Issue 
Acta Acust.
Volume 7, 2023



Article Number  52  
Number of page(s)  12  
Section  Audio Signal Processing and Transducers  
DOI  https://doi.org/10.1051/aacus/2023045  
Published online  27 October 2023 
Scientific Article
Theory of continuously curved and phased line sources for sound reinforcement
Institute of Electronic Music and Acoustics, University of Music and Performing Arts Graz, 8010 Graz, Austria
^{*} Corresponding author: goelles@iem.at
Received:
24
April
2023
Accepted:
6
September
2023
To supply large audience areas uniformly with amplified direct sound, largescale sound reinforcement often employs linesource loudspeaker arrays adapted to the listening area by either adjusting the angles or delays between their individual elements. This paper proposes a model for such or smaller linesource loudspeakers based on a delayed Green’s function integrated over an unknown contour. For a broad frequency range, stationary phase approximation yields a differential equation that we utilize to find a curve and delay progression providing direct sound levels rolling off with −6β dB per doubling of the distance; curve and phase designs can also be mixed to meet simultaneous targets using multiple design parameters β. The effectiveness of the formalism is proven by simulations of coverage, directivity, and discretization artifacts. Measurements on a miniature line array prototype that targets mediumscale immersive sound reinforcement applications verify the proposed theory for curvature, delay, and mixed designs.
Key words: Line source loudspeaker arrays / Sound reinforcement / Stationary phase approximation / Delay beamforming / Curvilinear arrays
© The Author(s), Published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
One of the big challenges for sound reinforcement is to provide highquality sound for the largest parts of a predefined audience area [1]. Highquality, highpower supply of large audience areas with consistently reinforced sound pressure levels has been considering linesource loudspeaker arrays driven by identical signals for several decades, nowadays. The wavefront sculpture technology (WST [2, 3]) as contemporary linesource theory is based on adjusting the length of the most relevant, binary Fresnel zone for all distances from the stage. Despite the simplicity of its model and a handful of criteria it needs, it provides a powerful and practical theory for linesource loudspeaker arrays. Furthermore, Straube et al. [4, 5], Hölter et al. [6] and Thompson et al. [7–9] discuss the modeling and optimization of the linesource array curvature to obtain tilt angles between the discrete elements, adapted to the listening area. Linesource arrays could also be applied in surround sound applications, and for instance Toole [10] of p. 330 describes the application of line sources compared to pointsource loudspeaker setups.
As an alternative way to obtain a desired sound level curve over the listening area, beamforming based on constrained least square optimization was suggested by Beuningen et al. [11]. Beamformers can be found in applications with microphone arrays [12] and with loudspeaker arrays [13–15], and they require individual delays and amplification per transducer, often also filtering.
Moreover, delay and sum beamformers, or phased arrays as they are called when used for narrowband signals, are also found in antenna theory, radar applications, or optics [16–18]. For a linear antenna array, Shanks [19] applied the stationary phase approximation to determine from the array’s radiation integral the phase and amplitude function required to form a cosecantsquared beam. Chiang et al. [20] used the same approximation method to estimate the beampatterns of inphase curvilinear arrays, and Chakraborty et al. [21] applied the approximation to a linear array, obtained and solved a firstorder differential equation, e.g., to form a cosecant beam. The cosecant beam is equivalent to a 0 dB attenuation target per doubling of the distance, when employing a vertical array below/above a horizontal plane.
Stationaryphase approximation has not only proven to be a powerful tool in antenna beamforming, but it is also used in wave field synthesis [22–24] and yields relatively simple signal processing (gains, delays, common prefilter). What is more, Schultz [25] suggested the Fourier transform and wave field synthesis as theoretical background for linesource loudspeaker arrays.
