# Diffuse Sound Field Synthesis

Franz Zotter, Stefan Riedel, Lukas Gölles, Matthias Frank
Institute of Electronic Music and Acoustics
University of Music and Performing Arts Graz, Austria

###### ?abstractname?

Can uncorrelated surrounding sound sources be used to generate extended
diffuse sound fields? By definition, targets are a constant sound pressure
level, a vanishing average sound intensity, uncorrelated sound waves arriving
isotropically from all directions. Does this require specific sources and
geometries for surrounding 2D and 3D source layouts?
As methods, we employ numeric simulations and undertake a series of
calculations with uncorrelated circular/spherical source layouts, or such with
infinite excess dimensions, and we point out relations to potential theory.
Using a radial decay $\nicefrac{{1}}{{r^{\beta}}}$ modified by the exponent
$\beta$, the representation of the resulting fields with hypergeometric
functions, Gegenbauer polynomials, and circular as well as spherical harmonics
yields fruitful insights.
In circular layouts, waves decaying by the exponent
$\beta=\nicefrac{{1}}{{2}}$ synthesize ideally extended, diffuse sound fields;
spherical layouts do so with $\beta=1$. None of the layouts synthesizes a
perfectly constant expected sound pressure level but its flatness is
acceptable. Spherical $t$-designs describe optimal source layouts with well-
described area of high diffuseness, and non-spherical, convex layouts can be
improved by restoring isotropy or by mode matching for a maximally diffuse
synthesis. Theory and simulation offer a basis for loudspeaker-based synthesis
of diffuse sound fields and contribute physical reasons to recent
psychoacoustic findings in spatial audio.

##  1 Introduction

The concept of a diffuse sound field refers to uncorrelated unit-variance
plane waves impinging from all directions. In an ‘ideal’, cylindrically or
spherically isotropic diffuse field, the sound pressure level is position-
independent and the time-averaged sound intensity vector vanishes at all
positions [1]. Measurement of sound field diffuseness has an extensive history
in acoustics, for instance based on the well-defined frequency-dependent
correlation between two points in space [2, 3, 4, 5]. In addition, the
behaviour of the sound intensity vector can be used to quantify the
diffuseness of the sound field [6, 7]. In particular, sound intensity
normalized by the energy density and speed of sound were defined as energy
vector [8, 9] when the diffuse sound field is synthesized by incoherently
driven loudspeakers, as it simplifies to the gain-square-weighted average of
the loudspeaker directions. The resulting vector is of unit length for fully
directional sound fields and zero length for fully diffuse sound fields. More
recent approaches employed spherical microphone arrays to measure the
diffuseness of the sound field, e.g. COMEDIE, which is based on an eigenvalue
decomposition of the spherical harmonics (SH) covariance matrix [10]. A
directional and temporal histogram of the energy decay in a simulated room was
proposed in auralization [11, 12]. Several recent works moreover showed ways
to retrieve or represent anisotropic reverberation properties based on
measurements [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

Synthesis of diffuse sound fields with loudspeaker arrays is of interest in
laboratory environments, for example to quantify the diffuse-field sound
transmission of building partitions [24] or acoustic absorption [25, 26], the
diffuse-field response of microphone arrays [27, 28, 29] and dummy heads [30,
31, 32, 33], or to measure the performance and perceptual quality of active
noise control/cancellation algorithms [34, 35, 36], etc.

Spatial sound reproduction targets a faithful reproduction of diffuse sound
fields that perceptually elicits a sensation of envelopment for the audience,
by suitably employing surrounding loudspeakers [37, 38, 39, 40, 41].

Theoretically, sound field synthesis techniques such as wave field synthesis
(WFS) [42, 43, 44, 45, 46, 47, 48, 49] or near-field compensated higher-order
Ambisonics (NFC-HOA) [50, 51, 52, 53, 46, 49] should allow to reproduce an
ideally diffuse sound field, given a continuous distribution of elementary
sources. In discrete-direction implementations, however, spatial aliasing
severely affect the sound field synthesis capacities of both WFS and HOA in
much of the auditory range. Nevertheless, it could be argued that the
fruitfulness of these approaches lies in much more tolerant psychoacoustic
effects of auditory localization [54], so that playback qualities allow for a
much larger sweet area, as analyzed in experiments and perceptual measures
[55, 56, 57, 58, 59].

In WFS, it was studied for diffuse-field synthesis how many uncorrelated
plane-wave sources are needed for a centered listener [60, 61]. In Ambisonics,
the required directional resolution or Ambisonic order for a large sweet area
was investigated [62], and a directionally non-uniform reverberation algorithm
was suggested [63].

The minimum number of loudspeakers and horizontal arrangements for reproducing
the spatial impression of diffuse sound fields in channel-based playback was
investigated in terms of interaural cross correlation [64] and listening
experiments [38] with uncorrelated signals.

Interaural coherence and interaural level differences were moreover considered
to evaluate the quality of diffuse-field reproduction capabilities [37, 40].

Adding height loudspeaker layers slightly increases perceived diffuseness,
when diffuseness is defined as ‘ _the sound coming from all directions with
equal intensity. Therefore, the sound should ideally be impossible to localise
and without any gaps […] in three dimensions_ ’ as shown in [39, 65]. Other
researchers showed that two distinct spatial attributes related to the
perception of diffuse sound fields can be separated, namely ‘ _envelopment_ ’
for being surrounded by sound and ‘ _engulfment_ ’ for being covered by sound
[41, 66]. _Isotropy_ as technical measure of directional uniformity was
recently discussed [15], and spherical $t$-designs were shown to be optimal
plane-wave layouts to synthesize isotropic sound fields [67]
Reverberant spatial audio objects and diffuse-field modeling were recently
proposed [19, 20] to model the typical/anisotropic characteristics of acoustic
rooms. Bandpass-based decorrelation for Diffuse Field Modeling that improves
diffuseness in measured $1^{\mathrm{st}}$-order Ambisonic room impulse
responses was proposed. By manipulation of the directional weights for front,
left, and right, the perception of envelopment was shown to be affected to
some degree by direction dependent diffuse fields [19]. Further investigating
the perceivability of anisotropy in reverberation, i.e., by recognizing a
rotation of the scene, revealed that an time average of the interaural energy
decay difference at 850 Hz of slightly more than
$1\text{\,}\mathrm{d}\mathrm{B}$ can already be heard [18]. Simplified
auralization of non-uniform room reverberation was moreover proposed and
tested [68], using a warped, sparse layout of virtual reverberation sources.

In loudspeaker-based reproduction [69], channel-based, scene-based
(Ambisonics), and object-based amplitude panning concepts mostly employ point-
source loudspeakers for playback, yielding a $1/r$ sound pressure decay with
distance and parallax with regard to the listening position. Object-based WFS
loudspeaker systems are typically 2.5D and based on dense horizontal point-
source loudspeaker arrays. When rendering plane-wave sound objects, they yield
a $1/\sqrt{r}$ sound pressure decay with distance, depending on the listening
position, cf. [46, 49]. Surprisingly, all typical loudspeaker-based
reproduction still exhibits pronounced shortcomings when used to synthesize
extended perceptually diffuse, and therefore consistently non-directional,
sound fields:

Object-based rendering using amplitude panning, channel-based rendering, or
scene-based Ambisonic rendering on moderateley dense loudspeaker layouts can
be interpreted to exhibit a notoriously small sweet area regarding their
capacity of rendering uniformly diffuse listener envelopment. Recent studies
revealed that instead of the conventional point-source loudspeakers, vertical
line-source loudspeakers could supply a larger sweet area with a perceptually
diffuse sound field [70, 71, 72, 73]. With conventional point-source
loudspeakers, the directional impression would often be dominated by the
loudspeakers closest to the off-center listener location, even if the off-
center displacement is limited to, say, a third of the layout radius.

A lot of research drive behind WFS was fueled by the idea that its many
loudspeakers and well-controlled time delays avoid a confined sweet area by
shaping consistent and extended wave fronts [54]. Plane waves were thought of
as an ideal virtual source type to consistently supply a large listening area.
And yet, experiments using 2.5D WFS showed a clear benefit in diffuse-field
rendering when the 2.5D reference line correcting the synthesized plane-wave
levels was specific to a certain listening position [74]. So also centered
2.5D wave-field synthesis appears limited in rendering a perceptually diffuse
sound field to off-center listeners. WFS theory suggests [43, 44, 46, 48, 75]
that vertical line sources loudspeakers are the optimal sound sources for 2D
WFS, of which the generated sound field is constant along the vertical
dimension. As a consequence, 2D WFS does not require any correction and
therefore no reference line, in contrast to 2.5D WFS. Nevertheless, the
associated effort in light of the already horizontally dense loudspeaker
arrangements appears enormous for 2D WFS compared to 2.5D WFS; this effort
might only be realistic with local WFS systems [48] that consider a reduced
number of loudspeakers while accepting a limited audience area.

(a) circle of point sources

(b) circle of vert. line sources

(c) circle of horiz. line sources

(d) sphere of point sources

?figurename? 1: Horizontal map of sound energy density (contours), normalized
intensity $\bm{I}/(c\,w)$ (arrows), and diffuseness $\psi$ (colors) with 100
regularly spaced sources on the azimuth for a circle (salmon), or for a sphere
with 480 sources at Chebyshev-type nodes from Gräf’s website [76], all playing
statistically independent signals of uniform variance.

If vertical line-source loudspeakers also enlarge the diffuse-field rendering
capacities in WFS, they appear to be similarly beneficial regardless of the
particular kind of 2D audio rendering method. However, considering that only
WFS synthesizes consistent parallel first wave fronts of large extent [54,
46], while wave fronts will rather be globally concentric otherwise, a common
conclusion might be coincidental. Suitable wave fields produced by the
playback loudspeakers therefore deserve a more fundamental investigation.
Preferably this yields an explanation technical thorough enough to serve as an
alternative to psychoacoustic experimentation or to psychoacoustic modeling
based on interaural level difference (ILD) and interaural coherence (IC) [73].
In this way it could also permit utilization, e.g., in technical acoustic
applications.

A systematic theory is desirable, and might contribute to a profoundly better
handling and understanding of diffuse sound field synthesis, which motivates
our paper.

###  1.1 Background and motivation

A noteworthy quote by Jacobsen and Roisin describes diffuse sound fields [77]:

> […M]ost acousticians would agree on a definition that involves sound coming
> from all directions. This leads to the concept of a sound field in an
> unbounded medium generated by distant, uncorrelated sources of random noise
> evenly distributed over all directions. Since the sources are uncorrelated
> there would be no interference phenomena in such a sound field, and the
> field would therefore be completely homogeneous and isotropic. For example,
> the sound pressure level would be the same at all positions, and temporal
> correlation functions between linear quantities measured at two points would
> depend only on the distance between the two points. The time-averaged sound
> intensity would be zero at all positions. An approximation to this
> “perfectly diffuse sound field” might be generated by a number of
> loudspeakers driven with uncorrelated noise in a large anechoic room […]

As metrics [78], we employ sound energy density $w$, sound intensity $\bm{I}$,
and diffuseness $\psi$ in the sound field [13, 7], which are described by the
sound pressure $p$ and sound particle velocity vector $\bm{v}$, according to

$\displaystyle w$
$\displaystyle=\frac{E\\{|p|^{2}\\}}{2\rho\,c^{2}}+\frac{\rho
E\\{\|\bm{v}\|^{2}\\}}{2},$ $\displaystyle{\bm{I}}$
$\displaystyle=\frac{E\\{\Re\\{p^{*}\bm{v}\\}\\}}{\rho\,c},$ (1)
$\displaystyle\psi$ $\displaystyle=1-\frac{\|\bm{I}\|}{c\,w},$ (2)

with the density of air $\rho=$1.2\text{\,}\frac{kg}{m^{3}}$$, the speed of
sound $c=$343\text{\,}\frac{m}{s}$$, and the statistical expectation
$E\\{\cdot\\}$. In a perfectly diffuse sound field, these quantities are
expected to ideally assume,

$\displaystyle{w}$ $\displaystyle=\text{const},$ $\displaystyle{\bm{I}}$
$\displaystyle=\bm{0},$ $\displaystyle\psi$ $\displaystyle=1.$ (3)

Otherwise, diffuseness becomes $\psi=0$ for a non-diffuse sound field from a
single source.

To observe whether $w$ as a measure of sound pressure level is constant and
whether the average sound intensity $\bm{I}$ vanishes, ideally, we present a
small simulation study motivating the particular research questions of this
paper.

Figure 1 shows a simulated free-field map of the sound intensity (vectors),
the diffuseness (colors), and the sound energy density (contours) along the
horizontal plane, for circular or spherical surfaces $S$ of acoustic sources
radiating uncorrelated signals of equal variance from many uniformly
distributed directions $-\bm{u}$. The simulation uses simplified integrals
[9], cf. subsection 2.1,

$\displaystyle\rho\,c^{2}w$
$\displaystyle=\frac{1}{\hat{N}}\sum_{l=1}^{\mathrm{L}}\sigma_{l}^{2}\,|G_{l}|^{2}\,$
$\displaystyle\rho\,c\,\bm{I}$
$\displaystyle=\frac{1}{\hat{N}}\sum_{l=1}^{\mathrm{L}}\bm{u}_{l}\,\sigma_{l}^{2}\,|G_{l}|^{2}$
(4)

determining $\psi$ in eq. (2). The Green’s function decays with
$|G_{l}|\propto\nicefrac{{1}}{{\sqrt{r_{l}}}}$ for a line source and
$|G_{l}|\propto\nicefrac{{1}}{{r_{l}}}$ for a point source, cf. eq. (7).
Distance $r_{l}$ and normalized direction vector $\bm{u}_{l}$ are measured
from each point or line source to the receiver; $\rho$ is the air density, $c$
the speed of sound, and $\hat{N}$ normalizes $\rho\,c^{2}\,w(\bm{0})=1$. With
the levels $\sigma_{l}=1$, vertical line sources arranged in a horizontal
circle yield the mapping in Figure 1 (b), tangentially arranged they yield
(c), and a circle of point sources yields (a). Point sources on a sphere yield
the mapping (d).

(a) $\sigma=1$

(b) $\sigma=\sqrt{R}$

(c) $\sigma=1$

(d) $\sigma=\sqrt{R}$

?figurename? 2: Diffuseness $\psi$ (colormap, solid contour: $90\%$) and level
$w$ normalized to origin (dotted $1\text{\,}\mathrm{d}\mathrm{B}$ contour) for
a 2D, 3:2 ellipse / rectangle ($L_{\mathsf{p}}$-superellipse with
$\mathsf{p}=10$) of 100 equi-angle uncorrelated vertical line sources
(salmon): (a,c) with unity gain $\sigma=1$, (b,d) isotropy-enforcing gain
$\sigma=\sqrt{R}$.

Circular arrangements of vertical line sources and spherical arrangements of
point sources yield perfect diffuseness $\psi=100\%$ everywhere inside, while
sound energy density $w$ varies and slightly increases outwards.

A more disturbing outwards increase of sound energy density is observed either
for the circle of point sources as in typical channel-, scene-, or panning-
based surround layouts, and even for the circle of horizontal line sources
that approximates diffuse-field rendering with 2.5D WFS [46, 79]. Either of
these cases seems incapable of rendering of a uniformly high diffuseness, and
sound intensity will be dominated by the nearest source.

