In personal audio systems, sounds should propagate toward the listening point and attenuate beyond the listening point. This study deals with controlling directivity and distance attenuation using loudspeaker arrays. The array signal processing is based on tangent line method (TLM), which can generate acoustic beams following arbitrary convex trajectories. A curvilinear acoustic beam is produced as an envelope of tangent lines, i.e., straight acoustic beams. Specifying the envelope length enables controlling distance attenuation while enhancing directivity. In this study, the TLM for arbitrary circular trajectories is formulated. Optimization algorithm is applied to identify the trajectory maximizing acoustic contrast.

Delivering sounds to a listener and maintaining silence elsewhere is beneficial when listeners and non-listeners share a common space. In such personal audio systems, sounds should propagate toward the listening point and attenuate beyond the listening point. Directivity and distance attenuation must be controlled using loudspeaker arrays. Delay-and-sum (DAS) beamformer is an established technique for controlling directivity.1 The basic idea of DAS is to generate a plane wave propagating toward a designated angle by aligning the wavefronts of spherical waves. The mechanism enhances directivity but reduces distance attenuation. The reduced distance attenuation is advantageous when it is desired to deliver sounds as far as possible. However, reduced distance attenuation is a disadvantage in the above personal audio systems. While enhancing the directivity, the distance attenuation beyond the listening point should be enhanced or maintained, at least in comparison with a spherical wave.

Such spatial distributions of the sound pressure level (SPL) can be realized by several methods. Acoustic contrast control (ACC),2–9 energy difference control (EDC),5,8,10 and pressure matching (PM)8,10–14 are well-known techniques for pursuing the designated SPL distributions by optimizing the amplitude and phase of each acoustic source. ACC maximizes the acoustic energy contrast between the listening and non-listening zones. EDC maximizes the acoustic energy difference between the listening and non-listening zones. PM minimizes the error between desired and reproduced pressures at the control points. In principle, these methods have no limitation on the shape of the SPL distribution. Though attentions were not explicitly paid to the distance attenuation in most studies, it was reported in the literatures9,13 that the distance attenuation can be enhanced by PM and ACC. In another approach, the near field of a loudspeaker array can be enhanced by maximizing the ratio of reactive acoustic power to active acoustic power.15–17 Possible applications of this method are headrest-type and wearable audio devices because the listening point must be very close to the loudspeaker array.

Alternatively, curvilinear acoustic beams can be used for controlling the directivity and distance attenuation. Generation of curvilinear acoustic beams is based on caustic theory and differently formulated by Zhang et al.18 and Zhao et al.7,19,20 The methodology is referred to as tangent line method (TLM),20 because a curvilinear acoustic beam is produced as the envelope of the tangent lines, i.e., straight acoustic beams. In principle, arbitrary convex trajectories are realizable by TLM.18,19 The envelope length is controllable. Therefore, it is possible to control the distance attenuation while enhancing the directivity, as opposed to DAS. In earlier studies by the authors,9,21 the TLM for circular trajectories has been applied to deliver sounds to a listening point. As expected, the sounds propagated toward the listening point and were attenuated beyond the listening point. However, 0 , R was specified as the center of the circular trajectories in the case study, where R is the radius and the acoustic sources are aligned on the x-axis. The employed trajectories were not optimized in terms of acoustic contrast between the listening point and other locations. Hence, the acoustic contrast can be improved by using other trajectories.

In this study, the TLM for circular trajectories with arbitrary centers and radii is formulated, and the driving signals of the acoustic sources are analytically derived. Moreover, TLM is applied to deliver sounds to a listening point. The optimal circular trajectory that maximizes the acoustic contrast between the listening point and the far field is identified through an iterative algorithm. Numerical examples are presented to compare the directivity and distance attenuation achieved by the optimal TLM with those achieved by DAS, ACC, and non-optimal TLM. The optimal TLM is also demonstrated experimentally to verify the validity of the theory. As mentioned previously, arbitrary convex trajectories, e.g., parabolic and Bezier curves, are realizable by TLM. There is no rational reason to specify circular trajectories. This study is a preliminary investigation on the sound delivery to a listening point using TLM.

In Sec. 2, the theory is described. In Sec. 3, the numerical examples are presented. In Sec. 4, the experimental data are presented, and in Sec. 5, the findings of this study are summarized.