In cinema sound practice, SMPTE RP 20961 [26] defines a rectangular area of roughly a third of the radius (±1/5 times the width by ±1/6 times the depth of a hall), in which the loudspeaker sound level should be flat within ±3 dB. This confines the maximum mixing imbalance of directional sounds to 6 dB in this area and is still soft enough to accept the mixing imbalance at offcenter listening positions accomplished by two ideal point source loudspeakers in the free field, at opposite sides. Providing for a more constant coverage with distance, Nettingsmeier et al. [27] demonstrated an enlarged listening area (sweet area), when using eight short linesource arrays for thirdorder ambisonic surround playback. Gölles et al. [28] showed that curved line sources designed for a directsound coverage of 0 dB attenuation per doubling of the distance potentially improves directional localization in a large listening area, above a lengthdependent frequency limit. For less perfect coverage targets, Zotter et al. [29] found that direct sound objects rendered with −1 dB rolloff per doubling of the distance may limit the mixing imbalances acceptably (≤3 dB) within 75% of the loudspeaker layout’s radius. For a large listening area preserving the envelopment of a diffuse sound scene, by contrast, a substantially different optimal coverage criterion was found by Riedel et al. [30, 31], requiring that horizontally surrounding loudspeakers should exhibit a −3 dB attenuation per doubling of the distance for optimal results.
We propose an extended theoretical basis for the design of linesource curving and phasing. Its result is broadband for line sources that are (i) sufficiently tall, cf. equation (A.1), and (ii) densely spaced, cf. equation (A.2), or waveguided. The target is to support desired soundpressure rolloff profiles of −6β dB per doubling of distance based on either curvature or delay, or both, and potentially multiple of such profiles, simultaneously, with the design parameter β. The proposed theory employs a stationaryphaseapproximated contour integral over delayed Green’s functions and yields a secondorder nonlinear differential equation. We propose two algorithms to numerically find optimal geometry and delay curves, followed by simulation studies and measurements on a miniature line array prototype as a proof of concept.
2 Continuous curvature and phase
We evaluate the sound pressure p of a curved source fed by a progressive time delay τ given as length w = c τ by an integral of a Green’s function over the natural length parameter s of the unknown source contour C,
The imaginary unit i yields i^{2} = −1, the wave number is the speed of sound is c = 343 m/s, and f is the regarded frequency.
2.1 Integrals defining contour and delay
We define the convex curve as parametric curve depending on the inclination angle at every length coordinate s on the source, cf. Figure 1,
and the distance to a receiver at is
Figure 1 Continuous arcshaped source (solid grey) with accompanying Frenet trihedron (top) at the point showing the steering angle and inclination ϑ; see Sections 2.2–2.4 for total inclination and the stationaryphase cone for containing the direction u of the polar angle φ_{w} (bottom: side view). 
Moreover, w shall integrate the sine of a local delayandsum steering angle measured down from the normal plane of t, hence with negative sign, cf. Figure 1,
The Frenet–Serret formulas describe as tangential vector , and as normal vector n scaled by curvature , in our case and a biorthogonal vector , so that t = n = b = 1 and t ⊥ n ⊥ b, cf. equation (2) and Figure 1.
2.2 Stationaryphase approximation
At high frequencies, the integrand of equation (1) oscillates rapidly. Therefore stationary phase approximation is appropriate, which evaluates the integrand’s stationaryphase points ; with derivatives abbreviated as , . In typical designs, there should only be a single stationaryphase point of minimum delay to the receiver and therefore the pressure is approximated by,
2.3 Target coverage
Our goal is to find the curvature and delay length w at each position s on the source such that the sound pressure
decays by −6β dB per doubling of distance on the listening plane that lies at z_{r} = 0. For an equalized magnitude square, we allow a gain g and desire
2.4 Stationaryphase direction
We derive with regard to s to describe stationaryphase directions pointing from the corresponding point on the array emitting the first wave front received to the receiver on the listening plane at z_{r} = 0,
We use , introduce a unitlength vector to stationaryphase receivers requiring , so , and these stationaryphase receivers lie on a cone of local stationaryphase directions, cf. Figure 1,
Below, we design the xaxis sound pressure contour at y_{r} = 0 for receivers at z_{r} = 0, simplifying the direction u with φ_{w} = 0 to
where the total inclination is gathered as .
2.5 Stationaryphase magnitude
We derive once more from equation (8) to describe ,
and with , , , t = 1, , , , as well as , we get a part of in equation (7)
and hereby the equalized magnitude squared
We assume receivers at z = 0, so retrieves r by the contributions of t, n to z in equation (9)
2.6 Optimum total curving
To design the sound pressure response at y_{r} = 0, we choose φ_{w} = 0, yielding with
For the current positions z, total angle , distance r from equation (15), the required angular curvature complies with the constraint equation (7) when it becomes
Its numerical integration using a phase/geometry split b, gain g^{2}, and start inclination defines Algorithm 1, where S denotes the source length and N is the number of discrete elements. The total inclination is split into geometry and delay parts by the factors a, b, linearly, and a constant inclination offset is additionally added to adjust the inclination for a straight linesource with phasing only. Geometry and phase inclination parts become