What seems ideal for circular/spherical source layouts is not ideal for non-
circular/non-spherical ellipsoid or superellipse/cuboid layouts. Figure 2 (a)
analyzes a 2:3 ellipse of 100 sources arranged evenly in azimuth, whose
diffuseness mapping still seems fair, and in (b) the $90\%$-diffuseness
contour increases in extent by using a direction-dependent gain that ensures
isotropy in the center by inverting the vertical line sources’ radial decay by
their distance $R$ from there: $\sigma=\sqrt{R}$. Figure 2 (c) analyzes what
happens when the equi-angle source arrangement is mapped onto a rectangle, in
particular a 2:3 superellipse of the $L_{10}$ norm. There, the result in (c)
is worse than with the ellipse (a) with unity gain, and improvement in (d) by
isotropy-enforcing gain $\sigma=\sqrt{R}$ is not as much as for the ellipse.
The level (sound energy density) that is plotted in dotted contours remains
nearly unaffected by the gain patterns.

When using uncorrelated vertical line sources evenly arranged in a circle as
the ideal case, but now discretized, simulation indicates in Figure 12 (a-d)
of subsection 6.1 a relative increase of the area within which diffuseness
exceeds $\psi\geq 90\%$ with the number of loudspeakers.

The observations in the above-mentioned numerical experiments motivate our
research questions:

  1. 1.

What is the physics behind ideal diffuseness with circular/spherical geometry
and matching sources?

  2. 2.

Can there be isotropy everywhere inside?

  3. 3.

Are there optimal sources to produce both, $100\%$ diffuseness and constant
sound pressure level?

  4. 4.

Can results be optimized for non-spherical layouts?

  5. 5.

What determines the radial limit of discrete layouts?

  6. 6.

Can 2.5D wave-field synthesis provide diffuseness?

###  1.2 Outline

section 2 introduces general uncorrelated source distributions in the volume,
their statistically expected sound energy density and sound intensity, and
Gauß’ divergence theorem that would measure the total enclosed variance of all
sources, cf. [80]. Here, the obvious relation to the solution of Green’s
functions of potential fields (gravity, electrostatics) is helpful to
calculate the particular result based on Green’s third identity [81, 82].

In particular dealing with _question 1_ , Gauß’ divergence can be used to
prove that sound intensity must vanish inside hollow source densities with
rotation invariance, or potentially with shift invariance. Such arrangements
permit reduction to continuous hulls of sources and can either be a pair of
infinite parallel source planes, a thin source shell in the shape of an
infinite cylinder, or of a finite sphere, cf. section 3. subsection 3.1
clarifies that both correlated and uncorrelated source distributions with
infinitely extended excess dimensions exhibit similar radial decays for sound
energy density and sound intensity. Expanding on _question 1_ and touching
_question 3_ , subsection 3.2 shows the formal equivalence to Newton’s
spherical shell theorem [83, prop.70, Sec.12], and explains the underlying
geometry to establish a solid understanding of its balance for directionally
opposing intensity contributions, for any direction and observer location. In
$D$ space dimensions, the right balance is accomplished by a source with the
radial decay $\nicefrac{{1}}{{\sqrt{r}^{D-1}}}$, whereas the far end would
dominate for any smaller exponent, the near end for any higher one.

Concerning _question 2_ on isotropy, subsection 3.3 uses the integration
elements of Newton’s spherical shell theorem to observe whether isotropic
contributions to sound intensity can be obtained everywhere inside the hull.

To get answers for _question 3_ , subsection 3.4 establishes the scalar
potential of sound intensity that becomes constant for vanishing sound
intensity. Comparison to the potential of the sound energy density reveals a
problematic dimensionality gap for typical physical sources.

Therefore section 4 introduces sources of modified decay
$\nicefrac{{1}}{{r^{\beta}}}$ more rigorously to study them. Numerical
evaluation will be used to find out which $\beta$ yields ideally large usable
listening areas. To prove which radial decay exponent $\beta$ is precisely
required for vanishing sound intensity or constant sound energy density,
subsection 4.1 expands the modified Green’s functions into Gegenbauer
polynomials [84, 85, 15.12.E4] by their generating function. Orthogonality
will be used to show that constant sound energy density level can be
synthesized by an exponent $\beta$ that is either trivial $0$ or smaller by
$\nicefrac{{1}}{{2}}$ than the exponent delivering a constant sound intensity
potential.

A more intuitive proof of the ideal $\beta$ could be successful as well based
on Newton’s shell theorem alone, or based on inspection of the properties of
hypergeometric functions [86, 87]. However, the sophistic and strict proof
based on Gegenbauer polynomials also permits to get answers about _question 4_
in section 5: subsection 5.1 addresses multi-axial ellipsoids with the
direction-dependent radius $R$ seen from their center. In particular,
different gain patterns $\sigma$ depending on $R$ will be shown to either
provide isotropy for a central observer (same level from all directions) or
accomplish perfect synthesis of diffuseness everywhere inside. As more general
shapes tunable from ellipsoidal to rectangular cuboid, subsection 5.2 will
discuss multi-axis $L_{\mathsf{p}}$-norm shells. Targeting ideal diffuseness
will be shown to work in exemplary layouts by a zeroth-order mode matching
technique proposed in subsection 5.3 and subsection 5.4. For non-circular/non-
spherical arrangements of suitable sources, the corresponding Gegenbauer
expansion is re-formulated to an equivalent expansion using circular/spherical
harmonics combined with the radial solution of the potential equation. In both
cases, mode matching will be shown to produce suitable amplitude patterns
$\sigma$ supporting perfect synthesis of diffuseness inside, however with a
negative effect on isotropy. As important practical case for 3D audio,
subsection 5.5 is dedicated to analyzing upper hemispherical source layouts by
numerical simulation, in order to describe how much sound intensity needs to
be accepted that vertically traverses the horizontal plane, and how intensity
develops when coming closer to the top.

Gegenbauer expansion will also be employed in section 6 to address _question
5_ and define the limitations of diffusneess synthesis with discrete source
layouts in terms of relative radius. To this end, spherical $t$ designs [88]
are used as sampling schemes for the circle and sphere, because they optimally
discretize Gegenbauer expansions.

section 7 deals with _question 6_ about whether 2.5D WFS can provide diffuse-
field synthesis. To this end, numerical simulations will be used to discuss
differently scaled virtual circular layouts of ideally uncorrelated sources,
based on their stationary-phase approximation.

And finally, section 8 will discuss the implications and connections of the
presented findings to the psychoacoustically relevant interaural metrics used
by literature to understand its related listening-experiment studies.

##  2 Distributed, uncorrelated sources

A sound field $p(\bm{x})$ that we aim to generate by diffuse sound field
synthesis shall be superimposed from a distribution $q(\bm{x})$ exciting the
Helmholtz equation, cf. [89],

$\displaystyle(\bigtriangleup+k^{2})\,p(\bm{x})$ $\displaystyle=-q(\bm{x}),$
(5)

with the Laplacian denoting the second order Cartesian derivatives
$\bigtriangleup=\sum_{i=1}^{D}\frac{\partial^{2}}{\partial x_{i}^{2}}$ for all
coordinates $x_{i}$, and $k=\frac{\omega}{c}$ is the wave number consisting of
the frequency argument $\omega=2\pi\,f$ and the speed of sound $c$.

Typically, the inhomogeneous problem is solved via the Green’s function of the
free field, cf. [89],

$\displaystyle(\bigtriangleup+k^{2})\,G(r)$
$\displaystyle=-\delta(\bm{x}-\bm{x}_{\mathrm{s}}),$ (6)

which is the wave of an excitation at a single source point
$\bm{x}_{\mathrm{s}}$ to which $r=\|\bm{x}-\bm{x}_{\mathrm{s}}\|$ is the
Euclidean distance. This excitation is expressed by the Dirac delta that is
zero everywhere else than $\bm{x}=\bm{x}_{\mathrm{s}}$, is normalized so that
$\int_{V}\delta(\bm{x}-\bm{x}_{\mathrm{s}})\,\mathrm{d}V=1$, and fulfills a
sifting property
$\int_{V}q(\bm{x})\delta(\bm{x}-\bm{x}_{\mathrm{s}})\,\mathrm{d}V=q(\bm{x}_{\mathrm{s}})$.
For a free sound field, the Green’s function must obey Sommerfeld’s radiation
condition
$\lim_{r\rightarrow\infty}\sqrt{r}^{\mathrm{D-1}}(\mathrm{i}k+\frac{\partial}{\partial
r})G=0$ [90, §28] so that its far-field/high-frequency expression $kr\gg 1$
that we are interested in yields, see derivations according to [90,
Ch.V(A.IV,C),E13] in Appendix A, eq. (74),

$\displaystyle G$
$\displaystyle=\sqrt{\frac{2\pi\,r}{\mathrm{i}k}}^{3-D}\frac{e^{-\mathrm{i}kr}}{4\pi\,r}=A(k)\,\frac{e^{-\mathrm{i}kr}}{\sqrt{r}^{D-1}}.$
(7)

The distance-independent term $A(k)$ is typically removed by equalization, and
$p(\bm{x})$ is obtained by superimposing Green’s functions $G$ weighted by
$q(\bm{x}_{\mathrm{s}})$ for every source point, implying
$q(\bm{x})=\int_{V}q(\bm{x}_{\mathrm{s}})\,\delta(\bm{x}-\bm{x}_{\mathrm{s}})\,\mathrm{d}V$
as right-hand side of $(\bigtriangleup+k^{2})p=-q$, implying the solution for
$p$ on the left

$\displaystyle p$
$\displaystyle=\int_{V}q(\bm{x}_{\mathrm{s}})\,G(r)\,\mathrm{d}V.$ (8)

The sound particle velocity is $\bm{v}=\frac{\nabla
p}{-\mathrm{i}\omega\,\rho}$, and accordingly

$\displaystyle\bm{v}$
$\displaystyle=\int_{V}\,a(\bm{x}_{\mathrm{s}})\,{\textstyle\frac{\nabla
G(r)}{-\mathrm{i}\omega\,\rho}}\,\mathrm{d}V=\int_{V}\,a(\bm{x}_{\mathrm{s}})\,{\textstyle\frac{\bm{u}\,G(r)}{-\mathrm{i}\omega\,\rho}}\,\mathrm{d}V,$
(9)

where the Green’s function $G\propto b\,e^{-\mathrm{i}k\,r}$ was approximated
for the far field/high frequencies $kr\gg 1$ by

$\displaystyle\nabla G(r)$ $\displaystyle=(\nabla
r)(G^{\prime}(r))=\bm{u}\,[{\textstyle\frac{1}{b}\frac{\partial b}{\partial
r}-\mathrm{i}k}]\,G$ $\displaystyle\approx-\mathrm{i}k\,\bm{u}\,G,$ (10)

and $\bm{u}=\frac{\bm{x}-\bm{x}_{\mathrm{s}}}{\|\bm{x}-\bm{x}_{\mathrm{s}}\|}$
is the direction vector to the receiver.

?figurename? 3: Differential surface elements in $\mathbb{R}^{3}$ for
symmetric arrangements of (left) infinite parallel planes with shift
invariance in $yz$, (middle) an infinite cylinder with and rotational symmetry
in $xy$ and shift invariance in $z$, and (right) a sphere with rotational
symmetry in $xyz$, according to Newton’s spherical shell theorem.

We assume an amplitude distribution $q(\bm{x}_{\mathrm{s}})$ of
uncorrelated/incoherent sources of the variance $\sigma^{2}$ in the
integration volume

$\displaystyle
E\\{q(\bm{x}_{1})^{*}q(\bm{x}_{2})\\}=\sigma^{2}\,\delta(\bm{x}_{1}-\bm{x}_{2})$
(11)

with the statistical expectation operator denoted as $E\\{\\}$. The Dirac
delta is zero unless $\bm{x}_{1}=\bm{x}_{2}$ and normalized.

###  2.1 Expected sound energy density, sound intensity, sound power

The sound energy density $\rho\,c^{2}\,w=E\\{|p|^{2}\\}$ scaled by
$\rho\,c^{2}$ equals the expected mangitude-square sound pressure, cf. [78],

$\displaystyle\rho\,c^{2}\,w$
$\displaystyle=\iint_{V}\,E\\{q(\bm{x}_{1})^{*}q(\bm{x}_{2})\\}\,G_{1}^{*}G_{2}\,\mathrm{d}V_{1}\mathrm{d}V_{2}$
$\displaystyle=\int_{V}\,\sigma^{2}\,|G(r)|^{2}\,\mathrm{d}V,$ (12)

and sound intensity $\rho\,c\,\bm{I}=E\\{p^{*}\bm{v}\\}$ times $\rho\,c$ is,
cf. [78],

$\displaystyle\rho\,c\,\bm{I}$
$\displaystyle=\iint_{V}\,E\\{q(\bm{x}_{1})^{*}q(\bm{x}_{2})\\}\,\bm{u}\,G_{1}^{*}G_{2}\,\mathrm{d}V_{1}\mathrm{d}V_{2}$
$\displaystyle=\int_{V}\,\sigma^{2}\,\bm{u}\,|G(r)|^{2}\,\mathrm{d}V.$ (13)

#### Gauß’ law

permits to estimate the sound power $\dot{W}$ of sources enclosed in a hull
$S=\partial V$ via net intensity flux out of the hull, or the divergence
$\nabla^{\intercal}\bm{I}=:\dot{w}$, the sound power density, integrated over
the enclosed volume $V$

$\displaystyle\dot{W}=\rho\,c\,\Re\Big{\\{}\int_{S}\bm{I}^{\intercal}\,\bm{n}\,\mathrm{d}S\Big{\\}}$
$\displaystyle=\rho\,c\,\Re\Big{\\{}\int_{V}\nabla^{\intercal}\bm{I}\,\mathrm{d}V\Big{\\}}.$
(14)

It is helpful to notice that squared free-field Green’s function of the
Helmholtz equation times direction

$\displaystyle\bm{u}\,|G(r)|^{2}$
$\displaystyle=\frac{\bm{u}}{\,r^{D-1}}{|A|^{2}}=-\mathcal{S}_{D-1}\,|A|^{2}\,\nabla\mathcal{G}(r)$
(15)
$\displaystyle=\mathcal{S}_{D-1}\,|A|^{2}\,\bm{u}\,\mathcal{G}^{\prime}(r),$

relates to the Green’s function $\mathcal{G}$ of the Laplace or potential
equation (and $\mathcal{S}_{D-1}\,\mathcal{G}^{\prime}=-\frac{1}{r^{D-1}}$)

$\displaystyle\bigtriangleup\,\mathcal{G}(r)$
$\displaystyle=-\delta(\bm{x}-\bm{x}_{\mathrm{s}})$
$\displaystyle\mathcal{G}(r)$
$\displaystyle=\begin{cases}-\frac{1}{\mathcal{S}_{D-1}}\ln r&D=2,\\\
\frac{1}{(D-2)\,\mathcal{S}_{D-1}}\frac{1}{r^{D-2}}&D\neq 2,\end{cases}$ (16)

and
$\mathcal{S}_{D-1}=\frac{2\,\sqrt{\pi}^{\mathrm{D}}}{\Gamma(\frac{\mathrm{D}}{2})}$
is the area of the unit sphere in $D$ dimensions [91, vol.2, p.387].
Divergence of gradient
$\nabla^{\intercal}(\nabla\mathcal{G})=\bigtriangleup\mathcal{G}=-\delta$
yields an integrand of eq. (14)

$\displaystyle\rho\,c\,\nabla^{\intercal}\bm{I}$
$\displaystyle=\int_{V}\sigma^{2}(\bm{x}_{\mathrm{s}})\,\delta(\bm{x}-\bm{x}_{\mathrm{s}})\,\mathcal{S}_{D-1}\,|A|^{2}\,\mathrm{d}V$
$\displaystyle=\mathcal{S}_{D-1}\,|A|^{2}\,\sigma^{2}(\bm{x}),$ (17)

and upon insertion, the total sound power of all sources enclosed in $V$ using
Gauß’ law eq. (14) becomes