This section begins with the formulation of TLM for arbitrary circular trajectories. Next, an iterative optimization algorithm is presented to identify the circular trajectory that maximizes acoustic contrast. In addition, the theories of DAS and ACC, which are examined for comparison, are briefly reviewed.

Figure 1(a) shows a schematic diagram of the TLM. Acoustic free space is assumed. The point acoustic sources are assumed to be aligned along the x-axis. A circular trajectory is produced as an envelope of the tangent lines. The formulation begins with describing the x-intercept and angle of each tangent line. Generating the tangent line toward the angle by acoustic sources around the x-intercept is then considered. The circle whose center is a , b and radius is R is considered as the prescribed trajectory of the acoustic beam. The circle is expressed as follows:
(1)
x c for the right and left half-circles is derived as follows:
(2)
Fig. 1.

(a) Arc, tangent line, angle, and x-intercept in TLM, (b) acoustic sources and listening point in TLM-based personal audio system.

Fig. 1.

(a) Arc, tangent line, angle, and x-intercept in TLM, (b) acoustic sources and listening point in TLM-based personal audio system.

Close modal
The angle between the tangent line and y-axis is denoted by θ, where π / 2 θ π / 2. The relationship between the x-intercept and θ for the right and left half-circles is expressed as follows:
(3)
Combining Eq. (3) with Eq. (2) and tan θ = y c b / ± x c a yields a quadratic equation for y c. The solutions are given by
(4)
Here, the listening point, x 0 , y 0, is assumed to be y 0 0. Accordingly, only y c 0 is of interest.
Combining sin θ = tan θ / 1 + tan 2 θ with tan θ = y c b / ± x c a and Eqs. (2) and (4) leads to
(5)
The time dependence is assumed to be harmonic and in the form of e j ω t, where ω is the angular frequency. The phase relationship depicted inside the subframes in Fig. 1(a) is described as follows:
(6)
which results in
(7)
where ϕ x is the phase of the point acoustic source at x , 0, d ϕ x , and d x are the phase shift and distance between two adjacent sources, respectively, and k is the wavenumber. Equation (7) can be regarded as the phase shift of DAS for producing the tangent line toward θ. Substituting Eq. (5) in Eq. (7) and integrating it along the x-axis yields the following equation:
(8)
Note that the signs inside and outside the square bracket are not directly linked to each other. The internal sign is related to the sign in Eq. (4). The outside sign is linked to the sign in Eq. (7). Equation (8) is the spatial phase profile for generating a circle as the envelope of tangent lines. The continuous function can be discretized for a linear array of N sources as follows:
(9)
where x n represents the position of the nth source. The delay time for each source is expressed as follows:
(10)
where c 0 represents the speed of sound. The delay time is independent of the frequency. Therefore, the TLM is realizable with delay-time control. The complex volume velocity for each source is expressed as q x n = Q e j ϕ x n = Q e j ω τ x n, where Q is constant.
Figure 1(b) shows a schematic of the personal audio system considered as an illustrative example in this study. Point acoustic sources are aligned along the x-axis, as in the preceding discussion. For simplicity, the distance between two adjacent sources is d, and the source position and array length are x n = n 1 d and L = N 1 d, respectively, where n = 1 , 2 , , N. A listening point, x 0 , y 0, depicted by a star, is located at an arbitrary position on the half-circle whose center is L / 2 , 0 and radius is R 0. The directional angle is denoted as θ 0, where 0 ° θ 0 180 °. Therefore, the listening point is written as x 0 , y 0 = L / 2 + R 0 cos θ 0 , R 0 sin θ 0. Arc-shaped acoustic beams are generated by the TLM with the aim of delivering sounds toward the listening point while attenuating sounds beyond the listening point as much as possible. The arcs to be generated are determined by substituting x 0 , y 0 in Eq. (1). However, the arcs are still not determined unless a , b are assigned. Once a , b are assigned, R is determined. The question is which a , b is the best in terms of the acoustic contrast between the listening point and other places. Acoustic contrast is defined here as the ratio of the squared pressure at the listening point to the active acoustic power:4 
(11)
where q is the column vector of the complex volume velocities of each source, z is the row vector of the radiation impedances from each source to the listening point, R is the square matrix of the radiation resistances between the source positions, and superscript H denotes the conjugate transpose. zq in the numerator is the complex acoustic pressure at the listening point, and q H z H z q is the squared pressure. q H R q is the active acoustic power, which is proportional to the sum of the squared pressures in the far field. Hence, A C indicates the acoustic contrast between the listening point and far field. a , b for maximizing A C is identified the following iterative algorithm.