2.7 Top inclination, gain, curving split
The aiming to the audience area typically lies within and reduces from a most horizontal aiming thinkable to a most upwards aiming 0 thinkable. For this reason, the total inclination decreases and clarifies the negative sign of . We recognize in equation (15) that yields the loudest sound pressure
Algorithm 1. Curving of geometry and/or phase
procedure CURVING(x_{r,0}, z_{0}, b, β, S, N, [g^{2}, ])
constant:
Δs = −S/N
= f(x_{r,0}, z_{0}) ▷ according to eq. (19)
a = 1 − b ▷ to ensure a + b = 1
▷ cf. Figure 2
Figure 2 Uniform starting conditions for both curved and phased sources: top end at x_{0} = 0 at s = 0, inclined by , supplying the most distant listening position at x_{r,0}, with the starting value . The lower end should lie at a length s = −S and supply the closest listeners at x_{r,S}. To start at the top point, it is useful to work with a reversed integration direction −ds. 
▷ default inclination offset
▷ default eq. (18)
initialization:
w = zeros(1,N)
x = zeros(1,N)
z = zeros(1,N)
r = r_{0}, , z[1] = z_{0}
for n = 1...N − 1 do
▷ cf. eq. (17)
▷ cf. eq. (17)
x[n + 1] ← x[n] + Δs ▷ num.int. (2)
z[n + 1] ← z[n] + Δs ▷ num.int. (2)
w[n + 1] ← w[n] − Δs ▷ num.int. (4)
▷ according to eq. (15)
▷ according to eq. (16)
▷ numerical integration
end for
return w, x, z
end procedure
We need to ensure that the curvature stays negative yielding a convex contour and a single stationaryphase point for each observing point as implied in equation (5). Moreover, a geometrically concave solution with would be impractical for linesource arrays whose splay angles are nonnegative. Choosing as a consequence both and at the top of the source, the beamsteering curvature must also vanish to preserve .
This total inclination at the top of the source at x = x_{0} = 0 and z = z_{0} should be adjusted to ensure the top is the minimumdistance point supplying the remotest position x_{r,0} in the audience as in Figure 2,
The top inclination is suggested to be purely geometrical per default, so that its beam steering is neutral and broadside for any choice of a, b.
The simplest choices for a, and b are: (i) a = 1, b = 0 for a line source without phasing but with curving , or (ii) a = 0, b = 1 for a straightline source without curving but with phasing . We will also demonstrate the usefulness of mixtures in a later section, targeting the case in which multiple criteria for β should be fulfilled. For instance, the control of the curved array with equal signals fulfills the criterion β_{1} = 0, and can be accomplished by additional delays.
3 Simulation studies
In this section, the differential equation will be solved numerically as outlined in Algorithm 1. It applies stepwise updates to based on from equation (16). The algorithm’s step size Δs should be appropriately small to solve the problem accurately, by choosing a large N. In default of the optional argument g, Algorithm 1 uses equation (18) to define g^{2}.
The simulations below show the result of summing individual point sources described by Green’s function positioned along the continuous source contour at 1 mm intervals. This summation is performed for frequencies ranging from 20 Hz to 20 kHz with a linear resolution of 248 points, to which and a thirdoctave band averaging was applied. To show broadband sound pressure curves over the listening area, results were Aweighted [32] and summarized. In addition to the Aweighted onaxis sound pressure curves, a level map of a phased line source with β = 0 demonstrates the coverage including offaxis listening positions, below. Furthermore the directivity factor is discussed, calculated as result of theoretically simplified considerations, and simulated by (delayed) point sources positioned along the desired contour.
3.1 Curved line source
For curved line sources without phasing (), a = 1 and b = 0, the design equation (15) simplifies to
and the differential equation (16) to
3.1.1 Comparison to literature
In a recent paper [28], we presented a secondorder differential equation for β = 0, yielding an optimally curved arc source. While its result is perfectly equivalent for β = 0, the main difference lies in the formulation of the integration with regard to the x axis instead of the natural parameter s, which complicates generalization to β > 0 or comparison to literature. Equation (20) is shown above as it matches the classic equation obtained by Urban et al. [3], which states after rewriting the variables STEP = Δs, , d = r, with ,
Remarkably, equation (22) resulting from a binary Fresnelzonebased approach only differs from stationaryphase approximation equation (20) by the missing limit Δs → 0.
A later publication [5] and variablecurvature linesource tutorials [33] chose to show a simplified relation
which is instructive and intuitively accessible. By letting and replacing , it implies
and hereby yields the expression