$\displaystyle\int_{S}\bm{I}^{\intercal}\bm{n}\,\mathrm{d}S=\mathcal{S}_{D-1}\,|A|^{2}\,\int_{V}\sigma^{2}\,\mathrm{d}V=\dot{W}.$
(18)

In this article, we will work with a volume $V$ that is free of sources
inside, so that sound power $\dot{W}$ vanishes inside

$\displaystyle\dot{W}$
$\displaystyle=\int_{S}\bm{I}^{\intercal}\bm{n}\,\mathrm{d}S=0,$
$\displaystyle\text{as }\sigma(\bm{x})$ $\displaystyle=0\;\forall\bm{x}\in V.$
(19)

In particular, we continue after re-defining the volume source distribution to
a thin shell $S$ of sources distributed tightly around $V$, just not contained
within $V$ anymore, so that the interior is source-free. Adapting eqs. (12)
(13), these shell integrals become

$\displaystyle\rho\,c^{2}\,w$
$\displaystyle=\frac{1}{\hat{N}}\int_{S}\sigma^{2}\,|G(r)|^{2}\,\mathrm{d}S,$
(20) $\displaystyle\rho\,c\,\bm{I}$
$\displaystyle=\frac{1}{\hat{N}}\int_{S}\sigma^{2}\,\bm{u}\,|G(r)|^{2}\,\mathrm{d}S,$
(21)

where $\hat{N}$ is used to normalize $\rho\,c^{2}\,w(\bm{0})=1$. Around the
line between any surface point $\bm{x}_{\mathrm{s}}$ and the observer
$\bm{x}$, defining the direction
$\bm{u}=\frac{\bm{x}-\bm{x}_{\mathrm{s}}}{\|\bm{x}-\bm{x}_{\mathrm{s}}\|}$,
the surface element $\mathrm{d}S$ is enclosed in an infinitesimally small
angular cone $\mathrm{d}\Omega$ intersecting with the surface normal $\bm{n}$
at an angle $\phi$, and it scales by projection
$\cos\phi=\bm{n}^{\intercal}\bm{u}$ and a distance-related area increase
$r^{D-1}$ so that $\mathrm{d}S=r^{D-1}\,\frac{\mathrm{d}\Omega}{\cos\phi}$.

Note that while $\mathrm{d}S$ is most suitable to cover the fundamentals, we
use instead of $\mathrm{d}S(\bm{u})$ a modified integral over
$\mathrm{d}\Omega_{0}(\bm{u}_{0})$ from subsection 3.4 on and in the
introductory examples, referring to an angular segment seen in the direction
$\bm{u}_{0}$ from the origin, yielding a wave towards $\bm{u}(\bm{u}_{0})$ at
the observer.

##  3 Optimal hulls of sources

A source-free interior of the volume $V$ as in eq. (19) can be combined with a
symmetry assumption that determines the intensity in eq. (19) and forces it to
zero. In particular, if the intensity:

  * •

has a constant magnitude at every surface position
$\|\bm{I}\|\neq\|\bm{I}(\bm{x}_{\mathrm{s}})\|$ and

  * •

is non-tangential, so strictly aligned with the surface normal for every
surface position $\bm{I}=I_{n}\,\bm{n}$

then intensity vanishes due to the source-free interior ($\dot{W}=0$) and the
non-zero hull area ($\oint_{S}\mathrm{d}S=S\neq 0$)

$\displaystyle\dot{W}=\oint_{S}\bm{I}^{\intercal}\bm{n}\,\mathrm{d}S$
$\displaystyle=I_{n}\oint_{S}\bm{n}^{\intercal}\bm{n}\,\mathrm{d}S=I_{n}\,S=0,$
$\displaystyle\Rightarrow I_{\mathrm{n}}\equiv 0\Rightarrow\bm{I}$
$\displaystyle\equiv\bm{0}\;\forall\bm{x}\in S.$ (22)

All-zero intensity components everywhere on $S$ entails zero intensity
$\bm{I}\equiv\bm{0}\;\forall\bm{x}\in V$ everywhere in the source-free field
enclosed. Similar considerations are employed in standard literature on
electrostatic fields, cf. [92].

The two constraints are fulfilled in particular if the variance of the sources
is invariant $\sigma^{2}(\bm{x})=\sigma^{2}(\bm{R}\,\bm{x})$ under rotation by
any matrix $\bm{R}\in\mathbb{R}^{D^{\prime}}$:
$\bm{R}^{\intercal}\bm{R}=\mathbf{I}$, $\mathrm{det}\\{\bm{R}\\}=1$. Moreover
rotation invariance can be limited to $D^{\prime}\leq D$ subspace dimensions
$D^{\prime}>0$, for which one could, e.g., define a spherical manifold of the
radius $\sum_{i=1}^{D^{\prime}}x_{\mathrm{s},i}^{2}=R^{2}$. Rotation
invariance will ensure there being no tangential components
$\bm{I}=I_{n}\bm{n}$, as long as there is shift invariance for any shift
$\Delta x\in\mathbb{R}^{D-D^{\prime}}$ along the excess dimensions
$i>D^{\prime}$, $\sigma^{2}(\bm{x})=\sigma^{2}(\bm{x}+\Delta\bm{x})$.
Considering a single spherical manifold radius $R$ as a shell of sources in
$\mathbb{R}^{3}$, requirements include, cf. Figure 3:

  1. 1.

an opposing pair of infinite planes, e.g., $x_{\mathrm{s},1}^{2}=R^{2}$

  2. 2.

an infinite cylinder, e.g., $x_{\mathrm{s},1}^{2}+x_{\mathrm{s},2}^{2}=R^{2}$

  3. 3.

a finite sphere,
$x_{\mathrm{s},1}^{2}+x_{\mathrm{s},2}^{2}+x_{\mathrm{s},3}^{2}=R^{2}$.

###  3.1 Reduced radial decay for infinite extent

When there are no infinitely extended excess dimensions $E=D-D^{\prime}=0$ of
the source distribution, the function
$\mathcal{G}^{\prime}(r)\propto\frac{1}{r^{D-1}}$ is responsible for the
ideally vanishing intensity in eq. (22). When there are $E>0$ excess
dimensions with uncorrelated sources being integrated over, we may use the
relations shown in Appendix B to find

$\displaystyle\lim_{M\rightarrow\infty}\underbrace{\int_{-M}^{M}\dots\int_{-M}^{M}}_{E}\frac{\prod_{i=D^{\prime}+1}^{D}\mathrm{d}x_{\mathrm{s},i}}{(2L)^{E}\,r^{D-1}}\propto\frac{1}{r^{D^{\prime}-1}}.$
(23)

This determines the function
$\mathcal{G}^{\prime}_{D^{\prime}}\propto\frac{1}{r^{D^{\prime}-1}}$ for the
remaining subspace. Constant multipliers and scaling by the size of the excess
surface in the limit towards infinity can be usefully suppressed by re-
defining $\sigma^{2}$ accordingly.

It is noteworthy that an equivalent reduction of the exponent occurs when
driving all sources in phase. Here, the infinite extent is obtained by
removing the independent dimension from the Helmholtz equation, yielding a
free-field Green’s function
$G_{D^{\prime}}\propto\frac{e^{-\mathrm{i}kr}}{\sqrt{r}^{D^{\prime}-1}}$ for
$kr\gg 1$, cf. eq. (7), of which the magnitude square in eq. (21) is
equivalent
$|G_{D^{\prime}}|^{2}\propto\mathcal{G}^{\prime}_{D^{\prime}}\propto\frac{1}{r^{D^{\prime}-1}}$.

Only the uncorrelated sources can create incoherence along the excess
dimension, but the metrics $w$, $\bm{I}$, $\psi$ are blind to this aspect. It
is formally equivalent when either the infinite dimensions stem from a
cylinder-wave or plane-wave radiator, or they stem from a linear or planar
distribution of uncorrelated point sources, respectively. Below, we simplify
$D^{\prime}$ to $D$ and skip explicit assumptions about correlation along
excess dimensions.

###  3.2 Newton’s spherical shell theorem

Newton’s spherical shell theorem [83, prop.70, Sec.12], concludes that the
gravitational force due to a thin hollow spherical shell vanishes inside. The
contribution $\mathrm{d}F$ of a surface element $\mathrm{d}S$ centered at
$\bm{x}_{\mathrm{s}}$ to the force $\bm{F}=\int\bm{u}\,\mathrm{d}F$ at
$\bm{x}$ is proportional to the factor $\frac{1}{r^{D-1}}$ (inverse square law
for $D=3$ of $\mathbb{R}^{3}$) of its distance
$r=\|\bm{x}-\bm{x}_{\mathrm{s}}\|$ times
$\mathrm{d}S=r^{D-1}\,\frac{\mathrm{d}\Omega}{\cos\phi}$, related to the
angular element $\mathrm{d}\Omega$ as described above. The force contributions
$\mathrm{d}F_{-}$ and $\mathrm{d}F_{+}$ of opposing intersections along
$\pm\bm{u}$ annihilate

$\displaystyle\frac{r_{+}^{D-1}}{r_{+}^{D-1}}\,\frac{\mathrm{d}\Omega}{\cos\phi_{+}}=\frac{r_{-}^{D-1}}{r_{-}^{D-1}}\,\frac{\mathrm{d}\Omega}{\cos\phi_{-}},$
(24)

i.e. whenever both intersection angles match $\phi_{+}=\phi_{-}$. This is
obviously the case for parallel dimensions. Also, any opposing pair of
intersection points $\bm{x}_{\mathrm{s}}^{+}$ and $\bm{x}_{\mathrm{s}}^{-}$
with a spherical manifold ensures $\phi_{+}=\phi_{-}$ by the isosceles
triangle enclosed with the manifold’s origin $x_{i}=0\,\forall i\leq
D^{\prime}$, cf. Figure 3 (middle). With this match for any $\pm\bm{u}$,
integration over all directions $\bm{u}$ yields a vanishing net force
$\bm{F}=\bm{0}$. Despite a global sign difference,
$\mathrm{d}\bm{I}=-\mathrm{d}\bm{F}$, the formulation for sound intensity is
equivalent.

Assuming $r_{+}>r_{-}$ and symmetry $\phi_{+}=\phi_{-}$, Newton’s theorem
shows that imbalance can arise from a modified exponent
$\nicefrac{{1}}{{r^{\alpha}}}$, cf. Figure 4, between near-end
$\mathrm{d}I_{-}=\frac{r_{-}^{D-1}}{r_{-}^{\alpha}}\,\frac{\mathrm{d}\Omega}{\cos\phi_{-}}$
and far-end
$\mathrm{d}I_{+}=\frac{r_{+}^{D-1}}{r_{+}^{\alpha}}\,\frac{\mathrm{d}\Omega}{\cos\phi_{+}}$
whenever $\alpha\neq D-1$: An exponent that is too high causes the near end to
dominate intensity, one that is too small causes dominance of the far end, cf.
section 4.

###  3.3 Isotropic directional intensity

Nolan et al. [15] expressed isotropy as the uniformity of the wave-number
magnitude spectrum, i.e. the uniform magnitude for all plane-wave directions
received. Equivalently corresponding with the definitions used for the shell
theorem, we may define isotropy via the intensity contribution from an
infinitesimal angular cone $\mathrm{d}\Omega$ aligned with a variable
direction of arrival $-\bm{u}$,

$\displaystyle\mathrm{d}I_{\mathrm{u}}$
$\displaystyle=\frac{\sigma^{2}}{r^{D-1}}\,\frac{r^{D-1}\mathrm{d}\Omega}{\cos\phi}=\sigma^{2}\,\frac{\mathrm{d}\Omega}{\cos\phi},$
(25)

so that isotropy can be measured by how constant

$\displaystyle\frac{\mathrm{d}I_{\mathrm{u}}}{\mathrm{d}\Omega}=\frac{\sigma^{2}}{\cos\phi}=\text{const}$
(26)

evolves across directions $\bm{u}$. This is a more strict constraint than
Newton’s spherical shell theorem that just requires for every direction
$\bm{u}$ the annihilation with the opposing direction $-\bm{u}$ by
$\frac{\mathrm{d}I_{\mathrm{u}}}{\mathrm{d}\Omega}=\frac{\mathrm{d}I_{\mathrm{-u}}}{\mathrm{d}\Omega}$.
The strictness of isotropy lies in direction-independence, which is only
accomplished (i) at the center of a perfectly spherical layout,
$D^{\prime}=D$, $\sigma=1$, and $r=R=\text{const}$, where $\cos\phi\equiv 1$,
or (ii) at a particular point of observation within an arbitrary convex layout
for which the source variance is adjusted to

$\displaystyle\sigma^{2}\propto\cos\phi,$ (27)

for every direction $\bm{u}$. Assuming to accomplish perfect isotropy by this
fixed $\sigma^{2}(\bm{u}_{\mathrm{s}})$ that matches the respective direction
cosine $\cos\phi_{0}$ at every source to a central observer at
$\bm{x}=\bm{0}$, an observer shifted off center will yield a different
direction cosine $\cos\phi$ at each source, in general. This contradicts a
fixed choice of $\sigma^{2}$, which can only enforce isotropy for a specific
point inside $V$. We will therefore restrict our subsequent discussion of
isotropy to a central observer in subsection 5.1 and subsection 5.2.

###  3.4 Intensity potential, energy density

For a thin source shell in the subspace $\mathbb{R}^{D}$, we fix the variance
to a constant value $\sigma^{2}$ that is non-zero only at a fixed radius, e.g.
$R=1$ of the unit sphere, denoted as $\mathbb{S}$. According to eq. (22),
symmetry ensures vanishing sound intensity $\bm{I}=0$ inside $\bm{x}\in V$

$\displaystyle\rho\,c\,\bm{I}$
$\displaystyle=\frac{\sigma^{2}}{\tilde{N}}\,\int_{\mathbb{S}}\nabla^{\intercal}\mathcal{G}_{D}(r)\,\mathrm{d}\Omega_{0}=\bm{0}.$
(28)

Here, we used a surface element
$\mathrm{d}S=\mathcal{S}_{D-2}\,R^{D-1}\,\mathrm{d}\Omega_{0}$ of an isotropic
angular segment $\mathrm{d}\Omega_{0}$ seen from the origin, the area of the
unit sphere $\mathcal{S}_{D-2}$ reduced by a dimension (the area of the unit
sphere is
$\mathcal{S}_{D-1}=\frac{2\,\sqrt{\pi}^{\mathrm{D}}}{\Gamma(\frac{\mathrm{D}}{2})}$,
cf. [91, vol.2, p.387]), a constant source variance $\sigma^{2}=\text{const}$,
and a new normalizer $\tilde{N}$ to hide some of the constant multipliers. To
relate Laplacian and Helmholtz Green’s functions, eq. (2.1), we used

$\displaystyle\mathcal{S}_{D-1}|A|^{2}\;\mathcal{G}^{\prime}_{D}(r)=|G_{D}(r)|^{2}.$
(29)

?figurename? 4: Point source at $\bm{x}_{\mathrm{s}}=\bm{0}$ with radial decay
exponent modified from left to right $\beta=\\{1,0.5,0\\}$, at
$f=$1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$.