The iterative optimization algorithm was inspired by the literature22 in which the source placement was optimized for the ACC. In this study, the arc geometry is optimized for the TLM. The iteration for a min a < a max , b min b b max is performed, where a min, a max, b min, and b max are the computing ranges. For each a , b, R is determined using Eq. (1). The delay time required to generate the arc is obtained from Eq. (10), where the right and left half-circles are dealt with individually. The resultant A C is calculated according to Eq. (11) and compared among all cases of a , b. Finally, a , b achieving the greatest A C is identified.

The DAS theory is reviewed first followed by the ACC theory. Point acoustic sources and a listening point are arranged, similar to the TLM [Fig. 1(b)]. The basic idea of DAS is to produce a plane wave propagating toward a designated angle by aligning the wave fronts of spherical waves. The phase for each source is given by1 
(12)
The delay time for each source is written as follows:
(13)
As shown in Eq. (13), the delay time is independent of the frequency. DAS can be realized simply by delay-time control. The complex volume velocity of each source is expressed as q x n = Q e j ϕ ¯ x n = Q e j ω τ ¯ x n.
ACC is an optimization in terms of the acoustic contrast and array effort. Array effort is defined as follows:
(14)
where q ref is the complex volume velocity required to drive a single point acoustic source at L / 2 , 0 so that the SPL at the listening point can be the same as that when the array is driven by q. The ACC is formulated as a constrained quadratic optimization problem, as follows:4,6
(15)
where λ 1 and λ 2 are Lagrange multipliers, and C 1 and C 2 are constant values. q H R q is the active acoustic power. q H z H z q is the squared pressure at the listening point. q H q is the sum of squared volume velocities. The active acoustic power is minimized, while maintaining that the squared pressure at the listening point is equal to C 1 and the sum of the squared volume velocities is equal to C 2. Differentiating J with respect to q and equating it to zero yields the following equation:6 
(16)
where I denotes an identity matrix. The optimal value of q is proportional to the eigenvector associated with the smallest eigenvalue of z H z 1 R + λ 2 I. This is similar to the eigenvector associated with the largest eigenvalue of R + λ 2 I 1 z H z. The matrix inversion occasionally suffers from singularity problems. λ 2 regularizes R + λ 2 I and prevents an increase in A E, at the cost of a decrease in A C. The amplitude and phase of each source depend on the frequency. For implementation, the frequency-domain filters are converted to the time-domain filters via inverse Fourier transform. The time-domain filters are usually implemented as finite impulse response filter (FIR) filters. Because the number of the taps is limited in practice, control frequencies are discretized and there exist non-control frequencies. The time-domain performance of the ACC is degraded at non-control frequencies.20 

In this section, the optimal TLM is compared with the monopole, DAS, ACC, and non-optimal TLM. As an example, the case of d = 0.05 m, N = 21, L = N 1 d = 1 m, and R 0 = L = 1 m was investigated. Because d = 0.05 m is almost equal to a half wavelength at 3400 Hz, spatial aliasing is avoidable up to this frequency.23 The upper limit of the speech frequency band is 3400 Hz. Respectively, 3400 Hz, 1700 Hz, and 850 Hz were examined as high, middle, and low frequencies. Moreover, the listening points corresponding to θ 0 = 30 ° , 60 ° , and 90 ° were considered as examples. In this regard, however, only the results of 1700 Hz and 60 ° are presented here for avoiding repetition. The results of the other cases are provided as supplementary material.25 

In the case of the monopole, a single source at L / 2 , 0 = 0.5 m , 0 m was driven. The resultant sound pressure distribution is shown in Fig. 2. The sound pressure level is normalized by referring to the listening point and is referred to as the normalized sound pressure level (NSPL). The NSPL at the listening point is 0 dB. The contour interval of all contour plots in this paper is 10 dB. The listening point is indicated by a star. In the case of the monopole, the contour plot is identical, regardless of the frequency and θ 0. The omnidirectionality and distance attenuation of a spherical wave were observed.

Fig. 2.