The equation demands for cabinets pointing at large distances that the splay angle between them must be small so that more cabinets radiate into the same direction. The exponent β defines the desired distance decay. Despite its easier accessibility, however, equation (25) is inaccurate compared to equation (22) whenever β → 0 gets small. While for β = 0.5 and small Δs, it makes no substantial difference which formula is used for calculation, because either or approximate , only with a different overall gain.
Without simplification, the equation (22) as defined in [3] yields upon targeting an approximation
of equation (21), as long as STEP = Δs stays sufficiently small. In particular, the original paper used the cabinet height for STEP and the splay angles for , i.e. a fixed geometric discretization determined by the hardware elements, as opposed to keeping it a free parameter adjusted for fineenough discretization Δs → 0, when numerically solving a differential equation.
3.1.2 Curved line source: results
The differential equation of equation (21) is solved numerically by the proposed Algorithm 1 with uniform parameters x_{r,0} = 10 m, z_{0} = 2.072 m, b = 0 and source length S = 1.312 m for all decays, giving the number of discrete elements N = 1313 for a step size Δs = 1 mm. Equation (19) specifies the total inclination on top of the source and for β = 0, the gain parameter is calculated by equation (18) yielding a value of . For the other decay values, g was set manually to g = 0.443 for β = 0.25 and g = 0.599 for β = 0.5, in order to supply the same listening positions 0 ≤ x_{r} ≤ 10 m with unchanged source length S.
Figure 3 shows the source contour for different decays and Figure 4 the resulting Aweighted sound pressure profiles with dB offsets to make the β = {0, 0.25, 0.5} curves to go through {0, −3.5, −7} dB at the distance of x_{r} = 4 m. The levels do not follow the theoretical rolloff curves −6β dB per doubling of the distance beyond x_{r} > 7 m as stationary phase approximation assumes an infinite line integral although the actual length is finite.
Figure 3 Curved line source contours for different decays β. 
Figure 4 Aweighted sound pressure levels of curved line sources for different decays β with mixed xaxis scaling, linear for x_{r} ≤ 1 and logarithmic for x_{r} > 1, with the target range 0 ≤ x_{r} ≤ 10 m. 
Another parameter to be considered for radiation characteristics is the directivity factor Q. Figure 5 shows the directivity index DI = 10 lg Q for both cases, curved and phased line source with different decays β. The radiation of the source was calculated at 10 m distance as summation of individual point sources with Δs = 1 mm. For low frequencies, omnidirectional radiation is seen, which coincides with theoretical considerations for frequencies below 130 Hz (). At frequencies above 500 Hz, the curves for different decay values differ, with small decay values leading to higher directivity. With Sections A.3 and A.4 we find
The results are shown as dotted lines in Figure 5 and show the same trends as the simulated solid lines, where a = 7 corresponds to the ratio of distances, within which the level curves fulfill the decay profile, cf. Figure 4. The horizontal extent Δx is 9.83 cm for β = 0, it is 19.82 cm for β = 0.25, and 32.14 cm for β = 0.5.
Figure 5 Directivity index for curved (thin) and phased (bold) line sources with different decays β compared to equation (27) (dotted). 
Figure 6 Discretization of a continuously curved line source (A) according to the β = 0 example from Figure 3 to: a polygon of straightline sources with splay angles rounded to integer degrees (B) and gaps between these straightline segments (C); graphs are rotated by and limited to z < 1.15 m for easy readability. 
Professional linesource array systems typically contain multiway transducers, of which the larger layout of the low and midfrequency transducers is horizontally directional, instead of radiating axisymmetrically. Similarly, their vertical high frequency linesource transducers, e.g. f > 1 kHz, are typically equipped with horizontal waveguides or wedgeshaped horns that limit horizontal radiation to 110°, so roughly a third of the 360° panorama, or to 70° which is roughly its fifth. The modeled DI can be adapted correspondingly with an increase by +5 dB for 110° or +7 dB for 70° horizontal coverage.
3.1.3 Discretization of source and splay angles
Later we will use a prototype of a line array to show the effectiveness of the formalism in practice. Since is solved with the same parameters as in Section 3.1.2 and the resulting continuous source is then composed by point sources that are positioned along the contour, cf. Figure 7A. To simulate a practical device built from discrete elements, the contour was discretized into a polygon of 8.2 cm straightline segments in Figure 7B. The splay angles are chosen so that the error made in the total inclination caused by using integer ° values for the splay angles is kept minimal. The same procedure is executed for Figure 7C but with shorter 6.2 cm straightline segments with the same splay angles but leaving gaps in between. Due to discretization with integerdegrees splay angles, the lost control of small curvatures yields a boost of the sound pressure level between 7 m and 8 m for β = 0 in our example, cf. Figure 7. Gaps between the straightline source elements weaken the attenuation of spatial aliasing. It occurs at frequencies above where the interelement spacing exceed a wavelength, which causes a noticeable downwards leakage of sound to listeners at x_{r} < 3 m with broadside steering .