We can conceive a potential field $-U_{\mathrm{I}}$ whose gradient delivers
the intensity $-\nabla U_{\mathrm{I}}=\rho\,c\,\bm{I}$. Because
$\bm{I}=\bm{0}$, the related potential field is spatially constant

$\displaystyle-U_{\mathrm{I}}$
$\displaystyle=\frac{\sigma^{2}}{\tilde{N}}\int_{\mathbb{S}}\mathcal{G}_{D}(r)\,\mathrm{d}\Omega_{0}=\text{const}.$
(30)

While for the synthesis of vanishing sound intensity, eq. (22) and Newton’s
spherical shell theorem eq. (24) are proof that spherical shells describe the
ideal geometry, these relations may indicate that sound-energy density, based
on $\mathcal{G}^{\prime}$ instead of $\mathcal{G}$, is not constant,

$\displaystyle\rho\,c^{2}\,w$
$\displaystyle=\frac{\sigma^{2}}{\tilde{N}}\int_{\mathbb{S}}\mathcal{G}^{\prime}_{D}(r)\,\mathrm{d}\Omega_{0}\neq\text{const?},$
(31)

unless $D=1$, i.e., a shell of two infinite planes, which is a trivial case;
$\tilde{N}$ normalizes $\rho\,c^{2}\,w(\bm{0})=1$ for
$\mathcal{G}_{D}^{\prime}$.

##  4 Source of modified decay $\beta$

To find conditions for constant sound energy density, we test a free-field
Green’s function with manipulated radial decay, cf. Figure 4, for which
practical implementation in 2D layouts was suggested using curved/phased
vertical line-source arrays [93, 94],

$\displaystyle G_{\beta}(r)$
$\displaystyle=\frac{e^{-\mathrm{i}kr}}{r^{\beta}},$
$\displaystyle|G_{\beta}(r)|^{2}$
$\displaystyle=\frac{1}{r^{2\beta}}=:\mathcal{G}_{2\beta}^{\prime}(r),$ (32)

and it modifies the related Laplacian Green’s function to

$\displaystyle\mathcal{G}_{2\beta}(r)$
$\displaystyle=\begin{cases}\frac{1}{-(2\beta-1)r^{2\beta-1}},&2\beta\neq
1,\\\ \ln r,&2\beta=1.\end{cases}$ (33)

We test a normalized displacement $-1\leq x\leq 1$ for the coordinate
$x_{1}=x$ and $x_{i}=0$ else for the shell radius $R=1$ of a unit sphere.
Integrals for $\rho\,c^{2}\,w$, $\rho\,c\,\bm{I}$, $\psi$ use the angular
fraction
$\frac{\mathrm{d}\Omega_{0}}{\mathcal{S}_{D-1}}=\frac{\mathcal{S}_{D-2}}{\mathcal{S}_{D-1}}\sqrt{1-z^{2}}^{D-3}\mathrm{d}z$
related to the direction cosine $z=\cos\phi$ between source and observer
directions from the origin, the unit-sphere areas $\mathcal{S}_{D-1}$,
$\mathcal{S}_{D-2}$ [91, vol.2, p.387], the weight $\sqrt{1-z^{2}}^{D-3}$, for
generalization111Altogether, this ensures
$\frac{\mathrm{d}\Omega_{0}}{\mathcal{S}_{D-1}}=\frac{2\pi\,\mathrm{d}\cos\phi}{4\pi}$
for $D=3$ and
$\frac{\mathcal{S}_{D-2}}{\mathcal{S}_{D-1}}\sqrt{1-z^{2}}^{D-3}\mathrm{d}z=\frac{\mathrm{d}\phi}{2\pi}$
for $D=2$, cf. [91]., and the corresponding distance $r=\sqrt{1-2zx+x^{2}}$
between source and observer. Appendix C describes the integrals in greater
detail and substitutes $z=2t-1$ within $-1\leq z\leq 1$ with the variable $t$
within $0\leq t\leq 1$ to make them fit the integral representation [85,
15.6.E1],

$\displaystyle F(a,b;c;z)$
$\displaystyle=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}\frac{t^{b-1}\,(1-t)^{c-b-1}}{(1-zt)^{a}}\;\mathrm{d}t,$

of Gauß’ hypergeometric function $F(a,b;c;z)$ [85, 15.2.E1]. Sound energy
density $\rho\,c^{2}\,w$ and the only non-zero sound intensity component
$\rho\,c\,I_{\mathrm{x}}$ along $x_{1}$ become

$\displaystyle\rho\,c^{2}\,w=$
$\displaystyle\frac{F\textstyle(\beta,\frac{D-1}{2};D-1;\frac{4x}{(1+x)^{2}})}{(1+x)^{2\beta}},$
(34) $\displaystyle\rho\,c\,I_{\mathrm{x}}=$
$\displaystyle\frac{F\textstyle(\beta+\frac{1}{2},\frac{D-1}{2};D-1;\frac{4x}{(1+x)^{2}})}{(1+x)^{2\beta}}-$
$\displaystyle\qquad\frac{F\textstyle(\beta+\frac{1}{2},\frac{D+1}{2};D;\frac{4x}{(1+x)^{2}})}{(1+x)^{2\beta+1}},$
(35)

and correspondingly diffuseness becomes

$\displaystyle\psi=$ $\displaystyle
1-\bigg{|}\frac{F\textstyle(\beta+\frac{1}{2},\frac{D-1}{2};D-1;\frac{4x}{(1+x)^{2}})}{F\textstyle(\beta,\frac{D-1}{2};D-1;\frac{4x}{(1+x)^{2}})}$
$\displaystyle\qquad-\frac{\frac{1}{1+x}F\textstyle(\beta+\frac{1}{2},\frac{D+1}{2};D;\frac{4x}{(1+x)^{2}})}{F\textstyle(\beta,\frac{D-1}{2};D-1;\frac{4x}{(1+x)^{2}})}\bigg{|}.$
(36)

Python implements the hypergeometric function as `scipy.special.hyp2f1`, and
results were verified by numerical integrals. 5(a) displays the resulting
sound energy density $\rho\,c^{2}\,w$ eq. (34) and 5(b) the diffuseness $\psi$
eq. (36) contours for the displacements $x$ normalized to lie between $-1\leq
x\leq 1$ and $\beta$ varied between $-1.25\leq\beta\leq 1.25$, either for a 2D
circle arrangement of sources or a 3D sphere arrangement of sources. At the
center $x=0$, ideal values are always reached. In the circular arrangement
$D=2$, an extended spatial range requires $\beta=0$ for constant sound energy
density, so unphysical point sources without decay, while constant $100\%$
diffuseness requires $\beta=\frac{1}{2}$, so physical vertical line sources.
For the spherical arrangement $D=3$, constant sound energy density requires
$\beta=\\{0,\frac{1}{2}\\}$, so either unphysical point sources without decay
or unphysical ones with the decay of a line source, while $100\%$ diffuseness
requires physical point sources with $\beta=1$.

(a) sound energy density for circular (left) / spherical (right) layout

(b) diffuseness for circular (left) / spherical (right) layout

?figurename? 5: Evaluation of sound energy density (a) and diffuseness (b) for
$-1.25\leq\beta\leq 1.25$ (vertical axis) within $-1\leq x\leq 1$ (horizontal
axis; orange-red arrows display $\nicefrac{{I_{\mathrm{x}}}}{{c\,w}}$ at
$x=\pm 0.6$).

The sound energy density seems to vary only moderately with non-ideal, but
physical values for $\beta$ of the given $D$. It appears reasonable to focus
on accomplishing $100\%$ diffuseness with physical sources while accepting a
sound energy density that is not perfectly constant.

###  4.1 Optimal $\beta$ via Gegenbauer polynomials

As a strict proof of the optimal $\beta$, we regard the generating function of
the orthogonal Gegenbauer polynomials [85, 18.12.E4] that permits to expand
our modified Green’s functions in $r=\sqrt{1-2zx+x^{2}}$ as

$\displaystyle\mathcal{G}^{\prime}_{2\beta}(r)\propto\frac{1}{r^{2\beta}}=$
$\displaystyle\sum_{n=0}^{\infty}x^{n}\,C_{n}^{(\beta)}(z),$ (37)
$\displaystyle\mathcal{G}_{2\beta}(r)\propto\frac{1}{r^{2\beta-1}}=$
$\displaystyle\sum_{n=0}^{\infty}x^{n}\,C_{n}^{(\beta-\frac{1}{2})}(z),$ (38)

where $\mathcal{G}^{\prime}_{2\beta}$ is integrated for $\rho\,c^{2}\,w$, cf.
eq. (31) and $\mathcal{G}_{2\beta}$ for $U_{\mathrm{I}}$, cf. eq. (28). As
above, integration of these modified Green’s functions over the shell uses the
axisymmetric surface element $\mathcal{S}_{D-2}\,\varpi_{\alpha}\,\mathrm{d}z$
for integration between $-1\leq z\leq 1$ along the direction of the shift $x$,
and the surface weight $\varpi_{\alpha}(z)=\sqrt{1-z^{2}}^{2\alpha-1}$, cf.
[91]. It matches the manifold dimensions when $2\alpha=D-2$. We choose a
constant variance $\sigma^{2}$ to eliminate independent scalars in the sound
energy density and sound intensity potential

$\displaystyle\rho\,c^{2}\,w$
$\displaystyle=\sum_{n=0}^{\infty}x^{n}\,\int_{-1}^{1}C_{n}^{(\beta)}(z)\,\varpi_{\alpha}(z)\,\mathrm{d}z,$
(39) $\displaystyle-U_{\mathrm{I}}$
$\displaystyle\propto\sum_{n=0}^{\infty}x^{n}\,\int_{-1}^{1}C_{n}^{(\beta-\frac{1}{2})}(z)\,\varpi_{\alpha}(z)\,\mathrm{d}z.$
(40)

As the zeroth degree Gegenbauer polynomial is both constant
$C_{0}^{(\nu)}(z)=1$ and orthogonal to any higher-degree Gegenbauer polynomial
of the specific family $\nu$, integrals for $n>0$ must vanish
$\int_{-1}^{1}C_{n}^{(\nu)}(z)\,\varpi_{\nu}\,\mathrm{d}z=0$, as long as the
surface weight $\varpi_{\nu}$ matches the family $\nu$.

(a) ellipse $\sigma=1$

(b) ellipse $\sigma=\sqrt{R}$

(c) ellipse $\sigma=R$

(d) CH mode-match $\sigma$

(e) ellipsoid $\sigma=1$

(f) ellipsoid $\sigma={R}$

(g) ellipsoid $\sigma=\sqrt{R}^{3}$

(h) SH mode-match $\sigma$

?figurename? 6: Diffuseness $\psi$ (colormap, solid contour: $90\%$) and level
$w$ normalized to origin (dotted $1\text{\,}\mathrm{d}\mathrm{B}$ contour) for
2D, 3:2 ellipse of 100 equi-angle uncorrelated vertical line sources (a-d) or
6:4:3 ellipsoid of 2500 maximum determinant nodes with uncorrelated point
sources (e-h): (a,e) with unity gain $\sigma=1$, (b,f) isotropy-enforcing gain
$\sigma=\sqrt{R}$ or $R$, (c,g) gain $\sigma=R$ or $\sqrt{R}^{3}$, and (d,h)
mode-matched $\sigma$ for ideal diffuseness.

This proves with $\alpha=\beta$ that constant sound energy density
$\rho\,c^{2}\,w=\text{const}$ of $2\alpha=D-2$ so $2\beta=D-2$ requires
$G_{D}\propto\frac{1}{\sqrt{r}^{D-2}}$; or trivially
$G_{D}\propto\frac{1}{r^{0}}$ of $\beta=0$.

By contrast, this proves with $\alpha=\beta-\frac{1}{2}$ that constant sound-
intensity potential $U_{\mathrm{I}}=\text{const}$ of $2\alpha=D-2$, so
$2\beta=D-1$, requires $G_{D}\propto\frac{1}{\sqrt{r}^{D-1}}$, as illustrated
before.

?figurename? 7: Directional intensity
$\mathrm{d}I/\mathrm{d}\Omega=\sigma^{2}/R$ in 2D, normalized at $0^{\circ}$,
observed at $\bm{x}=\bm{0}$ for different azimuth angles to between
$0^{\circ}$ and $90^{\circ}$ (quadrants are symmetric) for 2:3 2D horizontal
ellipse/rectangle of uncorrelated vertical line sources (superellipse
$\mathsf{p}=10$) with different gain patterns $\sigma$; isotropy requires
$\sigma/\sqrt{R}=\text{const}.$

(a) superellipse $\sigma=1$

(b) superellipse $\sigma=\sqrt{R}$

(c) superellipse $\sigma=R$

(d) CH mode-match $\sigma$

(e) s.ellipsoid/cut1 $\sigma=1$

(f) s.ellipsoid/cut1 $\sigma={R}$

(g) s.ellipsoid/cut1 $\sigma=\sqrt{R}^{3}$

(h) SH mode-match $\sigma$

(i) s.ellipsoid/cut2 $\sigma=1$

(j) s.ellipsoid/cut2 $\sigma={R}$

(k) s.ellipsoid/cut2 $\sigma=\sqrt{R}^{3}$

(l) SH mode-match $\sigma$

?figurename? 8: Diffuseness $\psi$ (colormap, solid contour: $90\%$) and level
$w$ normalized to origin (dotted $1\text{\,}\mathrm{d}\mathrm{B}$ contour),
for the superellipse of the ratio 3:2 (2D), and for the superellipsoid of the
ratio 6:4:3 (3D), both with $\mathsf{p}=10$, using 100 equi-angular nodes as
uncorrelated line vertical sources (2D) or 2500 maximum determinant nodes as
uncorrelated point sources (3D), respectively: (e,i) with unity gain
$\sigma=1$, (f,j) isotropy-enforcing gain $\sigma=R$, (g,k) gain
$\sigma=\sqrt{R}^{3}$, and (h,l) mode-matched $\sigma$ for ideal diffuseness.
(i-l) are as (e-h) but plotted along an inclined plane cf. Figure 9.

##  5 Suboptimal source layers

The previous section discussed the shift and rotation invariance of optimal
source layers. This section discusses other geometries (ellipsoids, cuboids,
incomplete hull).