SPL distributions generated by different methods, where SPL is normalized by reference to listening point at 60 ° and frequency is 1700 Hz.

Fig. 2.

SPL distributions generated by different methods, where SPL is normalized by reference to listening point at 60 ° and frequency is 1700 Hz.

Close modal

In the case of DAS, 21 sources located within 0 m x 1 m were driven according to Eq. (13). The resulting NSPL distribution in the case of 1700 Hz and 60 ° is shown in Fig. 2. Directionality was achieved. Moreover, distance attenuation in the DAS was less than that in the monopole. This is because the distance attenuation of a plane wave, which is mimicked by DAS, is less than that of a spherical wave. Owing to the poor distance attenuation, loud sounds were undesirably delivered beyond the listening point. The results of other frequencies and angles were in line with the above tendency (see supplementary material [SuppFig1.tiff]). In addition, the acoustic beams became sharper as the frequency increased. It is well known that the width of an acoustic beam produced by DAS decreases with the ratio of the array length to the acoustic wavelength.24 

In the case of ACC, the 21 sources were driven according to the eigenvector with the largest eigenvalue of R + λ 2 I 1 z H z. The regularization parameter was primarily set as λ 2 = 0 to maximize A C without considering A E. The numerical results of these indices achieved by the ACC and other methods are listed in Table 1 for the case of 1700 Hz and 60 °. A E in the monopole is 0 dB, because it is the reference. A E in the DAS was less than that in monopole. A E in the ACC was less than that in the monopole, even with λ 2 = 0. The NSPL distribution produced by the ACC is shown in Fig. 2. Directionality in the ACC was higher than that in the DAS. Distance attenuation in the ACC was steeper than that in the monopole. In other words, the sounds were successfully delivered toward the listening point and attenuated beyond the listening point. The results of other frequencies and angles were in line with this tendency (see supplementary material [SuppTable1.tiff and SuppFig2.tiff]). However, the difference between the ACC and DAS decreased as the frequency decreased in terms of A C, A E, and the NSPL distributions. At 850 Hz, the A E in the ACC with λ 2 = 0 was greater than that in the monopole. Therefore, non-zero values were applied to λ 2 at this frequency. The values were determined on a trial and error basis so that A E in the ACC could almost be equal to that in the single-frequency optimal TLM, which is discussed later. The non-zero λ 2 significantly improved A E while scarcely degrading A C. A E in the ACC with non-zero λ 2 was less than that in the monopole.

Table 1.

Numerical results of different methods, where listening point is located at 60 ° and frequency is 1700 Hz.

Method N λ2 a (m) b (m) R (m) AC (dB) AE (dB)
Monopole  NA  NA  NA  NA  18.1  0.0 
DAS  21  NA  NA  NA  NA  28.4  −8.4 
ACC  21  NA  NA  NA  33.1  −13.1 
Non-optimal TLM  21  NA  0.000  1.010  1.010  27.4  −8.2 
Single-frequency optimal TLM  21  NA  1.150  0.800  0.164  33.0  −13.1 
Frequency-averaged optimal TLM  21  NA  1.050  0.850  0.053  33.0  −13.1 
Method N λ2 a (m) b (m) R (m) AC (dB) AE (dB)
Monopole  NA  NA  NA  NA  18.1  0.0 
DAS  21  NA  NA  NA  NA  28.4  −8.4 
ACC  21  NA  NA  NA  33.1  −13.1 
Non-optimal TLM  21  NA  0.000  1.010  1.010  27.4  −8.2 
Single-frequency optimal TLM  21  NA  1.150  0.800  0.164  33.0  −13.1 
Frequency-averaged optimal TLM  21  NA  1.050  0.850  0.053  33.0  −13.1 

