Directional sound radiation from a rectangular panel and the high frequency limit

Directional sound radiation aims to focus sound energy in a given target direction such that the sound energy is greater in that direction than in other directions. This technology can be used in personal audio systems. Many studies have achieved good sound focusing performance using loudspeaker arrays. Sound focusing control methods that use loudspeaker arrays include beamforming,¹ pressure matching,^2,3 and acoustic contrast control.^4,5 In addition to loudspeaker arrays, flat-panel loudspeakers can be designed for sound field control. Flat-panel loudspeakers have received much attention because of the small space required for installation.⁶ Flat panel loudspeakers produce a sound field by exciting the bending vibration of a thin panel through actuators in place of conventional loudspeaker cones. Distributed mode loudspeakers (DMLs) have been proposed⁷ and applied in wave field synthesis (WFS).⁸ DMLs with smaller volumes and lighter weights than conventional loudspeaker arrays are suitable for WFS. However, because of the modal vibration of the flat panel, the radiation efficiency decreases at low frequencies. To overcome this limitation, multi-actuator panels (MAPs) have been proposed,^9,10 in which actuator arrays are distributed on highly damped panels. The low frequency response can be improved due to the larger surface area of the MAPs. In addition to the WFS method, Gauthier reproduced the sound field in an aircraft cabin using the least squares method with an actuator array.¹¹ 

The sound field directivity of a flat panel can be controlled using actuator arrays. Bai¹² used a genetic algorithm to optimize the positions of actuators and the delays of input signals in terms of omni-directionality. Jeon¹³ used the acoustic energy difference method for beamforming. The actuator positions were optimized to allow the panel vibration energy to be concentrated near the actuators. The optimized actuator array minimized the bending mode effect of the panels, thereby reducing abrupt changes in the beamforming performance at some frequencies. Kournoutos¹⁴ applied the acoustic contrast control method to achieve directional sound radiation by maximizing the acoustic energy contrast in two control zones.

When using loudspeaker arrays for directional sound field control, the discrete distribution of the loudspeaker array results in spatial aliasing effects at higher frequencies, thus imposing a high frequency limit.¹⁵ Actuator arrays generate sound fields by driving panel vibrations. Considering that the panel is a continuous source, a vibrating panel may reduce aliasing effects. However, a finite number of actuators on the panel used to excite the desired modes simultaneously excite a number of other modes.¹⁶ This leads to the aliasing effects of the panel vibration, which can influence the radiated sound field at higher frequencies.

In this study, directional sound field control in the elevation and azimuth directions is achieved using arrays of two-dimensional planar actuators uniformly distributed on square panels. Acoustic contrast control method is selected to realize directional sound radiation. This method is described in Sec. 2, and the theoretical model of a panel driven by an actuator array is presented. The aliasing effects of the two arrays are described in Sec. 3. In Sec. 4, the high frequency limit of directional sound radiation using actuator arrays due to the aliasing effect is analyzed based on simulations, and the acoustic contrast frequency responses of the two arrays are compared. Conclusion is given in Sec. 5.

Sound radiation based on an actuator array is achieved by exciting the bending vibrations of a panel.

As illustrated in Eq. (7), the displacement amplitude of a mode is formed as the sum of all actuator contributions to that mode. For $N × N$ uniformly distributed actuators on a panel, the amplitudes of the $( 1 , 1 )$ to $( N , N )$ modes can be specified by designing the driving forces of the actuators.¹⁶ According to Eq. (7), given the magnitudes of these modes, the driving forces can be solved.

In Eq. (7), $sin ( k m x i + k m ( a / 2 ) ) sin ( k n y i + k n ( b / 2 ) )$ is the coupling factor, which represents the contribution of the driving force of the ith actuator on the $( m , n )$ mode. With the designed driving forces exciting the desired lower order modes, the amplitudes of the higher order modes can be calculated using the corresponding coupling factors. This indicates that the higher order modes can be driven alongside the lower order modes. The coupling factors are determined by the actuator array distribution. The coupling factors of the higher order modes can be expressed by the coupling factors of the $( 1 , 1 )$ to $( N , N )$ modes. Therefore, the amplitudes of the higher order modes depend on those of the lower order modes.