###  5.1 Ellipsoidal source layer

Ellipsoidal shapes for $D\geq 2$ do not provide the above-mentioned symmetry.
Their similarity to spherical symmetry bears a chance that source variances
can be designed to provide $100\%$ diffuseness. A multi-axial ellipsoid with
semi-axes lengths $\bm{a}=[a_{i}]$ aligned with the Cartesian axes is
described by
$\sqrt{\frac{x_{\mathrm{1,s}}^{2}}{a_{1}^{2}}+\frac{x_{\mathrm{2,s}}^{2}}{a_{2}^{2}}+\dots}=1$,

$\displaystyle\sqrt{\bm{x}_{\mathrm{s}}^{\intercal}\,\mathrm{diag}\\{\bm{a}\\}^{-2}\bm{x}_{\mathrm{s}}}=1.$
(41)

Using a normalized direction vector $\bm{u}_{0}$ from the origin,
$\|\bm{u}_{0}\|=1$, we may write the coordinates as
$\bm{x}_{\mathrm{s}}=R\,\bm{u}_{0}$ to get the direction-dependent radius via
eq. (41),

$\displaystyle
R=\frac{1}{\sqrt{\bm{u}_{0}^{\intercal}\,\mathrm{diag}\\{\bm{a}\\}^{-2}\bm{u}_{0}}}.$
(42)

The distance between source $\bm{x}_{\mathrm{s}}$ and receiver at
$\bm{x}=x\,\bm{u}$, displaced from $\bm{0}$ into the direction $\bm{u}$ by
$x$, becomes

$\displaystyle r$
$\displaystyle=\sqrt{R^{2}-2\,R\,x\,\bm{u}_{0}^{\intercal}\bm{u}+x^{2}}$
$\displaystyle=R\,\sqrt{1-2\,\left(\frac{x}{R}\right)\,\bm{u}_{0}^{\intercal}\bm{u}+\left(\frac{x}{R}\right)^{2}}$

Correspondingly, expansion of
$\mathcal{G}(r)\propto\frac{1}{r^{2\beta-1}}=\frac{1}{r^{D-2}}$ that was
physical and ideal for $U_{\mathrm{I}}$ above into Gegenbauer polynomials
yields

$\displaystyle\frac{1}{r^{D-2}}$
$\displaystyle=\frac{1}{R^{D-2}}\sum_{n=0}^{\infty}\left(\frac{x}{R}\right)^{n}C_{n}^{(\frac{D-2}{2})}(\bm{u}_{0}^{\intercal}\bm{u})$
(43)
$\displaystyle=\sum_{n=0}^{\infty}x^{n}\,R^{-(n+D-2)}\,C_{n}^{(\frac{D-2}{2})}(\bm{u}_{0}^{\intercal}\bm{u}).$

We test integrating all sources weighted by the variance

$\displaystyle\sigma^{2}=R^{D-2+\mu},$ (44)

by integrating over the element
$\mathrm{d}\Omega_{0}=\mathrm{d}\Omega_{0\setminus
z}\varpi_{\frac{D-2}{2}}\mathrm{d}z$ seen from the origin, with
$\varpi_{\frac{D-2}{2}}=\sqrt{1-z^{2}}^{D-3}$. The idea is to separate the
integral over the direction cosine $z=\bm{u}_{0}^{\intercal}\bm{u}$ to the
axis $\bm{u}$ from other angles $\Omega_{0\setminus z}$ to get

$\displaystyle U_{\mathrm{I}}$
$\displaystyle\propto\sum_{n=0}^{\infty}x^{n}\,\int_{-1}^{1}\overline{R^{q}}\;C_{n}^{(\frac{D-2}{2})}(z)\,\varpi_{\frac{D-2}{2}}\mathrm{d}z,$
(45) $\displaystyle\text{with }\overline{R^{q}}$
$\displaystyle=\int_{S\setminus z}R^{-(n-\mu)}\;\mathrm{d}\Omega_{0\setminus
z},$ $\displaystyle\text{and }-q$ $\displaystyle=n-\mu.$

The integral $\overline{R^{q}}$ is an even function in $z$ as also $R$ is
symmetric with regard to a flipped axis $R(\bm{u}_{0})=R(-\bm{u}_{0})$, cf.
eq. (42). Consequently, odd $n$ do not contribute, we re-write $n=2l$ and
$-q=2l-\mu$, as well as

$\displaystyle U_{\mathrm{I}}$
$\displaystyle\propto\sum_{l=0}^{\infty}x^{2l}\,\int_{-1}^{1}\overline{R^{q}}\;C_{2l}^{(\frac{D-2}{2})}(z)\,\varpi_{\frac{D-2}{2}}\,\mathrm{d}z.$
(46)

$U_{\mathrm{I}}=\text{const}$ requires
$\int_{-1}^{1}\overline{R^{q}}\,C_{2l}^{(\nu)}\,\varpi_{\nu}\,\mathrm{d}z=\kappa\,\delta_{l}$
to avoid any degree $l>0$ of variation $x^{2l}\neq 1$. Integers
$-\frac{q}{2}=k\geq 0$ and $\frac{\mu}{2}=m$ make $\overline{R^{q}}$ a degree
$2k$ polynomial $\mathscr{P}_{2k}$,
$\overline{(\bm{u}_{0}^{\intercal}\mathrm{diag}\\{\bm{a}\\}^{-2}\bm{u}_{0})^{k}}$,
cf. eq. (42), or $\sum_{j=0}^{k}a_{2j}\,C_{2j}^{(\nu)}(z)$ . Orthogonality
$\int_{-1}^{1}C_{n}^{(\nu)}\,\,C_{m}^{(\nu)}\,\varpi_{\nu}\,\mathrm{d}z=N_{n}^{2}\,\delta_{nm}$
ensures
$\int_{-1}^{1}\mathscr{P}_{2k}\,C_{2l}^{(\nu)}\,\varpi_{\nu}\,\mathrm{d}z=0$
for $l>0$ within $0\leq 2k<2l$. With $2k=2l-2m$, these limits are $2l\geq
2m>0$ and get most rerstrictive for $l=1>0$, only allowing $m=1$,

$\displaystyle\mu$ $\displaystyle=2m=2,$ $\displaystyle\Rightarrow\sigma$
$\displaystyle=\sqrt{R}^{D}.$ (47)

This gain pattern $\sigma$ perfectly accomplishes a constant sound intensity
potential $U_{\mathrm{I}}=\text{const}$ for ellipsoids.

By contrast, the radial decay of the physical Green’s function of the
Helmholtz equation is proportional to
$|G|\propto\nicefrac{{1}}{{\sqrt{r}^{D-1}}}$, cf. (6), therefore _isotropy_
[15] at $\bm{x}=\bm{0}$ refers to constant directional sound intensity
$\frac{\mathrm{d}I}{\mathrm{d}\Omega_{0}}$, see subsection 3.3. For layers of
constant angular density $\mathrm{d}\Omega_{0}$ instead of constant surface
density $\mathrm{d}S$, the term $\frac{\mathrm{d}I}{\mathrm{d}\Omega_{0}}$
does not contain $\frac{1}{\cos\phi}$ anymore. Only the radiation decay
$\frac{1}{\sqrt{R}^{D-2}}$ and source variance $\sigma^{2}$ remain, so that
the intensity arriving from $\mathrm{d}\Omega_{0}$ around $\bm{u}_{0}$ becomes

$\displaystyle\frac{\mathrm{d}I}{\mathrm{d}\Omega_{0}}=\frac{\sigma^{2}}{{R}^{D-1}}.$
(48)

This reveals that sotropy can be enforced by setting the levels $\sigma$ of
the surrounding sources to

$\displaystyle\sigma=\sqrt{R}^{D-1}.$ (49)

For the 2D case ($D=2$), Figure 6 analyzes the diffuseness and sound energy
density of a 3:2 horizontal ellipse of vertical line sources (a-c). Obviously,
the unity gain $\sigma=1$ in (a) is not ideal for extended diffuseness greater
than $90\%$; yet the diffuseness level already reaches a high level with
$>80\%$ everywhere inside. The isotropy gain $\sigma=\sqrt{R}^{D-1}$ in (b)
greatly improves the extent and appears useful, and $\sigma=\sqrt{R}^{D}$ in
(c) readily accomplishes ideal diffuseness everywhere inside, and the
equivalent results are accomplished by the mode-matching method (d) presented
further below; the sound energy density levels $w$ shown in the figure remain
largely unaffected.

Figure 7 analyzes isotropy by the directional contributions observed at
$\bm{0}$. For the ellipse of the ratio $a_{1}:a_{2}=2:3$, the distant sources
are under-represented by the factor $\sqrt{\nicefrac{{2}}{{3}}}$, i.e.
$-1.76\text{\,}\mathrm{d}\mathrm{B}$, with unity gain, see _ell $\sigma=1$_
curve. The ideal isotropy weight $\sigma=\sqrt{R}^{D-1}$ removes this
shortcoming in the _ell $\sigma=\sqrt{R}$_ curve that is consistently flat,
while the ideal diffuseness gain $\sigma=\sqrt{R}^{D}$ is _not isotropic_ for
the elliptical geometry, see _ell $\sigma=R$_ curve. It over-emphasizes
distant sources and exaggerates their level by the factor
$\sqrt{\nicefrac{{3}}{{2}}}$, i.e. $1.76\text{\,}\mathrm{d}\mathrm{B}$.

For the 3D case ($D=3$), Figure 6 shows in (e-g) a similar analysis for an
ellipsoid with the axis ratio 6:4:3. The cases in (e-g) are convincing already
from (f) on and perfect with (g) and (h). The sound energy density level $w$
would be nearly perfectly flat for unity gain $\sigma=1$ in (e), but these
cases are not so much interesting for their diffuseness: only values slightly
higher than $70\%$ are reached in (e) off center. In the other cases, a bit
more than a $1\text{\,}\mathrm{d}\mathrm{B}$ increase towards the boundaries
has to be accepted.

###  5.2 Towards rectangular cuboids:
$L_{\mathsf{p}}$-norm ellipsoid/superellipsoid

To find out if more rectangular cuboid layouts could work similarly, we re-
define the multi-axial ellipsoid as superellipsoid [95] with its manifold
defined as weighted $L_{\mathsf{p}}$-norm reaching unity, with semi-axes
lengths $\bm{a}=[a_{i}]$ as before,

$\displaystyle\left[\frac{|x_{\mathrm{1,s}}|^{\mathsf{p}}}{|a_{1}|^{\mathsf{p}}}+\frac{|x_{\mathrm{2,s}}|^{\mathsf{p}}}{|a_{2}|^{\mathsf{p}}}+\dots\right]^{\frac{1}{\mathsf{p}}}$
$\displaystyle=1.$ (50)

?figurename? 9: Cross-section: 6:4:3 rectangular cuboid (superellipse
$\mathsf{p}=10$): cut rotated by $0^{\circ}$ or $36.9^{\circ}$ wrt. $x$.

The higher $2<\mathsf{p}\rightarrow\infty$, the more cuboid the surface gets.
With the entries $u_{0,i}$ of the direction vector $\bm{u}_{0}$, we write the
coordinates as $x_{\mathrm{s},i}=R\,u_{0,i}$ and get

$\displaystyle
R=\frac{1}{\left(\sum_{i=1}^{D}\frac{|u_{\mathrm{0,i}}|^{\mathsf{p}}}{|a_{i}|^{\mathsf{p}}}\right)^{\frac{1}{\mathsf{p}}}}.$
(51)

And this effectively shows why it is not as easy as before to find an ideal
weight $\sigma$ as for the ellipsoid: Here,
${R^{\mu-n}}=\left(\sum_{i=1}^{D}\frac{|u_{\mathrm{0,i}}|^{\mathsf{p}}}{|a_{i}|^{\mathsf{p}}}\right)^{\frac{n-\mu}{\mathsf{p}}}$
becomes a $\mathsf{p}k$-degree polynomial with integer $k$ and
$n-\mu=\mathsf{p}k>0$ to remove the root, and even $\mathsf{p}>0$ to avoid the
absolute value. As $\overline{R^{\mu-n}}$ is even, $n=2l$ and
$2l-\mu=\mathsf{p}k$ must hold for integer $l>0$ and constant $\mu$, we get
$\mathsf{p}=2$ and $\mu=\mathsf{p}m$. This only leaves the ellipsoid case
discussed in subsection 5.1 to provide a simple solution for diffuseness with
$\sigma=\sqrt{R}^{D}$.

For the 2D case ($D=2$), Figure 8 analyzes in (a-c) the diffuseness
accomplished with the directional gains $\sigma=1$, $\sigma=\sqrt{R}^{D-1}$,
$\sigma=\sqrt{R}^{D}$ for a 3:2 horizontal superellipse with $\mathsf{p}=10$
and uncorrelated vertical line sources. In fact, the drop of diffuseness in
the corners underlines that there is no simple ideal solution. And yet, ideal
isotropy gains $\sigma=\sqrt{R}^{D-1}$ in (b) produce diffuseness inside that
already exceeds $70\%$; the sound energy density levels $w$ shown in the
figure remain largely unaffected (a-c). The isotropy gain
$\sigma=\sqrt{R}^{D-1}$ in (b) greatly improves the extent, slightly more does
$\sigma=\sqrt{R}^{D}$ in (c), but only the mode-matching solution (d)
presented below also reaches the corners; the sound energy density levels $w$
behave similarly for (e-h) as for the elliptic cases (a-d).

Figure 7 also analyzes isotropy by the corresponding directional contributions
in the superellipse layout, see _rect_ curves. The noticable difference to the
_ell_ curves of the ellipse is around $56^{\circ}$, the angle targeting the
corner of the circumscribed 3:2 rectangle, at which the maximum radius $\max
R$ is reached. There, $\sigma=1$ yields the minimum directional contribution,
and $\sigma=R$ a maximum one, while $\sigma=\sqrt{R}$ is the ideal isotropy
solution $\sigma=\sqrt{R}^{D-1}$ for $D=2$.

The 3D ($D=3$) cuboid case is analyzed in Figure 8(e-l) for the superellipsoid
with the axes ratio 6:4:3. The $xy$ cross-section (e-h) only cuts the vertical
edges, which seem to be uncritical compared to the 2D case. Therefore, most of
the diagrams already look convincing (f-h). For instance the levels
$\sigma=\sqrt{R}^{D}$ in (g) appear to be a perfect choice, and mode-matching
in (h) does not yield further improvement on the $xy$ cut. However the
inclined cross-section (i-l), cf. Figure 9, cuts the corners of the cuboid and
reveals that full coverage with $90\%$ diffuseness is also problematic in the
cases (i-k).

Figure 10 analyzes the isotropy for the ellipsoid in (a-d) by the directional
contributions for different choices of $\sigma$. The isotropy-enforcing levels
$\sigma=R$ accomplish perfectly flat results, while for $\sigma=1$ the most
distant direction is quieter by about $-$6\text{\,}\mathrm{d}\mathrm{B}$$, and
in (c) $\sigma=\sqrt{R}^{3}$, it is louder by
$4\text{\,}\mathrm{d}\mathrm{B}$.

If the resulting diffuseness is insufficient for a given, arbitrary non-
circular or non-spherical layout, despite the promising examples in Figure
8(f) or (g). Only mode-matching (l) proposed in subsection 5.3 and subsection
5.4 enforces ideal diffuseness reaching the corners.

(a) ellipsoid $\sigma=1$

(b) ellipsoid $\sigma=R$

(c) ellipsoid $\sigma=\sqrt{R}^{3}$

(d) ellipsoid $\sigma$ modematch

(e) cuboid $\sigma=1$

(f) cuboid $\sigma=R$

(g) cuboid $\sigma=\sqrt{R}^{3}$

(h) cuboid $\sigma$ modematch

?figurename? 10: Directional intensity
$\mathrm{d}I/\mathrm{d}\Omega=\sigma^{2}/R^{2}$ in 3D mapped as
$(\varphi\cdot|\sin\vartheta|^{0.4}$, $\frac{\pi}{2}-\vartheta)$ on
$\varphi\in[-\pi,\pi]$, $\vartheta\in[0,\pi]$, normalized at
$(\varphi,\vartheta)=(0,\frac{\pi}{2})$, for $\bm{x}=\bm{0}$, cf. eq. (48),
with different choices of directional magnitude $\sigma=\\{1,R,R^{1.5}\\}$ or
mode matched, for a 6:4:3 ellipse (top) or superellipsoid cuboid (bottom) in
3D, i.e. with $D=3$.