Figure 7 Aweighted sound pressure level for different decays β and different discretization with mixed xaxis scaling, linear for x_{r} ≤ 1 and logarithmic for x_{r} > 1, as above; continuous line source (dotted), discrete point sources with 8.2 cm (dashed), straight 6.2 cm linesources at the same spacing with inclination increments rounded to integer ° values (solid). 
Figure 8 Simulated A weighted sound pressure map of a phased line source for β = 0 over a listening area of 10 m × 10 m. 
Figure 9 Aweighted sound pressure levels of a phased source with different decays β for a target range 0 ≤ x_{r} ≤ 10 m, with a continuous source Figure 6A or discrete segments Figure 6C of integer sample delays (48 kHz sample rate); the xaxis is scaled linearly for x_{r} ≤ 1 and logarithmically for x_{r} > 1. 
3.2 Phased straightline source
With a = 0, we get the differential equation for the beamforming angle of a straightline source from equation (16),
3.2.1 Phased straightline source: results
The differential equation (28) is solved numerically by Algorithm 1 with uniform parameters x_{r,0} = 10 m, z_{0} = 2.117 m, b = 1, and source length S = 1.312 m for all decay values, resulting in the number of discrete elements N = 1313 for a step size Δs = 1 mm. Equation (19) defines the inclination on top of the source ° and for β = 0, the gain parameter is calculated by equation (18) yielding a value of . For the other decay values, g was set manually, g = 0.449 for β = 0.25 and g = 0.614 for β = 0.5. Figure 10 shows the delay lengths for different decay values, and Figure 9 describes the resulting Aweighted sound pressure curves for a continuously phased source (dotted lines) compared to direct sound pressures curves of a discrete phased source with enclosure height 8.2 cm, gaps in between and rounded delays being accurate to 1 sample at f_{s} = 48 kHz. The SPL plots for the continuous source exhibit the same trend as those for the purely curved source. For very close listening positions, the curves differ due to discretization and spatial aliasing. The delays also cause a comb filter, which is greatest at β = 0 due to the temporal structure of the summed impulse response and has the opposite effect to spatial aliasing. The Aweighted sound pressure for β = 0 including offaxis listening positions can be found as a level map in Figure 8 over a 10 m × 10 m listening area. The coverage for which level changes stays below 1 dB reaches a limit of φ_{w} = 27.5° for closer observation points. For farther observation points (5 m < x_{r} < 7.8 m) the coverage gets wider up to maximum of φ_{w} = 35° for r = 8.5 m.
Figure 10 Delay length w for different decays β. 
3.2.2 Comparison to literature
Electronically steered line arrays for sound reinforcement can be found in literature, however not fixed to using delays, only. Meyer [14, 34] employs individual filters to control each array loudspeaker and proposes to mount the line arrays along the ceiling. Different designs are used with regard to the arrays’ throw distance, shortthrow vs. longthrow array, and to keep the direct sound coverage of the audience flat. Van der Werff [15] discusses wallmounted line arrays with nonuniformly spaced transducers, as nested array, with individual filtering with IIR lowpass filter designs. The goal of his work is to increase the directtoreverberant ratio in large halls by achieving a flat direct sound pressure level profile of 0 dB per distance doubling, and the design was verified in EASE 1.2. And finally, Duran Audio’s Digital Directivity Synthesis is a wellknown DSPassisted method based on leastsquares for uniformly spaced line arrays [11]. Its optimization of the desired direct sound pressure level profiles takes transducer directivity into account. The method requires the implementation of FIR filters as signal processing done for each loudspeaker.
Algorithm 2. Twotarget curving of geometry and phase
procedure TWOBETA(x_{r,0}, z_{0}, β_{1}, β_{2}, S, N, , )
constant:
Δs = −S/N
= f(x_{r,0}, z_{0}) ▷ according to eq. (19)
▷ cf. Figure 2
initialization:
w = zeros(1,N)
x = zeros(1,N)
z = zeros(1,N)
r_{1} = r_{2} = r_{0}, , z[1] = z_{0}
for n = 1...N − 1 do
w[n + 1] ← w[n] − Δs ▷ num.int. (4)
x[n + 1] ← x[n] + Δs ▷ num.int. (2)
z[n + 1] ← z[n] + Δs ▷ num.int. (2)
▷ cf. eq. (15)
▷ according to eq. (21)
▷ numerical integration
▷ acc. to eq. (29)
▷ numerical integration
end for
return w, x, z
end procedure
3.3 Curved and phased line source
To support two simultaneous coverage designs (preserving mixingbalance and envelopment) for a large audience area, phasing delays are used in combination with geometrical curving. In addition to a decay β_{1} accomplished by driving the cabinets of a curved array with identical signals, delays between individually driven cabinets can be used to satisfy an alternative decay β_{2}.