The higher order aliased modes can influence the sound field controlled by the actuator array. The high frequency limits of actuator arrays will be investigated through simulations in Sec. 4 and estimated based on the aliased modes analyzed above.

In this section, a panel driven by an actuator array is simulated for directional sound radiation. The acoustic contrast is calculated as a function of frequency and analyzed to estimate the high frequency limit of the actuator array. Simulations are also performed using arrays with different parameters to validate the proposed estimation method and compare the directivity performance with the loudspeaker arrays.

The simulation uses a square aluminum panel with dimensions 0.5 m × 0.5 m × 0.003 m. The Young's modulus, panel density and Poisson's ratio of the aluminum panel are 70 GPa, 2700 kg/m³, and 0.334, respectively. The damping factor is $η = 0.01$⁠. The boundary condition of the panel is simply supported. A two-dimensional array of 4 × 4 evenly distributed actuators is attached to the panel, as shown in Fig. 2(a). The spacings of $N x$ actuators in the x direction and $N y$ actuators in the y direction are $d x = a / ( N x + 1 )$ and $d y = b / ( N y + 1 )$⁠, respectively, where $a = b$ and $N x = N y = N$ in simulations. In the forward half-space relative to the panel, the sound field control points are distributed on a hemispherical surface with a 3 m radius, spaced at 5° in both the elevation and azimuth directions. The control zone is illustrated in Fig. 2(b). For the target direction at angle $θ = 0 °$⁠, the bright zone comprises the control points in the elevation angle range of $θ ≤ 20 °$ and azimuth angles of $0 ° ≤ ϕ ≤ 360 °$⁠. The dark zone comprises the remaining control points. The acoustic contrast response within the frequency range between 100 and 4000 Hz is shown in Fig. 3(a).

For better visualization of the directivity performance of the optimized actuator array by ACC method, the normalized sound pressure level (SPL) on the hemispherical surface at 1000 Hz is shown in Fig. 3(b). The sound pressure is maximum at $θ = 0 °$ for all $ϕ$ from $0 °$ to $360 °$⁠. We calculate the sound pressure at the control points in the xOz plane, that is, the sound pressure on the semi-circle marked in red in Fig. 2(b). The normalized SPL as a function of frequency is shown in Fig. 3(c). As shown in Fig. 3(c), the sound focusing performance of the panel first improves with increasing frequency, and then decreases because of the higher sound energy in other directions.

The radiated sound fields are calculated when mode (1,1) is driven. According to the aliasing effect described in Sec. 3.2, the amplitudes of the other modes between (1,1) and (4,4) can be zero, however, the higher order modes can be driven alongside (1,1) mode. The acoustic contrast response of the radiated sound field is also shown in Fig. 3(a).

The acoustic contrast response of the (1,1) mode, including its aliased modes, exhibits a trend similar to that of the actuator array optimized by ACC method, as shown in Fig. 3(a). Figure 3(d) illustrates the sound field distribution radiated from the (1,1) mode as a function of frequency, similar to that shown in Ref. 19. Above the frequency at which the wavelength of the (1,1) mode coincides with the acoustic wavelength in air, the radiation starts to beam. The aliasing effect occurs at high frequencies. The directivity of the sound field produced by (1,1) mode is consistent with the simulation objective of this study. As shown in Fig. 3(a), the acoustic contrast of (1,1) mode is similar to that of the ACC method at frequencies above 1000 Hz. The optimized array force vector tends to increase the contribution of the (1,1) mode at these frequencies. Therefore, the aliasing frequency of actuator array for ACC method is analyzed based on the sound radiation of the (1,1) mode.