In the case of the non-optimal TLM, the right half-circles, whose center was 0 , R, were adopted. These were chosen as examples because such circles were investigated in an earlier work.9 The arc to be generated for 60 ° is shown in Fig. 3. R of the arc is listed in Table 1. The 21 sources were driven according to Eq. (10). The resulting NSPL distribution is shown in Fig. 2. The arc-shaped contour plot indicates that prescribed arc (Fig. 3) was generated. The directionality of the non-optimal TLM was higher than that of the monopole. The distance attenuation in the non-optimal TLM was steeper than that in DAS. Nevertheless, ACC outperformed the non-optimal TLM in terms of directivity and distance attenuation. The results of other frequencies and angles were in line with the above trend (see supplementary material [SuppTable1.tiff, SuppTLM3a.tiff, and SuppTLM3b.tiff]). Moreover, the widths of the arc-shaped acoustic beams decreased with increasing frequency. This is reasonable because an arc consists of tangent lines produced by the principles of DAS. Moreover, the TLM is based on the caustic theory, which is a high-frequency approximation.18,19 The TLM is also regarded as the stationary phase method, which works well, especially at high frequencies.20 These explanations are in line with the observed relationship between the beam width and frequency. Since the TLM works better at higher frequencies, the applicable scope is not limited to the speech frequency band.

Fig. 3.

Arc, tangent lines, and acoustic sources in different TLMs, where listening point is located at 60 °.

Fig. 3.

Arc, tangent lines, and acoustic sources in different TLMs, where listening point is located at 60 °.

Close modal

In the case of the optimal TLM, the iteration for a min a < a max , b min b b max was performed, where a min = L / 2 2.5 m, a max = L / 2 + 2.5 m, b min = 1 m, and b max = 2 m. The intervals for a and b were 0.05 m. The optimal arc for 1700 Hz and 60 ° is shown in Fig. 3. The values of a , b and R are listed in Table 1. The optimal arcs are dependent on the frequency because the optimizations were performed for each frequency. The 21 sources were driven according to Eq. (10). The resulting NSPL distribution is shown in Fig. 2. The directivity and distance attenuation were significantly improved compared to the non-optimal TLM and were comparable to those of the ACC. According to Table 1, the single-frequency optimal TLM was outperformed at some extent by ACC in terms of A C and A E, because ACC optimizes these indices. The results of other frequencies and angles were in line with the above trend (see supplementary material [SuppTable1.tiff, SuppFig4a.tiff, and SuppFig4b.tiff]). As mentioned previously, the difference between the DAS and ACC decreased with decreasing frequency in terms of A C, A E, and the NSPL distributions. In summary, the performances of the DAS, ACC, and single-frequency optimal TLM became similar as the frequency decreased. The ACC and single-frequency optimal TLM gained advantages over the DAS especially at the middle and high frequencies. As mentioned above, the principle of the TLM explains better performances at higher frequencies. Moreover, the optimal arcs for each angle depend on the frequency. For cases where R was too small to graphically show the arc, an enlarged view is presented in the figure (see supplementary material [SuppFig4b.tiff]). In the case of 90 °, R was almost zero. Such extremely small arcs can be interpreted as concentrating all tangent lines, that is, all straight acoustic beams, on the listening point. However, the relationship between the optimized circular trajectories and wavelength is not clear and will be investigated in the future work. The TLM can be realized simply by delay-time control, if—and only if—the arcs to be generated are specified, regardless of the frequency. The advantage of simplicity cannot be achieved in the single-frequency optimal TLM. Therefore, the frequency-averaged optimal TLM is considered.

The performance index in the iterative optimization algorithm is averaged for frequency, and is defined as follows:
(17)
where f m denotes the mth frequency. Here, A C was averaged over the speech frequency band from 300 3400 Hz at an interval of 10 Hz. The computing ranges, such as a min, a max, b min, and b max, were similar to those of the single-frequency optimization. The frequency-averaged optimal arc for 60 ° is shown in Fig. 3. The values of a , b and R are listed in Table 1. The 21 sources were driven according to Eq. (10). The resulting NSPL distribution is shown in Fig. 2. The directivity and distance attenuation were barely degraded from the single-frequency optimal TLM. According to Table 1, the frequency-averaged optimal TLM is comparable to the single-frequency optimal TLM in terms of A C and A E. The results of other frequencies and angles were in line with the above tendency (see supplementary material [SuppTable1.tiff, SuppFig5a.tiff, and SuppFig5b.tiff]). In summary, the frequency-averaged optimal TLM achieved a performance comparable to that of ACC and was straightforward to implement. Therefore, this method is a suitable option.