###  5.3 Mode matching: non-circular

For $D=2$, Green’s function $\mathcal{G}=-\frac{\ln r}{2\pi}$ of the potential
is with $r=R\sqrt{1-2z\rho+\rho^{2}}$, $z=\cos(\varphi-\varphi_{0})$,
$\rho=\frac{x}{R}$,

$\displaystyle-\mathcal{G}$ $\displaystyle=\frac{\ln
r}{2\pi}=\lim_{\nu\rightarrow
0}\sum_{n=1}^{\infty}\rho^{n}\frac{C_{n}^{(\nu)}(z)}{2\pi}=2\sum_{m=1}^{\infty}\rho^{m}\frac{T_{m}(z)}{2\pi\,m}$
$\displaystyle=\sum_{m=-\infty}^{\infty}\frac{x^{|m|}}{R^{|m|}}\frac{(1-\delta_{m})\,\Phi_{m}(\varphi)\Phi_{m}(\varphi_{0})}{|m|},$
(52)

for a source at polar coordinates $R,\varphi_{0}$ evaluated at $x,\varphi$,
with Chebyshev polynomials $T_{m}(z)$ and circular harmonics

$\displaystyle\Phi_{m}(\varphi)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\begin{cases}\sqrt{2}\sin(m\varphi),&m<0\\\
1,&m=0\\\ \sqrt{2}\cos(m\varphi),&m>0.\end{cases}$ (53)

Superimposing multiple uncorrelated sources with regard to the angle
$\varphi_{0}$ via an integral, their angle-dependent radius $R$ and variance
$\sigma^{2}$, we can represent the total contribution by their coefficient
$a_{m}$

$\displaystyle U_{\mathrm{I}}$
$\displaystyle=2\sum_{m=-\infty}^{\infty}\rho^{m}\,\Phi_{m}(\varphi)\,(1-\delta_{m})\,a_{m}$
(54) $\displaystyle a_{m}$
$\displaystyle=\int_{0}^{2\pi}\frac{\Phi_{m}(\varphi_{0})}{|m|\,R(\varphi_{0})^{|m|}}\;\sigma(\varphi_{0})^{2}\;\mathrm{d}\varphi_{0}.$

For an ideally diffuse intensity potential $U_{\mathrm{I}}=0=\text{const}$, we
try to find variances $\sigma^{2}$ providing $a_{m}=0$ for $m>0$.

Numerically, we can implement the search for a limited $|m|\leq N$ with
reasonably large maximum degree $N$ to avoid as many $|m|\neq 0$ terms getting
synthesized as possible. We may accordingly uniformly sample the source angles
$\varphi_{0,l}=\frac{2\pi}{L}\,l$ and write the integral for $a_{m}$ as
summation

$\displaystyle
a_{m}=\sum_{l=1}^{L}\frac{\Phi_{m}(\varphi_{0,l})}{|m|\,R(\varphi_{0,l})^{|m|}}\,\sigma_{l}^{2}\;\frac{2\pi}{L}.$

It can be defined as matrix vector product equation with the regularization
$|m|\rightarrow\max\\{|m|,1\\}$ to find the source variances
$\bm{\sigma}_{2}=[\sigma_{l}^{2}]$

$\displaystyle\mathbf{i}_{0}$ $\displaystyle=\bm{M}\,\bm{\sigma}_{2},$
$\displaystyle\mathbf{i}_{0}$ $\displaystyle=[\delta_{m}]_{m},$
$\displaystyle\bm{\sigma}_{2}$ $\displaystyle=[\sigma_{l}^{2}]_{l}$
$\displaystyle\bm{M}$
$\displaystyle=\bigg{[}\frac{\Phi_{m}(\varphi_{0,l})}{|m|\,R_{l}^{|m|}}\bigg{]}_{m}^{l},$
$\displaystyle\bm{\sigma}_{2}$ $\displaystyle=\bm{M}^{-1}\mathbf{i}_{0}.$ (55)

For regularization, either the expression $R(\varphi_{0,l})^{-|m|}$ should be
damped towards $|m|\rightarrow\infty$, or $N$ should be limited at some point.
The simulations below still worked fine without additional regularization with
$L=100$ and selecting the modes $-49\leq m\leq 50$ to get a $100\times 100$
square matrix $\bm{M}$.

Figure 6(d) shows the mode-matching solution for the horizontal 3:2 ellipse of
uncorrelated vertical line sources, which is obviously the optimum $\sigma=R$
in (c) obtained analytically, before. For the $\mathsf{p}=10$ superellipse,
Figure 8(d) shows the improvement to ideal diffuseness. While this might be
helpful, the isotropy of the ideal solution might not always be satisfactory:
Figure 7 shows in its _rect $\sigma$ mode-m._-curve that the contribution of
any corner source is largely over-emphasized. Over emphasis gets even worse
for $\mathsf{p}>10$. Nevertheless, on sparse discrete-direction layouts, the
behavior may depend on the detail, i.e., how corners are sampled.

###  5.4 Mode matching: non-spherical

For $D=3$, the Green’s function $\mathcal{G}=\frac{1}{r}$ of the potential is
with $r=R\sqrt{1-2z\rho+\rho^{2}}$, $z=\cos(\bm{u}^{\intercal}\bm{u}_{0})$,
$\rho=\frac{x}{R}$,

$\displaystyle\mathcal{G}$ $\displaystyle=\frac{1}{4\pi
r}=\frac{1}{R}\sum_{n=0}^{\infty}\rho^{n}\frac{C_{n}^{(\frac{1}{2})}(z)}{4\pi}=\frac{1}{R}\sum_{n=0}^{\infty}\rho^{n}\frac{P_{n}(z)}{4\pi}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\frac{x^{n}}{R^{n+1}}\,\frac{Y_{n}^{m}(\bm{u})\,Y_{n}^{m}(\bm{u}_{0})}{2n+1},$
(56)

for a source at $R\,\bm{u}_{0}$ evaluated at $\bm{x}=x\,\bm{u}$, with the
spherical harmonics $Y_{n}^{m}$ at the azimuth
$\varphi=\arctan\frac{u_{2}}{u_{1}}$ and zenith $\vartheta=\arccos u_{3}$
observed in the entries of $\bm{u}=[u_{i}]_{i}$

$\displaystyle Y_{n}^{m}(\bm{u})$
$\displaystyle=N_{n}^{m}\,\Phi_{m}(\varphi)\,P_{n}^{|m|}(\cos\vartheta)$ (57)
$\displaystyle N_{n}^{m}$
$\displaystyle=(-1)^{m}\sqrt{\frac{2n+1}{2}\frac{(n-|m|)!}{(n+|m|)!}}.$

Integrating over uncorrelated sources that are uniformly distributed across
all directions $\bm{u}_{0}$ with their radius $R$ and source variance
$\sigma^{2}$, we get

$\displaystyle U_{\mathrm{I}}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}x^{m}\,Y_{n}^{m}(\bm{u})\,a_{nm}$
(58) $\displaystyle a_{nm}$
$\displaystyle=\int_{\mathbb{S}^{2}}\frac{Y_{n}^{m}(\bm{u}_{0})\,\sigma(\bm{u}_{0})^{2}}{(2n+1)\,R(\bm{u}_{0})^{n+1}}\;\mathrm{d}\bm{u}_{0}.$

We can find suitable variances $\sigma_{l}^{2}$ that ensure $a_{nm}=0$ for
$n>0$ to get a constant intensity potential $U_{\mathrm{I}}=\text{const.}$
Similar as above, we define the integral for $a_{nm}$ as sum for numeric
evaluation

$\displaystyle a_{nm}$
$\displaystyle=\sum_{l=1}^{L}\frac{Y_{n}^{m}(\bm{u}_{0,l})\,\sigma_{l}^{2}}{(2n+1)\,R_{l}^{n+1}}\;\frac{4\pi}{L}.$

And ideal discretization for numerical integration can be found in spherical
$t$-designs, for instance. The squared gains $\sigma_{l}^{2}$ are found by
their modeling in terms of a spherical harmonics expansion with the
coefficients $\gamma_{n}^{m}$ writing the summation as matrix-vector product

$\displaystyle\bm{\sigma}_{2}$ $\displaystyle=\bm{Y}\,\bm{\gamma}_{2},$
$\displaystyle\bm{\sigma}_{2}$ $\displaystyle=[\sigma_{l}^{2}]_{l},$
$\displaystyle\bm{Y}$ $\displaystyle=[Y_{n}^{m}(\bm{u}_{0,l})]_{nm}^{l},$
$\displaystyle\bm{\gamma}_{2}$ $\displaystyle=[\gamma_{n}^{m}]_{nm},$
$\displaystyle\bm{M}$
$\displaystyle=\bigg{[}\frac{Y_{n}^{m}(\bm{u}_{0,l})}{(2n+1)R_{l}^{n+1}}\bigg{]}_{nm}^{l}.$
(59)

The procedure to synthesize only a zeroth-order component is different here,
as $\bm{M}$ has a poor numerical conditioning. Instead, to avoid such
coefficients to get synthesized, the $n=0$ row of $\bm{M}$ is removed
$\bm{M}_{\setminus 0}$ and should vanish when multiplied with solution
variances $\bm{\sigma}_{2}=\bm{Y}\,\bm{\gamma}_{2}$,

$\displaystyle\bm{M}_{\setminus 0}\,\bm{Y}\,\bm{\gamma}_{2}=\bm{0},$ (60)

and we may chose to regularize the result by also minimizing higher-order
coefficients in $\gamma_{n}^{m}$ by a weight $w_{n}$ in $\bm{W}=[w_{n}]_{nm}$

$\displaystyle\tilde{\bm{M}}_{\mathrm{W}\setminus
0}=\begin{bmatrix}\bm{M}_{\setminus 0}\,\bm{Y}\\\
\bm{W}\end{bmatrix}\,\bm{\gamma}_{2}\rightarrow\min.$ (61)

In the analysis in Figure 8(d), (h), (l), the maximum-determinant points [96]
from the website [97] were chosen for $N=49$ and $L=2500$, with the idea to
get a square matrix $\bm{M}$. However conditioning was
$\mathrm{cond}\\{\bm{M}\\}>10^{22}$, so regularization is necessary in the
spherical case, for which the order was limited to $N=17$, despite $49$ would
be possible. For further regularization that avoids unnecessary oscillation in
$\bm{\sigma}_{2}$, a suitably normalized, wiggle-suppressing weight

$\displaystyle
w_{n}={\textstyle\frac{0.015}{\sqrt{4\pi\,\sum_{n}^{\prime}[n^{\prime}(n^{\prime}+1)]^{2}}}}\;n\,(n+1)$
(62)

was used, cp. [98, p.428,Eq.(18.74)], so that of the SVD
$\tilde{\bm{M}}_{\mathrm{W}\setminus
0}=\bm{U}\mathrm{diag}_{L\times(N+1)^{2}}\\{\bm{s}\\}\bm{V}^{\intercal}$ the
vector $\bm{\gamma}_{2}=\bm{v}_{\mathrm{end}}$ of the smallest singular value
$s_{\mathrm{end}}$ could be used as minimizer that yields
$\bm{\sigma}_{2}=\bm{Y}\,\bm{\gamma}_{2}$.

Same as for the elliptic case (2D) but in the ellipsoidal 3D case, the mode-
matching solution in Figure 6(h) brings no further improvement over the
analytic solution $\sigma=\sqrt{R}^{D}$ in (g). An improvement is obtained in
the superellipsoidal case, at least when inspecting the differences in the
cross sections Figure 8 (k) and (l) including the corners of the cuboid.
Particularly the regularized mode-matching solution pushes the $90\%$
diffuseness synthesis outwards to the corners of the cuboid, while sightly
increasing the level towards them, as seen in the more rounded shape of the
$1\text{\,}\mathrm{d}\mathrm{B}$ contour for $\rho\,c^{2}\,w$ in (l).

Isotropy of the mode-matching solution is analyzed in Figure 10(d) for the
ellipsoid case, with not much difference to the analytic $\sigma=\sqrt{R}$.

###  5.5 Hemisphere

As practical realizations of spherical (3D) loudspeaker layouts often only
cover the upper hemisphere with point sources $\beta=1$, a numerical
simulation was carried out to analyze the resulting sound field metrics.

?figurename? 11: Sound energy density (dotted contours) and diffuseness $\psi$
(colors+contours) for 2598 of 5200 upper-hemisphere point sources (Gräf,
Chebyshev-type [76]).

(a) circle of 4 vert. line srcs.

(b) circle of 6 vert. line srcs.

(c) circle of 8 vert. line srcs.

(d) circle of 12 vert. line srcs.

?figurename? 12: Diffuseness $\psi$ (colormap and $90\%$ contour) and level
$w$ normalized to origin (blue dotted $1\text{\,}\mathrm{d}\mathrm{B}$
contour) of a unit circle of $L=\\{4,6,8,12\\}$ equi-angle uncorrelated
vertical line sources (dots), compared against a circle of the radius
$r=\\{\nicefrac{{1}}{{2}},\nicefrac{{2}}{{3}},\nicefrac{{3}}{{4}},\nicefrac{{5}}{{6}}\\}$
(orange red, dotted).

(a) sphere of 6 pt. srcs.

(b) sphere of 12 pt. srcs.

(c) sphere of 24 pt. srcs.

(d) sphere of 70 pt. srcs.

?figurename? 13: Diffuseness $\psi$ (colormap and $90\%$ contour) and level
$w$ normalized to origin (blue dotted $1\text{\,}\mathrm{d}\mathrm{B}$
contour) of a unit sphere of $L=\\{6,12,24,70\\}$ point sources (dots) from
spherical $\\{3,5,7,11\\}$ designs (Gräf, Chebyshev-type [76]) compared
against a radius
$r=\\{\nicefrac{{1}}{{2}},\nicefrac{{2}}{{3}},\nicefrac{{3}}{{4}},\nicefrac{{5}}{{6}}\\}$
(orange red, dotted).

At the horizontal cutting plane that bounds the upper hemisphere, the missing
lower hemisphere causes a loss of up-down symmetry. Where up-down symmetry
cancelled the vertical intensity component before, it becomes non-zero
$I_{\mathrm{z}}<0$ along a horizontal observer position at $z=0$. The
hemispherical integrals

$\displaystyle\rho\,c^{2}\,w$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{1}\,\int_{0}^{2\pi}\frac{\mathrm{d}\cos\vartheta\,\mathrm{d}\varphi}{\|\bm{x}-\bm{u}(\varphi,\vartheta)\|^{2}},$
(63) $\displaystyle\rho\,c\,\bm{I}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{1}\,\int_{0}^{2\pi}\frac{[\bm{x}-\bm{u}(\varphi,\vartheta)]\,\mathrm{d}\cos\vartheta\,\mathrm{d}\varphi}{\|\bm{x}-\bm{u}(\varphi,\vartheta)\|^{3}},$
(64)

are easy to solve at $\bm{x}=\bm{0}$ where the denominator is unity, sound
energy density is unity $\rho\,c^{2}\,w=\int_{0}^{1}\mathrm{d}z=1$, horizontal
sound intensities vanish because of rotational symmetry, and the vertical one
is $\rho\,c\,I_{z}=-\int_{0}^{1}z\,\mathrm{d}z=-\nicefrac{{1}}{{2}}$, yielding
a diffuseness of $\psi=50\%$. For Figure 11, the integrals were solved
numerically via a spherical $t$-design of high value for $t=100$ with $L=5200$
nodes, of which about $\nicefrac{{L}}{{2}}$ lie on the upper hemisphere.
Diffuseness stays roughly $50\%$ on the horizontal cut. At vertical shifts
into the hemisphere, diffuseness exceeds $75\%$ at about half of the height.