With increasing frequency, the excitation frequency is far from the resonant frequency of the (1,1) mode. Consequently, the amplitude of the (1,1) mode decreases. However, the excitation frequency is near the resonant frequencies of its aliased modes; therefore, the amplitudes of the aliased modes increase. The sound fields radiating from the aliased modes have large sidelobes; thus, the radiated sound field of the ACC method exhibits large sidelobes when the amplitudes of the aliased modes are large.

According to the above analysis, the spatial aliasing effect occurs at an excitation frequency near the resonant frequency of the aliased mode. Thus, the high frequency limit of the actuator array for ACC method is estimated using the resonant frequency of the first aliased mode of the (1,1) mode. The aliased modes of the (1,1) mode can be determined according to Eq. (7). In the sound field calculation for the (1,1) mode, the amplitudes of the other modes between (1,1) and (4,4) are set to zero by designing the driving force $F i$⁠. Substituting the designed $F i$ into Eq. (7), the modes with non-zero amplitude are the aliased modes which are excited simultaneously. For the array distribution in this section, the (1,9) or (9,1) mode is the first aliased mode and the corresponding resonant frequency is 2411 Hz. This frequency is denoted by a circle in Fig. 3(a) and a dashed line in Figs. 3(c), 3(d). As shown, the focusing performance begins to deteriorate near the denoted frequency.

Simulations will be performed using different array parameters to verify the proposed method for estimating the high frequency limit. The acoustic contrast responses of the actuator arrays will be compared with those of the loudspeaker arrays. The high frequency limit can be used as a quantitative parameter for the comparison. The loudspeaker arrays are two-dimensional planar arrays with the loudspeakers evenly distributed in both the x and y directions. The loudspeaker and actuator arrays in comparison have the same spacing and number of array elements. The material of panel is fixed. The distributions of the bright and dark zones do not change.

The number of array elements is 4 × 4, which is kept fixed. The lengths of the panel sides are 0.5, 0.8, and 1 m. Because the array elements are evenly distributed, the array element spacings for actuator arrays are 0.10, 0.16, and 0.20 m. The compared loudspeaker arrays have the same spacings, and loudspeaker positions are the same as actuators. The distribution of loudspeaker array can also be denoted using the circles in Fig. 2(a). Using the ACC method for directional sound control, the acoustic contrast responses of the actuator and loudspeaker arrays with different spacings are shown in Figs. 4(a)–4(c). The acoustic contrast response of the (1,1) mode of the panel driven by each actuator array is also shown in Figs. 4(a)–4(c).

As shown in Figs. 4(a)–4(c), the acoustic contrast responses at low frequencies are similar for the actuator and loudspeaker arrays. With increasing frequency, the acoustic contrasts of the actuator arrays start to be higher than the loudspeaker arrays because the loudspeaker array sizes are smaller than the panel sizes. At higher frequencies, the acoustic contrasts of the actuator and loudspeaker arrays decrease due to the aliasing effects. In each simulation shown in Figs. 4(a)–4(c), the aliasing frequency of the actuator array is lower than that of the loudspeaker array. When the panel length is increased, the spacing of the loudspeaker is increased such that the aliasing frequency is reduced. The aliasing frequency of the actuator array is also reduced because the resonant frequencies of the panel modes decrease with increasing panel length. Based on Eq. (6), the resonant frequency of a mode is inversely proportional to the square of the panel side length. Because of the actuator spacing $d = a / ( N + 1 )$⁠, the resonant frequency of a mode is also inversely proportional to the square of the actuator spacing. The aliasing frequency of the loudspeaker array is inversely proportional to the spacing. The rate of decrease in the aliasing frequency for the actuator array is greater than that of the loudspeaker array as the spacing increases, which is consistent with the results shown in Figs. 4(a)–4(c).

The side lengths of the arrays are fixed at 0.5 m, and the number of array elements is 5 × 5 and 6 × 6. The actuator spacings are 0.08 and 0.07 m. The compared loudspeaker arrays have the same spacings. The acoustic contrast responses are shown in Figs. 4(d), 4(e).

