Controlling sound radiation through an opening with secondary loudspeakers along its boundaries

We propose a virtual sound barrier system that blocks sound transmission through openings without affecting access, light and air circulation. The proposed system applies active control technique to cancel sound transmission with a double layered loudspeaker array at the edge of the opening. Unlike traditional transparent glass windows, recently invented double-glazed ventilation windows and planar active sound barriers or any other metamaterials designed to reduce sound transmission, secondary loudspeakers are put only along the boundaries of the opening, which provides the possibility to make it invisible. Simulation and experimental results demonstrate its feasibility for broadband sound control, especially for low frequency sound which is usually hard to attenuate with existing methods.


Planar active sound barrier
According to the Huygens' Principle, every point at a wave front may be considered as the source of secondary wavelets with a speed equal to the speed of the waves. This is the theoretical basis of planar active sound barriers. In a planar active sound barrier system, the loudspeakers are distributed over the entire opening as shown in Supplementary Fig. S1a, and they can be regarded as the sources of secondary wavelets, thus sound radiation through the opening can be reduced as long as there are sufficient of them. In the simulations, 32 loudspeakers are evenly distributed in x-y plane at the height of 0.448 m, as shown in Supplementary Fig. S1b, and the picture of the experimental setup is in Supplementary Fig. S1c.

Number of modal terms used in the simulations
In the simulations, we used the sound power of the system as the cost function, which can be calculated by S1 In Eq. (S-2), Rss is the real part of the acoustic transfer function matrix between the secondary loudspeakers, Rsp is the real part of the acoustic transfer function vector from the primary sound source to the secondary loudspeakers, and qp is the strength of the primary sound source. I is an identity matrix and β is a positive real number to constrain the strengths of secondary loudspeakers. In the simulations, we set qp as 10 4 m 3 /s, so qs is a real vector.
Since qp is a real number and all the elements in qs are real, whether the sound power of the system (Eq. (S-1)) can be accurately calculated only depends on the real part of pp and ps. The modal superposition method is applied to obtain the sound pressure. x y x y G k z z Sk N is the number of modal terms that we used for the calculation of sound pressure.
In the near field of a point source, the sound pressure becomes intense, but we only focus on the real part of sound pressure, which converges with N. Supplementary Figure   S2 shows the real and imaginary part of the sound pressure generated by a point source at

The advantages of a double layered loudspeaker system
We employed a double layered loudspeaker system because it performs much better than a single layer system with the same number of secondary loudspeakers. We compared a double layered and a single layer system by simulations and experiments.
Supplementary Figure Supplementary Fig. S4. It is clear that the sound power reduction the double layered loudspeaker system achieved is much higher than the single layer system, and that's why we employed a double layered loudspeaker array instead of a single layer system.
Supplementary Figure S5 shows the sound power level without and with 48 secondary loudspeakers at the edge when they are divided into different numbers of layers. We can see that multi-layer loudspeaker arrays achieve much better sound reduction performance than a single layer; however, it is not that the more layers, the better.

Noise reduction with different layer heights
No matter which heights the 2 loudspeaker arrays are set at, the virtual sound barrier can achieve satisfactory sound power reduction over a wide frequency band.

More details about the experiments
The input port of The virtual error sensor arrangement S3-S4 will be investigated in the future to facilitate access through the opening. In the virtual error sensor arrangement, physical error microphones are implemented remote from the opening and the sound pressure at the opening is predicted by analytical or numerical methods and minimized to reduce the sound power radiation through the opening.
In the experiments, the expectation of the sum of the squared amplitude of the sine and cosine component of all the 32 error signals is defined as the cost function S5-S6 The lth output of the active controller is cs cc where l = 1, 2, …, 32, and Acsl(n) and Accl(n) are updated by 32 cs cs es ss ec sc 1 where Csslm, Csclm, Ccslm and Ccclm are the acoustic transfer functions between the mth error microphone to the lth loudspeaker, which can be obtained with the online cancellation path modelling method.
With an appropriate step size μ, Equation (S-8) converges to the optimized solution.
In the experiments, 10 microphones are located on a hemisphere frame of 1.5 m radius according to ISO 3744 to measure the sound power level of the system, and the positions are listed in Supplementary Table S1  where Acs(n) and Acc (n The internally synthesized tonal signal is synthesized as the reference signal, so we don't need a reference microphone. Advantages of the algorithms also include fast convergence speed and low computation load.