First, equation (21) is solved for a purely curved source (b = 0) with β_{1} as shown in Algorithm 1, and afterwards the required delay lengths for another choice of β_{2} is calculated preserving the geometrical curvature ,
which is summarized in Algorithm 2.
As this design and algorithm will be analyzed in measurements using prototypical hardware in the next section, graphical analysis of the results will be shown below.
4 Experimental setup
To show the effectiveness of the formalism in practice, the reproducible miniature line array presented in [35] was used, cf. Figure 11. The trapezoidal shape was chosen so that a maximum tilt angle of 10 degrees can be set. SB Acoustics SB65WBAC254 were used as the 2.5inch fullrange speakers.
Figure 11 Miniature Line Array of eight 3D printed enclosures (closed 0.4 l boxes). 
For the measurements, 12 enclosures are lined up for which we obtain the continuous source contour and delay lengths from Algorithm 1. The measurements took place in an auditorium (30 m × 9 m × 3.5 m) with a reverberation time of T_{60} ≈ 0.82 s. To focus on verifying the direct sound design, the impulse responses were truncated to the first 300 samples (6.25 ms at f_{s} = 48 kHz). The gain values g for β = 0 are calculated by equation (18), for the other decay values g was set manually, g = 0.46 for β = 0.25 and g = 0.65 for β = 0.5. The solution is then discretized using a step size of 8.2 cm which corresponds to the distance between two neighbouring drivers. In order to record the positiondependent direct sound pressure level curves, impulse responses were measured along 20 positions (onaxis) starting at x_{r} = 1 m and ending at x_{r} = 10.5 m. Pressure zone microphones were positioned on the ground to avoid floor reflections in the measurements. We applied equalization to obtain a flat frequency response between 200 Hz and 20 kHz for the single array element. Moreover, a filter with magnitude increasing by until the spatial aliasing frequency was employed to equalize the otherwise pink sound pressure of a linear arrangement of ideal point sources, cf. equation (7). The results are averaged in third octaves and the curves are plotted as Aweighted sum.
4.1 Curved line source
Figure 12 shows the A weighted sum over distance (onaxis) for different decays β compared to the simulated curves with the same discretization as in Figure 6C. The measured results show almost identical curves to the simulation and the artefacts caused by discretization of source and splay angles are observed.
Figure 12 Aweighted measured sound pressure of curved arrays (solid) with different decays over distance compared to the simulated discretized source (dashed) with rounded splay angles keeping the error made in total inclination minimal. 
Figure 13 Measured A weighted sound pressure map of a phased array with β = 0 compared to simulated contours in steps of 1 dB (dashed). 
4.2 Phased line source
Figure 14 shows the Aweighted onaxis sound pressure curves in comparison to the simulated for different decays β. Also here, the measured results follow almost the theoretical simulated based on point source summation. The noticeable differences are close to the source where the beamforming no longer reaches closer listening positions and spatial aliasing affects the results. Figure 13 presents the measured coverage (solid) including offaxis listening positions of a phased array with β = 0 compared to the simulated (dashed). We observe a lower sound pressure for lateral listening positions because the linearly arranged point sources in the simulation do not take the horizontal directivity of the drivers into account.
Figure 14 Aweighted measured sound pressure of phased arrays (solid) with different decays over distance compared to the simulated discretized source (dashed) with rounded delays at f_{s} = 48 kHz. 
4.3 Mixed curve and phase design
For the mixed design, we take the curved line source from Section 4.1 with β_{1} = 0 and use the corresponding time of flight to satisfy β_{2} = 0.5 according to Algorithm 2. The rounded delays at f_{s} = 48 kHz were inserted in the offline analysis. Figure 15 shows the Aweighted sound pressure profile of a purely curved source with β = 0 and compares the Aweighted profile of a curved line source with β = 0.5 with the curve of the mixed design.
Figure 15 Aweighted measured sound pressure of curved arrays (solid) with decays β = 0 and β = 0.5 and mixed array with β_{1} = 0 and β_{2} = 0.5 over distance compared to simulated discretized sources (dashed). 
The profile for a purely curved source and the one of the mixed design differ in the vicinity of the source only, because the impulse responses of the enclosures differ in time structure in this area, yielding a different sum. Nevertheless, both follow the simulated −3 dB per doubling of distance.
5 Conclusion