The aliasing frequency increases with the increasing number of actuators. The number of array elements is represented by $N × N$⁠. The first aliased mode for the actuator arrays is $( 1 , 2 N + 1 )$ or $( 2 N + 1 , 1 )$⁠. The aliasing frequency increases quadratically with $N$⁠. When the array spacing is 0.07 m, the aliasing frequency of the actuator array is higher than that of the loudspeaker array. Comparing the results of Figs. 4(d), 4(e), and 4(a), it is also shown that when the panel material and thickness are constant the rate of decrease in the aliasing frequency for the actuator array is greater as the spacing increases.

The panel length and the number of array elements are the same as in Fig. 4(b), varying the panel thickness to 0.0077 m. The acoustic contrast responses are shown in Fig. 4(f). The panel thickness is adjusted so that the panel has the same resonant frequencies as the panel used in Fig. 4(a). With the same number of actuators, Fig. 4(f) shows the same aliasing frequency of the actuator array compared to Fig. 4(a). The aliasing frequency of the actuator array is increased compared to the case of 0.003 m thickness in Fig. 4(b), which is higher than that of the loudspeaker array. It is indicated that the aliasing frequency of the actuator array can be made higher than that of the loudspeaker array by increasing the panel thickness with the same number and spacing of array elements. However, at low frequencies below 200 Hz, the acoustic contrast decreases with increasing thickness. Because of the increase in the resonant frequencies, the decrease in the amplitudes of the vibration modes limits the performance of ACC method. In all simulations, the acoustic contrast responses of (1,1) modes exhibit trends similar to those of the actuator arrays optimized by ACC method.

The target direction of directional sound radiation in simulations is directly in front of the array. It is consistent with the directivity of the sound field produced by the (1,1) mode. Thus, the resonant frequency of the first aliased mode of the (1,1) mode is used as the aliasing frequency of the ACC optimized array. Consider the coincidence frequency at which the bending wavelength on a panel equals the acoustic wavelength in air. At the coincidence frequency of the aliased mode of the (1,1) mode, the directivity of the sound field radiated from the aliased mode is in other directions than directly in front of the panel. This influences the result of directional sound radiation. When the coincidence frequency of the aliased mode is lower than the resonance frequency, aliasing may occur at a lower frequency than the estimated aliasing frequency. However, since the sound field radiated by the optimized actuator array is a superposition of the sound fields from multiple modes. The resonant frequency of the aliased mode is chosen as the estimate of the aliasing frequency. At the resonant frequency, the aliased mode influences the radiated sound field of the array most. More accurate estimate requires further study.

Comparing the results of directional sound radiation with loudspeaker arrays, it is found that the increase rate in the aliasing frequency of the actuator array is greater with decreasing spacing. The aliasing frequencies of the actuator array and loudspeaker array differ because the former depends on the bending wavelength and physical parameters of the panel, while the latter depends on the wavelength in air. Thus, varying the thickness can lead to an increase in the aliasing frequency with the same actuator distribution.

In this study, two-dimensional directional sound radiation is realized using the actuator array driving panel by the ACC method. The transfer functions from the actuators to the sound field control zones are calculated according to the modal superposition method. Based on the modal vibration of the panel, the aliasing effect of the panel driven by the actuator arrays is analyzed. The high frequency limit of the directional sound radiation is estimated from the resonant frequency of the first aliased mode of (1,1) mode. The acoustic contrast responses between the loudspeaker and actuator arrays with the same parameters are compared through simulations. The acoustic contrast is similar between the actuator and loudspeaker arrays at low frequencies. As the frequency increases, both arrays become aliased, thereby reducing the focusing performance of the sound field. The aliasing frequencies of the actuator and loudspeaker arrays depend on the bending wavelength and physical parameters of the panel and the wavelength in air, respectively.

This work was supported by National Key Research and Development Project under Grant No. 2022YFB2602000.

The authors declare no conflict of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.