In this section, experimental results are presented to validate the theory of frequency-averaged optimal TLM. As shown in Fig. 4(a), N = 16 loudspeakers were used in the experiment. The gap between the loudspeakers was d = 0.05 m. The array length was L = N 1 d = 0.75 m. The case of R 0 = L = 0.75 m and θ 0 = 60 ° was observed. The iterative optimization algorithm was performed, where a min = L / 2 2.5 m, a max = L / 2 + 2.5 m, b min = 1 m, and b max = 2 m. The intervals for a and b were 0.05 m. A C was averaged between 300 and 3400 Hz at an interval of 10 Hz. The frequency-averaged optimal arc is shown in Fig. 4(b). The center and radius were a , b = 0.775 m , 0.650 m and R = 0.025 m, respectively. The 16 loudspeakers were driven by a digital signal processor, according to Eq. (10), where the sampling rate was 44.1 kHz. The source signal was white noise. The sound pressure radiating from the loudspeaker array was measured in an anechoic room at Kogakuin University using a sound level meter, with a sample rate of 48 kHz. The measurement points were positioned on two half-circles. Their centers were equally L / 2 , 0, and their radii were R 0 = 0.75 m and 2 R 0 = 1.5 m. The interval between the neighboring measurement points was 15 °. The 1/3 octave band analysis was performed on the measured data. The bands centered at 3150, 1600, and 800 Hz, and the overall from 315 to 3150 Hz were evaluated. In this regard, however, only the result at 1600 Hz is presented here for avoiding repetition. The results of the other frequencies are provided with supplementary material.

Fig. 4.

(a) Loudspeaker array in anechoic room, (b) arc, tangent lines, and acoustic sources in frequency-averaged optimal TLM, where listening point is located at 60 °, (c) SPL distributions generated by frequency-averaged optimal TLM, where SPL is normalized by reference to listening point at 60 ° and central frequency is 1600 Hz.

Fig. 4.

(a) Loudspeaker array in anechoic room, (b) arc, tangent lines, and acoustic sources in frequency-averaged optimal TLM, where listening point is located at 60 °, (c) SPL distributions generated by frequency-averaged optimal TLM, where SPL is normalized by reference to listening point at 60 ° and central frequency is 1600 Hz.

Close modal

The measured NSPL distribution at 1600 Hz is shown in Fig. 4(c). The blue and red circles in the figure depict the measured values at R 0 = 0.75 m and 2 R 0 = 1.5 m, respectively. The simulated results for 1600 Hz is also plotted for comparison. The blue and red lines depict the simulated values for R 0 = 0.75 m and 2 R 0 = 1.5 m, respectively. The single-frequency analysis was conducted in the simulation. The measured and simulated results agreed with each other. The directivity toward 60 ° was successfully realized. The achieved distance attenuation was equal to that of a spherical wave or more. Theoretically, the distance attenuation in the spherical wave is 6.0 dB when the distance is doubled from R 0 to 2 R 0. On the other hand, the measured distance attenuation in the TLM was 7.0 dB at 1600 Hz. The results of other frequencies were in line with this trend (see supplementary material [SuppFig6.tiff]). As expected, the gained directivity and distance attenuation decreased with decreasing frequency.

In this study, the TLM for circular trajectories with arbitrary center and radius has been formulated, where point acoustic sources are assumed to be aligned. The delay time for each source was derived analytically. Arc-shaped acoustic beams with arbitrary centers and radii can be generated by driving a linear loudspeaker array, according to the delay times. Delivering sounds to a listening point using the TLM has also been discussed. Arc-shaped acoustic beams were used to deliver sounds toward the listening point while attenuating sounds beyond the listening point. The center and radius of the arcs were optimized to maximize the acoustic contrast between the listening point and the far field. The optimization was performed on a frequency-averaged basis by using an iterative algorithm. In the presented numerical examples, the directivity and distance attenuation achieved by the frequency-averaged optimal TLM were comparable to those achieved by the ACC. The advantage of the proposed method over the ACC is the simplicity in implementation. Multiple channels of FIR filters are required for the ACC but not for the frequency-averaged optimal TLM. This is because the TLM is a delay-time control, like the traditional DAS. Determining a number of the FIR taps and evaluating performance degradations at discretized non-control frequencies are no concern of the TLM. The frequency-averaged optimal TLM was also demonstrated in an experiment. The measured data corresponded to the simulated results.

This study is related to collaborative research with CASIO COMPUTER CO., Ltd. Moreover, Kogakuin University applied for a patent related to this study.

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See supplementary material at https://doi.org/10.1121/10.0020812 for the results of various frequencies and angles.

Supplementary Material