This paper presented design equations for continuously curved and phased line sources with the target to accomplish desired sound pressure rolloffs. The stationary phase approximation was applied to the contour integral of a Green’s function with delay (phasing) to obtain a secondorder nonlinear differential equation for curving and phasing. The discretized results are directly applicable to design curvature and phasing of discrete line source arrays. Simulations of the continuous source based on summation of point sources positioned along the line source contour showed the ideal −6β dB profiles per doubling of the distance. Furthermore, the simulations revealed the effects caused by the discretization of the source. The theoretical results were underlined by measurements of a line array prototype. Moreover, the results showed that a twotarget design based on curving and phasing is feasible.
Future work should consider psychoacoustic evaluations of line sources with different β profiles for mediumsized immersive sound reinforcement for 50–250 listeners to assess improvements in envelopment and directsound mix. Experiments should also take frequencyresponse homogeneity into account, which is often optimized as a tradeoff with the coverage parameter β, see also [33].
Acknowledgments
Our research was funded by the Austrian Science Fund (FWF): P 35254N, Envelopment in Immersive Sound Reinforcement (EnImSo). The authors thank Thomas Musil for his design improvements on our miniature line array.
Conflict of interests
The authors declared no conflicts of interests.
Data Availability Statement
The research data associated with this article are included in the supplementary material of this article [36].
Appendix
A.1 Line array near field
When observing a line source of the length S at the normal distance r to one of its ends, the distance to its other end is . A distance difference to both ends that may exceed a phase shift towards r → 0 defines the acoustic near field, within which the distance decay can differ from , with and ,
Example: The near field extends to 10 m for a source of the length 1 m at a frequency of f = 5c = 1.7 kHz.
A.2 Line array aliasing
A line array composed of discrete elements of the height and spacing h driven in phase radiates waves focused to directions perpendicular to the line. It exhibits uncontrolled radiation due to spatial aliasing at and above the frequency at which also waves propagating along the line fit the inphase control,
Example: The frequency should stay below 3 kHz for to avoid spatial aliasing.
A.3 Directivity index of a line source
For very low frequencies f → 0, any phased/curved line source has vanishing extent in terms of wavelengths and the directivity index is zero DI → 0 dB.
Any phased/curved line source exhibits a length S as its larger geometric extent. Its directivity index first rises when the length S exceeds half a wavelength
Around this frequency, the directivity function is well approximated by a line source of the length S, with . The directivity factor of an axisymmetric radiation is 2 divided by the squared directivity pattern integrated along the inclination d sin ϑ,
and at low frequencies we approximate the sine integral Si(kS) by kS and cosine cos(kS) by 1, yielding Q ≈ 1, and the asymptotic value of Si(kS) is at high frequencies, where we may neglect cos(kS) + 1 and get , or in total
After the linear increase of the directivity factor above , a slight saturation will be reached when the second extent of the curved/phased line source or Δx = w_{S} exceeds half a wavelength
which is conveniently denoted as frequency f_{1} scaled by the approximate aspect ratio of the source.
A.4 Directivity index of a directional point source with predefined coverage
To estimate the directivity index, we consider the farfield sound pressure of a directional point source,
mounted at the height z_{0} above the audience and exhibiting a roughly axisymmetric directivity function g around the vertical axis. Listeners at ear height are reached if the distance r relates to the variable inclination angle by
and a correspondingly reformulated directivity factor depending on this r can be tailored so that listeners receive
and hereby a wellcontrolled directsound level decaying with −6β dB per doubling of the distance. To avoid radiation to angles outside , the directivity function must vanish elsewhere, accordingly, and we may substitute and choose r_{s} = z_{0} for simplicity,
The characterization of a directivity pattern relies on observation at a directionindependent distance, e.g. r_{0}, but we may keep r as unrelated variable defining the directivity function g by the angledependent distance to the listeners,
With the directivity pattern integrated along the inclination the directivity factor of the axisymmetric radiation is
For curved/phased linesource arrays, this rough asymptotic approximation can be assumed to hold for high frequencies f > f_{2}, where the directivity function vanishes outside the angular range .
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All Figures
Figure 1 Continuous arcshaped source (solid grey) with accompanying Frenet trihedron (top) at the point showing the steering angle and inclination ϑ; see Sections 2.2–2.4 for total inclination and the stationaryphase cone for containing the direction u of the polar angle φ_{w} (bottom: side view). 