##  6 Discrete source layouts: $t$-designs

The integral over the Gegenbauer expansions of eqs. (39) (40) on the unit
sphere can be ideally discretized and holds for a maximum degree $N\leq t$
with discrete sourced arranged in a spherical $t$-design [88]. Knowing that
only the $n=0$ component is present around $x\rightarrow 0$, and gradually
with rising $x$ the next higher expansion powers emerge, the first
uncontrolled component $x^{N+1}C_{N+1}^{(\nu)}(z)$ defines the error. Assuming
the RMS value of $C_{N+1}^{(\nu)}(z)$ in $z$ is sufficiently close to unity,
we get an approximate radial limit of an error-free sweet area

$\displaystyle x^{N+1}$ $\displaystyle<e^{-1}$ $\displaystyle x$
$\displaystyle<e^{-\frac{1}{N+1}}\approx 1-{\textstyle\frac{1}{N+1}}.$ (65)

This implies that the synthesized diffuse field gets larger by increasing the
number of spherical $t$-design nodes which reach a higher values for $t\geq
N$, see also [67].

###  6.1 Discretization study 2D

Uncorrelated vertical line sources evenly arranged in a circle are the ideal
case, and discretized to only $L=\\{4,6,8,12\\}$ evenly spaced sources,
simulation yields the maps in Figure 12 (a-d). Obviously, the relative sweet
area within which diffuseness exceeds $\psi\geq 90\%$ is reduced and scales
with the number of sources. Scaling is $100\%\cdot\frac{L-2}{L}$ as drawn in
orange red. For a radius $r$ in eq. (65) following this contour, $t$ is chosen
to be $t=2N+1$ for the $2N+2$ equi-angle nodes.

###  6.2 Discretization study 3D

For the sphere, spherical $t$-designs [88] are useful discretization schemes.
For the purpose to define the sweet area for diffuse-field synthesis with
discrete sources, the Chebyshev-type quadratures from [99, 100, 76] were
chosen for discretization. The parameter $t=2N+1$ is chosen twice as high as
$N$ plus one, consistently with the $t$ chosen for discretizing the circle
before.

Simulation yields results in Figure 13 that are similar to those for the
circle discretized by a $t=2N+1$ design in Figure 12. Only now for the sphere,
the number of sources needs to be much larger with $L=\\{6,12,24,70\\}$ for
$N=\\{1,2,3,5\\}$ and $t=\\{3,5,7,11\\}$, when compared to $L=\\{4,6,8,12\\}$
for the circle. Hereby, the sweet-area radius estimated in eq. (65) also
nicely models the $\psi=90\%$ diffuseness contour, as indicated by the orange
red, dotted circle. Concerning the sound energy density, the
$1\text{\,}\mathrm{d}\mathrm{B}$ contour stays outside this sweet area until
Figure 13(c) with a discretization by $24$ sources. The slightly non-constant
sound pressure level measured by $\rho\,c^{2}\,w$ appears to be negligible in
practical implementations using $N<4$ or $t$-designs with $t<9$.

##  7 Diffuse sound fields of 2.5D WFS

Wave field synthesis (WFS) [42, 43, 54, 46, 75] with a spherical/circular
loudspeaker system of the radius $R_{\mathrm{s}}$ is able to reproduce
primary, virtual sound sources at multiples of this radius
$R_{0}=m\,R_{\mathrm{s}}$, below the spatial aliasing frequency for which the
loudspeaker density of the practical implementation is higher than half the
acoustic wavelength. One would expect improved diffuse field rendering from a
virtual distance increase. We assume a circular layout of such uncorrelated
virtual sources to represent a diffuse sound field, using the global direction
$\bm{u}_{0}$ from the origin to define their positions
$\bm{x}_{0}=R_{0}\,\bm{u}_{0}$.

?figurename? 14: 2.5D WFS scenario with uncorrelated virtual primary sources
at the circular radius $R_{0}$, of which one is at $\bm{x}_{0}$ and reproduced
by secondary sources at the radius $R_{\mathrm{s}}$. Reproduction at
$\bm{x}_{1}$ is dominated by the stationary-phase point
$\bm{x}_{\mathrm{s}}^{*}$ and a reference contour through
$\bm{x}_{1,\mathrm{ref}}$.

A similar aspect as discussed above is relevant here: When the space
dimensions $D^{\prime}$ occupied by the loudspeaker layout are fewer than the
actual dimensions $D$ of the space, then primary sound sources are reproduced
with erroneous decay $\frac{1}{\sqrt{r}^{D^{\prime}-1}}$ that does not match
$\frac{1}{\sqrt{r}^{D-1}}$. 2D WFS would be the ideal choice in theory,
requiring a horizontal circle of vertical line sources, but the affordable
practical choice usually is a circle of point sources for 2.5D WFS, cf. [43].
While a plane-wave sound field has a constant amplitude, its 2.5D WFS
reproduction exhibits some amplitude decay, cf. [46, 79]. The introductory
example Figure 1(c) crudely approximated it by the amplitude decay of a
horizontal line source.

More precisely, as sketched in Figure 14, the part $r_{1}^{*}$ of the distance
$r=\|\bm{x}_{1}-\bm{x}_{0}\|=r_{0}^{*}\pm r_{1}^{*}$ between observer at
$\bm{x}_{1}=R_{1}\,\bm{u}_{1}$ and primary source at
$\bm{x}_{0}=R_{0}\,\bm{u}_{0}$ often defines the 2.5D WFS local amplitude
decay more than the distance $r$ to the primary source. In particular,
$r_{1}^{*}$ is the distance $r_{1}^{*}=\|\bm{x}_{\mathrm{s}}^{*}-\bm{x}_{1}\|$
to the stationary-phase source that is located at $\bm{x}_{\mathrm{s}}^{*}$,
or for a focused source $\bm{x}_{\mathrm{s,foc}}^{*}$, see [43, 75]. This
source dominates the amplitude received at $\bm{x}_{1}$ for the virtual source
direction $\bm{u}_{0}$, and its amplitude can be corrected for observation at
a reference distance $r_{1,\mathrm{ref}}^{*}$ along the direction $\bm{u}$.
Doing so for all pairs of stationary-phase points $\bm{x}_{\mathrm{s}}^{*}$
and associated reference points defines a reference contour of correct
synthesis [47]. We choose a reference circle of the radius $R_{0}$ around the
primary source, through the origin $\bm{0}$, here. To lie on this contour, the
distance $r$ must be $R_{0}$.

Appendix D outlines the underlying calculations and yields the stationary-
phase approximated 2.5D WFS of a Green’s function at
$\bm{x}_{0}=R_{0}\,\bm{u}_{0}$ in eq. (85),

$\displaystyle
G_{\mathrm{2.5DWFS}}=\sqrt{\frac{R_{0}-r_{0}^{*}}{r-r_{0}^{*}}}\sqrt{\frac{R_{0}}{r}}\;e^{\mp\mathrm{i}k\,r},$
(66)

where $r_{0}^{*}$ yields with the constraint
$\|\bm{x}_{\mathrm{s}}^{*}\|^{2}=R_{\mathrm{s}}^{2}$ and its coordinates
$\bm{x}_{\mathrm{s}}^{*}=R_{0}\bm{u}_{0}\mp r_{0}^{*}\,\bm{u}$,

$\displaystyle r_{0}^{*}$
$\displaystyle=-R_{0}\,\bm{u}_{0}^{\intercal}\bm{u}\mp\sqrt{R_{\mathrm{s}}^{2}-R_{0}^{2}\,[1-(\bm{u}_{0}^{\intercal}\bm{u})^{2}]}.$
(67)

We apply the integrals of eqs. (28) and (31) to superimpose such uncorrelated
2.5D WFS sources. The resulting expressions are too elaborate for analytic
integration, so our investigation relies on numerical integration over the
angular integration variable $\mathrm{d}\Omega_{0}(\bm{u}_{0})$, where
$\bm{u}_{0}$ defines the primary sources $\bm{x}_{0}=R_{0}\bm{u}_{0}$,
$R_{\mathrm{s}}=1$, $R_{0}=m\,R_{\mathrm{s}}$,
$\bm{u}=\frac{\bm{x}_{0}-\bm{x}_{1}}{r}$, and eq. (67) defines the magnitude
square of eq. (66), as used in $\mathcal{G}\propto|G_{\mathrm{2.5DWFS}}|^{2}$,
using the integrals derived for $\rho\,c^{2}\,w$ eq. (20) and
$\rho\,c\,\bm{I}$ eq. (21), so that

$\displaystyle\rho\,c^{2}\,w$
$\displaystyle=\frac{1}{L}\sum_{l=1}^{L}|G_{\mathrm{2.5DWFS},l}|^{2},$ (68)
$\displaystyle\rho\,c\,\bm{I}$
$\displaystyle=\frac{1}{L}\sum_{l=1}^{L}\bm{u}_{l}\,|G_{\mathrm{2.5DWFS},l}|^{2},$
(69)

and hereby $\psi=1-\frac{\rho\,c\,\|\bm{I}\|}{\rho\,c^{2}\,w}$. Figure 15
displays the resulting sound energy density (dotted contour) and diffuseness
(color/solid contour) for varying relative size $m$ of an uncorrelated virtual
source circle, either non-focused or focused. Non-focused sources are
equivalent to uncorrelated point-source loudspeakers without WFS when $m=1$,
and to the typical 2.5D WFS of virtual plane waves with $m\rightarrow\infty$,
cf. Appendix D.

It has been a common belief that 2.5D WFS with plane-wave sound objects has
benefits in producing diffuse sound fields. Listening experiments by Melchior
et al. [74] and the crude approximation Figure 1(c) provide reason for doubt
that the map in Figure 15 now clearly confirms: focused sources with $m<1$
synthesize a non-diffuse zone $|x|>m$ and can be disregarded, at small
displacements $|x|\leq 0.2$, diffuseness is still $\psi>90\%$ for any $m\geq
1$; elsewhere, any $m>1$ even reduces diffuseness as its $70\%$, $55\%$,
$33\%$ contours in Figure 15 shrink. Conversely, sound pressure levels get
more constant by the expanded $0.1\text{\,}\mathrm{d}\mathrm{B}$,
$1\text{\,}\mathrm{d}\mathrm{B}$, $3\text{\,}\mathrm{d}\mathrm{B}$ contours
for $m\rightarrow\infty$ of non-focused sources, and even more so for
$m\rightarrow 1$ with focused sources.

?figurename? 15: Diffuseness $\psi$ (colors/solid contours) at off-center
positions $x$ in 2.5D WFS of a circle of uncorrelated virtual sources
(focused/non-focused) at different radii $R_{0}=mR_{\mathrm{s}}$ and sound
energy density $w$ (dotted).  ?figurename? 16: Interaural coherence (IC) and
interaural level difference (ILD) evaluated for sound fields with variable
diffuseness $\psi$. The evaluation uses a variably-weighted superposition of a
2D/3D diffuse sound field and a single-DOA soundfield from a horizontal,
fromtal azimuth angle. Gray areas indicate the JND to a diffuse-field
reference.

At high frequencies, sound field synthesis exhibits spatial aliasing that
makes the observed loudspeaker phases appear random, so that the metrics for
$m=1$ and non-focused sources are expected to apply.

In summary, the effort of driving a ring of point-source loudspeakers via 2.5D
WFS to reproduce uncorrelated virtual sources at other radii is surprisingly
unsuccessful and even deteriorates diffuseness. While wave fronts of the
virtual sources are successfully reproduced, their $\nicefrac{{1}}{{r}}$ or
$\nicefrac{{1}}{{\sqrt{r}}}$ magnitude reproduces as
$\nicefrac{{1}}{{(\sqrt{r}\sqrt{r-r_{0}^{*}})}}$, instead. Loudspeakers should
be vertical line sources to enable 2D WFS that accurately reproduces the
target decay $\nicefrac{{1}}{{\sqrt{r}}}$.

##  8 Perceptual Aspects

The numerical simulations and analytical derivations could show which
constellations favorably achieve a maximum of diffuseness across the entire
audience area. Perceptually, this is equivalent to maintaining minimum the
interaural level difference (ILD) and interaural coherence (IC) across this
area. This section demonstrates the relation between diffuseness in the range
$0\%\leq\psi\leq 100\%$ and the resulting ILD/IC values and establishes the
relation to known perceptual thresholds [72, 101, 102]. To this end, the ear
signals of a uniformly diffuse sound field is simulated and superimposed with
a unidirectional sound field (single non-diffuse direction from horizontal
plane and frontal hemisphere), as numerically analyzed in Figure 16. The
interaural cues are computed from Neumann KU100 HRTFs [103], averaging across
gammatone frequency bands of one equivalent rectangular bandwidth (ERB)
between 200 Hz and 12.8 kHz. While the low-end JND $0.5$ of IC is uncritical
in most cases and not surpassed for diffuseness of values of $\psi>40\%$ for
all directions, the ILD JND of around $1\text{\,}\mathrm{d}\mathrm{B}$ is the
more critical cue. A perceptually irrelevant ILD requires a diffuseness as
high as $\psi>80\%$ when the unidirectional component is lateral. No ILD is
caused by unidirectional content from the median plane, where diffuseness
values $\psi>40\%$ are still sufficiently high for low IC.

It is a lucky practical coincidence that while the vertical intensity
component of a hemispherical loudspeaker layout drastically reduces the
diffuseness to $\psi\approx 50\%$, cf. Figure 11, it will still be providing
an acceptably small IC, and ILD will stay limited as long as intensity is not
inclined by more than $8^{\circ}$ with regard to the median plane, either
because of a listener who rolls the head sideways, or by a slightly inclined
intensity, cf. Figure 16.

The most suitable solutions providing $100\%$ diffuseness rendering
(circle/sphere of $\nicefrac{{1}}{{\sqrt{r}^{D-1}}}$) seem to be incapable of
supplying perfectly constant sound pressure level everywhere inside. However,
it seems from the respective simulations above that restricting the playback
area to about $r<80\%\,R$ in size will keep the level constant enough, with
variations below $1\text{\,}\mathrm{d}\mathrm{B}$, which resembles relevant
loudness JNDs [104, 105] or lies below.

Unless working with 2D WFS and plane-wave virtual sources, isotropy will
typically also not be reached everywhere, but it is at least apparent that the
more spherical and full-dimensional the layout becomes, and the larger in
loudspeaker count, the easier it is to maintain isotropy around the center
location, see also [72].

##  9 Conclusion

In this article we discussed theoretical requirements and limitations of
diffuse sound fields synthesized with uncorrelated sources in the free field.
The findings are of practical relevance when attempting to produce extended
diffuse fields with loudspeakers, either in the anechoic chamber or in typical
spatial audio rendering scenarios with loudspeakers or virtual sources.

We could prove there being optimal layouts that can synthesize vanishing sound
intensity in the entire field enclosed, if their surface normal is
rotation/shift invariant. This was accomplished by statistical expectation,
assuming a continuous layer of uncorrelated surrounding sources of equal
variance, relating sound intensity synthesized by the Green’s function of the
Helmholtz equation to Gauß’ divergence theorem and Newton’s spherical shell
theorem for the Green’s function of the potential equation. These are optimal
layouts:

  * •

A uniform spherical layer (3D) of uncorrelated point sources is able to
produce vanishing sound intensity, i.e. $100\%$ diffuseness, everywhere
inside.

  * •

A circular layout of uncorrelated (vertical line) sources produces $100\%$
diffuseness everywhere inside, in 2D. Embedded in 3D, it must be of infinite
extent in the axis perpendicular to the circle, forming a cylindrical source
distribution. Sources need to be uncorrelated along the circle, not
necessarily along the cylinder axis.

  * •

A pair of opposing (plane) sources produces $100\%$ diffuseness everywhere
inside in 1D. Embedded in 3D, they are two uncorrelated infinite-plane source
distributions, whose two sides need to be uncorrelated, not necessarily the
sources along either plane.

