,1) mode were calculated with the average surface vibration velocity as a reference velocity. Simultaneously, to verify the simulation results, 10-node Gauss–Legendre numerical integration was used to calculate the radiation resistance and reactance of the radiator in the same mode. In Fig. 11, the dashed lines represent the simulation results, and the solid lines represent the Gauss–Legendre integration results. The two methods produced very similar results. Detailed data are also given in Table 4, and the maximum relative errors of the radiation resistance and radiation reactance are 0.2246 and 2.3172% respectively.
Fig. 11.
Radiation resistance and reactance of a simply supported rectangular radiator.
Table 4.
Radiation resistance function and reactance function for a simply supported rectangular radiator
\(k\sqrt s \) | Integral results | Our results | (R2 – R1)/R1, % | (X2 – X1)/X1, % | ||
|---|---|---|---|---|---|---|
R1 | X1 | R2 | X2 | |||
0.1 | 0.0015892 | 0.0577687 | 0.0015906 | 0.0574044 | 0.0924 | −0.6307 |
0.2 | 0.0063467 | 0.1153085 | 0.0063526 | 0.1145798 | 0.0927 | −0.6320 |
0.3 | 0.0142426 | 0.1723924 | 0.0142558 | 0.1712992 | 0.0931 | −0.6341 |
0.4 | 0.0252272 | 0.2287974 | 0.0252509 | 0.2273398 | 0.0936 | −0.6371 |
0.5 | 0.0392321 | 0.2843065 | 0.0392691 | 0.2824844 | 0.0944 | −0.6409 |
1.0 | 0.1509319 | 0.5414513 | 0.1510833 | 0.5378039 | 0.1004 | −0.6737 |
2.0 | 0.5198068 | 0.8977362 | 0.5204452 | 0.8903807 | 0.1228 | −0.8193 |
3.0 | 0.9320934 | 1.0155781 | 0.9335378 | 1.0043004 | 0.1550 | −1.1105 |
4.0 | 1.2633679 | 0.9705233 | 1.2657675 | 0.9550112 | 0.1899 | −1.5983 |
5.0 | 1.4937380 | 0.8611323 | 1.4970934 | 0.8411783 | 0.2246 | −2.3172 |
Free-Boundary Stepped Radiator
The calculations and analyses described above, focusing on the radiation impedance of piston radiators and simple rectangular radiators with various geometries, emphasize the effectiveness of the proposed method for calculating radiation impedance. Subsequently, the radiation impedance of the free boundary rectangular stepped radiator shown in Fig. 2 was calculated.
As shown in Fig. 12, the radiation resistance of a rectangular stepped radiator is relatively small at low frequencies, resulting in a low degree of radiation efficiency and limited energy radiation. The resistance increases with frequency, reaching the maximum at a midfrequency, at which radiation efficiency is significantly improved, before finally stabilizing and approaching and fluctuating around a value of one at high frequencies. The radiation reactance of a rectangular stepped radiator increases linearly at low frequencies, fluctuates at mid frequencies, and finally approaches zero at high frequencies, indicating that the energy is primarily dominated by radiation in the high-frequency region. The calculation results indicate that the radiation impedance can experience small oscillations at certain frequency points owing to phase interference by acoustic waves. In practical applications, further analysis should be conducted based on specific situations.
Fig. 12.
Radiation resistance and reactance of the rectangular stepped radiator.
CONCLUSIONS
This study involves developing a simulation-based approach for calculating the sound field and radiation impedance of stepped sound radiators. Several calculation results were obtained; these results indicate that, although not optimal, the proposed method provides computational estimation for engineering applications to calculate and analyze parameters such as the radiation sound field and efficiency of radiators.
If the vibration velocity amplitude and phase are uniform across the radiator’s surface, the radiator can be modeled as a piston radiator. This transformation also allows us to convert the radiation impedance of the flexural vibration radiator into the radiation impedance of the piston radiator. It is indicated that the method can be used to calculate the radiation impedance of bending vibration radiator and piston radiator.
The proposed method requires two conditions. First, when calculating the radiation impedance, the radiator’s surface mesh must be refined for accurate calculations. Second, after meshing, because the finite element software cannot accurately locate the vibration velocity of the radiator’s center, the center’s vibration velocity can only be approximated with the velocity of the infinitesimal areas closest to the center. Therefore, calculations of radiation impedance that use the average surface vibration velocity as the reference velocity are more accurate.
CONFLICT OF INTEREST
The authors of this work declare that they have no conflicts of interest.
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