In the text 
Figure 2 Uniform starting conditions for both curved and phased sources: top end at x_{0} = 0 at s = 0, inclined by , supplying the most distant listening position at x_{r,0}, with the starting value . The lower end should lie at a length s = −S and supply the closest listeners at x_{r,S}. To start at the top point, it is useful to work with a reversed integration direction −ds. 

In the text 
Figure 3 Curved line source contours for different decays β. 

In the text 
Figure 4 Aweighted sound pressure levels of curved line sources for different decays β with mixed xaxis scaling, linear for x_{r} ≤ 1 and logarithmic for x_{r} > 1, with the target range 0 ≤ x_{r} ≤ 10 m. 

In the text 
Figure 5 Directivity index for curved (thin) and phased (bold) line sources with different decays β compared to equation (27) (dotted). 

In the text 
Figure 6 Discretization of a continuously curved line source (A) according to the β = 0 example from Figure 3 to: a polygon of straightline sources with splay angles rounded to integer degrees (B) and gaps between these straightline segments (C); graphs are rotated by and limited to z < 1.15 m for easy readability. 

In the text 
Figure 7 Aweighted sound pressure level for different decays β and different discretization with mixed xaxis scaling, linear for x_{r} ≤ 1 and logarithmic for x_{r} > 1, as above; continuous line source (dotted), discrete point sources with 8.2 cm (dashed), straight 6.2 cm linesources at the same spacing with inclination increments rounded to integer ° values (solid). 

In the text 
Figure 8 Simulated A weighted sound pressure map of a phased line source for β = 0 over a listening area of 10 m × 10 m. 

In the text 
Figure 9 Aweighted sound pressure levels of a phased source with different decays β for a target range 0 ≤ x_{r} ≤ 10 m, with a continuous source Figure 6A or discrete segments Figure 6C of integer sample delays (48 kHz sample rate); the xaxis is scaled linearly for x_{r} ≤ 1 and logarithmically for x_{r} > 1. 

In the text 
Figure 10 Delay length w for different decays β. 

In the text 
Figure 11 Miniature Line Array of eight 3D printed enclosures (closed 0.4 l boxes). 

In the text 
Figure 12 Aweighted measured sound pressure of curved arrays (solid) with different decays over distance compared to the simulated discretized source (dashed) with rounded splay angles keeping the error made in total inclination minimal. 

In the text 
Figure 13 Measured A weighted sound pressure map of a phased array with β = 0 compared to simulated contours in steps of 1 dB (dashed). 

In the text 
Figure 14 Aweighted measured sound pressure of phased arrays (solid) with different decays over distance compared to the simulated discretized source (dashed) with rounded delays at f_{s} = 48 kHz. 

In the text 
Figure 15 Aweighted measured sound pressure of curved arrays (solid) with decays β = 0 and β = 0.5 and mixed array with β_{1} = 0 and β_{2} = 0.5 over distance compared to simulated discretized sources (dashed). 

In the text 
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