Noise reduction for general noise sources
We presented the results under the simplest condition that the primary source is a point source because the loudspeaker can be regarded as a point source within relatively low frequency range and this enables us to quantify and compare the simulation and experiment results. The system is also effective when the primary source is more complicated. Supplementary Figure S8 shows the sound power level without and with the virtual sound barrier when the primary source is a dipole, longitudinal quadrupole, lateral quadrupole, a line source and plane source, respectively. The virtual sound barrier system still has 32 secondary sources at the edge of two layers as mentioned in the manuscript. It is clear in Supplementary Fig. S8 that the sound power reduction below 1000 Hz in all the cases are more than 30 dB.
We did 3 experiments on more complicated situations when there are multiple primary sources emitting sound energy at different frequencies: (1) A line primary source with 6 loudspeakers, as shown in Supplementary Fig. S9a.
These loudspeakers radiate sound energy at the same frequency. We still apply the waveform synthesis algorithm and tonal sound of the same frequency is used as the reference signal by the active controller S5-S6 . The sound power level without and with the virtual sound barrier and the sound power reduction are shown in Supplementary Table   S2.
(2) Two point sources at different locations simultaneously emitting acoustic energy at different frequencies. These two point sources are at (0.1, 0.1, 0.1) m and (0.2, 0.2, 0.2) m and the exciting frequency is f1 and f2, respectively. The photo of the experimental setup is shown in Supplementary Fig. S9b. The internally synthesized tonal signals at f1 and f2 are used as the reference signal S5-S6 . The sound power level at these two frequencies with and without the virtual sound barrier and the sound power reductions are shown in Supplementary Table S3.
(3) Three point sources at different locations emitting acoustic energy at 3 different frequencies 600 Hz, 700 Hz and 800. The experimental setup is shown in Supplementary   Fig. S9c. The sound power level at the 3 frequencies with and without the virtual sound barrier and the sound power reductions are shown in Supplementary Table S4.
We can see that all these sound power reductions with multiple sound sources in Supplementary Tables S2, S3 and S4 are more than 10 dB, but less than that with a single primary source. The reason is that dealing with 3 frequencies simultaneously puts very heavy computation load on the active controller used in the experiments, which deteriorates the performance of the system. Due to the constraints of the hardware and algorithm used in the experiments, 10-20 dB is almost the best result we could achieve with the current system. Although the noise reduction is not as good as that with a single primary source, 10 dB noise reduction is sufficient for demonstrating the feasibility of the concept proposed in the paper. These experiments demonstrate the feasibility of the virtual sound barrier system when the primary noise source is a more general one.
We did some further numerical simulations on the experimental setup with 6 primary sources as shown in supplementary Figure S10a shows clearly that the sound power reductions are almost the same when there are 1 or 6 primary sources. This demonstrates that the system works effectively in broad frequency range for tonal sound no matter how complicated the primary sound field is.
The double layered secondary loudspeaker system can achieve effective control of low frequency sound in experiments for complicated primary sound fields, but the experimental results at high frequencies is not as good as that in low frequency range due to the constraints of hardware at present. We believe its performance will be improved if more powerful active controller can be obtained.

Alternative error sensor strategies
The error microphones are distributed over the entire opening in the experiments in the paper, but there are other alternative error sensor arrangements to avoid putting them in the pathway of the opening. As long as double-layered secondary sources at the edge of the opening work effectively, the system has the possibility to be invisible.
Supplementary Figure