For these optimal source layouts, we could prove that energy density or sound
pressure level will only be constant inside the 1D layout of parallel planes.
A distance decay modified from the physical one
$\nicefrac{{1}}{{\sqrt{r}^{D-1}}}$ to $\nicefrac{{1}}{{r^{\beta}}}$ was
employed in the source integrals and solved by hypergeometric functions to
show that $\nicefrac{{1}}{{\sqrt{r}^{D-1}}}$ of physical sources yields
$100\%$ diffuseness, while $\nicefrac{{1}}{{\sqrt{r}^{D-2}}}$ or
$\nicefrac{{1}}{{r^{0}}}$ yields constant sound pressure level. Our formally
strict proof exploited $\nicefrac{{1}}{{r^{\beta}}}$ as generating function of
the orthogonal Gegenbauer polynomials, and in particular the orthogonality
between the constant potential of the zeroth-order polynomial to higher-order
polynomials, given the integration weight $\varpi_{\beta}$ of the
$D$-dimensional geometry.

We could moreover prove that it is impossible to construct a diffuse sound
field that is isotropic regardless of the observer position within a set of
uncorrelated sources at finite distance, with fixed source amplitude density
$\sigma$. Isotropy requires a constant ratio $\sigma^{2}/\cos\phi$ of squared
source amplitude density $\sigma^{2}$ and direction cosine $\cos\phi$ between
the sound ray observed and the surface normal at location of its source. While
$\sigma^{2}$ can be chosen to provide isotropy at a specific observer
position, the direction cosines $\cos\phi$ are always position-dependent.

Non-circular/non-spherical source layouts exhibit a varying distance $R$ to
the center of the layout. For elliptic 2D or ellipsoidal 3D layouts of
directionally uniform source density, we could show that driving their
uncorrelated sources by a gain $\sigma=\sqrt{R}^{D}$ produces $100\%$
diffuseness everywhere within. This choice differs from
$\sigma=\sqrt{R}^{D-1}$ that provides isotropy to a central observer.

For more arbitrary source layouts, we proposed a mode matching approach to
find $\sigma$ on the example of a rectangular/cuboid superellipse to produce
$100\%$ diffuseness inside, based on circular/spherical harmonics. The mode-
matching solutions of these cases differ from $\sqrt{R}^{D}$, even more so
from $\sigma=\sqrt{R}^{D-1}$, which still provides isotropy to a central
observer.

Hemispherical layouts are practically relevant in 3D audio and were shown to
only reach $50\%$ diffuseness, due to the intensity component leaving the
hemisphere. However, there will not be a tangential intensity component along
the terminating plane of the hemisphere.

For optimal distance decay $\nicefrac{{1}}{{\sqrt{r}^{D-1}}}$, we could show
that circular/spherical $t$-designs as discrete source layouts yield sweet
areas of $>90\%$ diffuseness within $r<\frac{N}{N+1}$, when choosing $t=2N+1$
for both cylindric and spherical layouts, using Gegenbauer polynomials.

Scaling the radius of the source layout while using only a limited central
area is obviously always a stabilizing option. This may succeed virtually by
array processing or soundfield synthesis. It fails in the particular practical
example of 2.5D WFS and a physical point-source loudspeaker layout:
Uncorrelated virtual plane waves approximated by virtual sources at the radius
$R_{0}\rightarrow\infty$ render limited diffuseness that is not better than
what uncorrelated physical sources $R_{0}\rightarrow R_{\mathrm{s}}$ at the
loudspeaker distance accomplish. Simulations of various scalings
$R_{0}=m\,R_{\mathrm{s}}$ indicated that the 2.5D WFS technique has no
benefits. Alternatively, theory suggests horizontal-only 2D WFS to work. It
employs vertical line-source loudspeakers. While these are also optimal for
any other 2D spatial audio rendering method, two extra benefits are accessible
via 2D WFS when driven by uncorrelated virtual plane-wave sources: (i) a flat
sound pressure level and (ii) 2D isotropy everywhere inside, cf. e.g. [67,
106].

Finally, the relation to relevant interaural measures as spatial auditory cues
and their JNDs was established. A simulation example showed that high
diffuseness $\psi$ enforces low interaural coherence (IC) and level difference
(ILD), both important for diffuse envelopment. It may therefore be beneficial
to work with $\nicefrac{{1}}{{\sqrt{r}^{D-1}}}$ sources for optimal
diffuseness and accept the synthesis of an imperfect but reasonably flat sound
pressure level varying by only $1$ or $2\text{\,}\mathrm{d}\mathrm{B}$. A
strict diffuseness criterion of $\psi\geq 80\%$ or $90\%$ is reasonable under
the presence of a lateral sounds, which easily trigger noticeable ILDs. By
contrast, vertical and hereby non-interaural directional content produced by
upper hemispherical layouts still appears acceptable, despite imposing an
upper limit $\psi\leq 50\%$ to diffuseness.

##  10 Acknowledgments

We gratefully received funding from the Austrian Science Fund (FWF): P 35254-N
(Envelopment in Immersive Sound Reinforcement, EnImSo).

##  11 Data Availability Statement

The jupyter lab python source code to create all sound field graphs shown in
this article is available online [107], and a brief paper on the WFS basics
that helped to develop the equations in section 7 is available online [106].

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##  ?appendixname? A High-frequency Green’s function

Knowing that the free-field Green’s function only depends on
$r=\sqrt{\sum_{i=1}^{D}x_{i}}$, we may define it via the Laplacian in terms of
the radius $r$, only

$\displaystyle\left[\frac{\partial^{2}}{\partial
r^{2}}+\frac{D-1}{r}\frac{\partial}{\partial
r}+k^{2}\right]G(r)=-\delta(\bm{x}).$ (70)

The corresponding homogeneous equation simplifies by multiplication with
$r^{2}$ and substition $x=kr$ to

$\displaystyle\left[x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+(D-1)x\frac{\mathrm{d}}{\mathrm{d}x}+x^{2}\right]y=0.$
(71)

According to Sommerfeld [90, Ch.V(A.IV,C)], it is useful to transform it into
a Bessel differential equation
$\left[x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+x\frac{\mathrm{d}}{\mathrm{d}x}+(x^{2}-\nu^{2})\right]v=0$,
which succeeds by a suitable choice of $\nu$ in $y=x^{-\nu}v$,

$\displaystyle\bigg{[}x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+(-2\nu+D-1)x\frac{\mathrm{d}}{\mathrm{d}x}+x^{2}$
$\displaystyle\qquad\quad-\nu(-\nu-1+D-1)\bigg{]}v=0,$

requiring the factor of $x\frac{\mathrm{d}}{\mathrm{d}x}$ to become unity
$D=2\nu+2$

$\displaystyle\left[x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+x\frac{\mathrm{d}}{\mathrm{d}x}+x^{2}-\nu^{2}\right]v=0,$

so $\nu=\frac{D}{2}-1$, making a suitably scaled and second-kind Hankel
function [85, 10.2.E6] a solution

$\displaystyle G=a\,x^{-\nu}\,H_{\nu}^{(2)}(x)$ (72)

that fulfills Sommerfeld’s radiation condition [90, §28]. The scalar $a$ is
found by letting $k\rightarrow 0$ and integrating eq. (6) over a small volume,
with $\mathcal{S}_{D-1}=\frac{2\,\pi^{\frac{D}{2}}}{\Gamma(\frac{D}{2})}$,

$\displaystyle\lim_{k,V\rightarrow
0}\int_{V}(\bigtriangleup+k^{2})G\,\mathrm{d}V=-\int\delta\,\mathrm{d}V$
$\displaystyle=-1$ $\displaystyle\lim_{k,V\rightarrow 0}\int_{V}\bigtriangleup
G\,\mathrm{d}V=\lim_{k,r\rightarrow 0}r^{D-1}\mathcal{S}_{D-1}\frac{\partial
G}{\partial r}$ $\displaystyle=-1,$ $\displaystyle\lim_{k,x\rightarrow
0}r^{D-1}\mathcal{S}_{D-1}\frac{k\,\partial G}{\partial x}$ $\displaystyle=-1$
$\displaystyle\lim_{k,x\rightarrow 0}xr^{D-2}\mathcal{S}_{D-1}\frac{\partial
G}{\partial x}$ $\displaystyle=-1$ $\displaystyle\lim_{k,x\rightarrow
0}xr^{2\nu}\frac{2\pi^{\nu+1}}{\Gamma(\nu+1)}\frac{\partial G}{\partial x}$
$\displaystyle=-1.$

The small-argument approximation of the Hankel function in
$G=\frac{a}{x^{\nu}}H_{\nu}^{(2)}(x)$ yields, cf. [85, 10.7.E7],

$\displaystyle\lim_{x\rightarrow
0}H_{\nu}^{(2)}(x)=\frac{\mathrm{i}\Gamma(\nu)}{\pi}\frac{2^{\nu}}{x^{\nu}},$
$\displaystyle\lim_{x\rightarrow
0}G=a\frac{\mathrm{i}\Gamma(\nu)}{\pi}\frac{2^{\nu}}{x^{2\nu}},$
$\displaystyle\lim_{x\rightarrow 0}\frac{\partial G}{\partial
x}=-2\mathrm{i}a\frac{\nu\Gamma(\nu)}{\pi}\frac{2^{\nu}}{x^{2\nu+1}},$

so that its scaling is found to be

$\displaystyle
xr^{2\nu}\frac{2\pi^{\nu+1}}{\Gamma(\nu+1)}2\mathrm{i}a\frac{\nu\Gamma(\nu)}{\pi}\frac{2^{\nu}}{x^{2\nu+1}}=1$
$\displaystyle(2\pi)^{\nu}4\mathrm{i}a\frac{r^{2\nu}}{x^{2\nu}}=1$
$\displaystyle a=-\frac{\mathrm{i}}{4}\left(\frac{k^{2}}{2\pi}\right)^{\nu}.$

This yields together with $x^{-\nu}$

$\displaystyle G$
$\displaystyle=-\frac{\mathrm{i}}{4}\left(\frac{k}{2\pi\,r}\right)^{\frac{D}{2}-1}\,H_{\frac{D}{2}-1}^{(2)}(kr),$
(73)

and the far-field approximation [85, 10.2.E6]

$\displaystyle\lim_{r\rightarrow\infty}H_{\nu}^{(2)}(z)$
$\displaystyle=\sqrt{\frac{2}{\pi\,z}}e^{-\mathrm{i}(z-\frac{(2\nu+1)\pi}{4})}$

yields the far-field Green’s function eq. (7) in section 2

$\displaystyle\lim_{r\rightarrow\infty}G$
$\displaystyle=\frac{1}{4}\left(\frac{k}{2\pi\,r}\right)^{\frac{D}{2}-1}\,\sqrt{\frac{2}{\pi\,kr}}e^{-\mathrm{i}kr+\mathrm{i}\frac{(D-3)\pi}{4}}$
$\displaystyle=\sqrt{\frac{2\pi
r}{\mathrm{i}k}}^{\frac{3-D}{2}}\,\frac{e^{-\mathrm{i}kr}}{4\pi\,r}.$ (74)

##  ?appendixname? B Reduced decay for infinite extent

The reduced distance decay distributed sources exhibit in terms of a parallel
distance $r$ with regard to $E$ infinitely extended excess dimensions is found
by first regarding a negative power of $r$

$\displaystyle
r^{-2\alpha}=\left[\sum_{j=1}^{D^{\prime}}(x_{i}-x_{\mathrm{s},i})^{2}+\sum_{i=D^{\prime}+1}^{D}x_{i}^{2}\right]^{-\alpha}.$
(75)

We differentiate $r$ it in all the $E$ excess dimensions

$\displaystyle\left[\prod_{i=1}^{E}\frac{\partial}{\partial
x_{D^{\prime}+i}}\right]\frac{1}{r^{2\alpha}}=\frac{(-2)^{E}(\alpha+E-1)!}{(\alpha-1)!}\frac{\prod_{i=1}^{E}x_{D^{\prime}+i}}{r^{2\alpha+E}}.$

To make the right hand side $\frac{1}{r^{2\alpha+E}}$ match
$\mathcal{G}^{\prime}_{D}\propto\frac{1}{r^{D-1}}$, its exponent needs to be
$2\alpha+E=D-1$. Because of the $E$ derivatives, integrating over the $E$
excess dimensions is trivial, just removes the differentials, and yields
$\frac{1}{r^{2\alpha}}$ on the left, so that the decay exponent
correspondingly reduces from $2\alpha+E=D-1$ to $2\alpha=D-E-1=D^{\prime}-1$.
This relation is used in subsection 3.1 eq. (23).

##  ?appendixname? C Modified decay exponent $\beta$

section 4 uses integrals to find $\rho\,c^{2}\,w$ and
$\rho\,c\,I_{\mathrm{x}}$ for an observer shifted along the first coordinate
$x_{1}=x$ within a spherical shell of $D$ dimensions, involving the Euclidean
distance $r=\|\bm{x}-\bm{x}_{\mathrm{s}}\|=\sqrt{1-2\cos\phi\,x+x^{2}}$, where
the direction cosine between source and observer location along the axis
$x_{1}$ is summarized as $z=\cos\phi$. Integration over the angular element
$\frac{\mathrm{d}\Omega_{0}}{\mathcal{S}_{D-1}}=\varpi\,\mathrm{d}z$ with
$-1\leq z\leq 1$ from the center of the sphere uses the weight
$\varpi=\frac{\mathcal{S}_{D-2}}{\mathcal{S}_{D-1}}\sqrt{1-z^{2}}^{D-3}$, cf.
[91]. The axisymmetric integral around the first dimension of the shift $x$
remains over the decay $\frac{1}{{r}^{2\beta}}$ for sound energy density and
additionally over the directional factor $\frac{z-x}{r}$ in
$\frac{z-x}{r^{2\beta+1}}$ for the intensity $\rho\,c\,I_{\mathrm{x}}$ along
$x_{1}$,

$\displaystyle\rho\,c^{2}\,w$
$\displaystyle=\frac{1}{\tilde{N}}\int_{-1}^{1}\frac{\sqrt{1-z^{2}}^{D-3}\mathrm{d}z}{\sqrt{1-2zx+x^{2}}^{2\beta}},$
(76) $\displaystyle\rho\,c\,I_{\mathrm{x}}$
$\displaystyle=\frac{1}{\tilde{N}}\int_{-1}^{1}\frac{(z-x)\,\sqrt{1-z^{2}}^{D-3}\mathrm{d}z}{\sqrt{1-2xz+x^{2}}^{2\beta+1}}.$
(77)

These two integrals are better solved analytically than numerically to obtain
numerically stable results. We substitute $z=2t-1$ and get

$\displaystyle\rho\,c^{2}\,w\propto$
$\displaystyle\frac{1}{(1+x)^{2\beta}}\int_{0}^{1}\frac{t^{\frac{D-3}{2}}(1-t)^{\frac{D-3}{2}}\mathrm{d}t}{[1-\frac{4x}{(1+x)^{2}}t]^{\frac{2\beta}{2}}},$
(78) $\displaystyle\rho\,c\,I_{\mathrm{x}}\propto$
$\displaystyle\frac{1}{(1+x)^{2\beta}}\int_{0}^{1}\frac{t^{\frac{D-3}{2}}(1-t)^{\frac{D-3}{2}}\mathrm{d}t}{[1-\frac{4x}{(1+x)^{2}}t]^{\frac{2\beta+1}{2}}}$
(79)
$\displaystyle-\frac{2}{(1+x)^{2\beta+1}}\int_{0}^{1}\frac{t^{\frac{D-1}{2}}(1-t)^{\frac{D-3}{2}}\mathrm{d}t}{[1-\frac{4x}{(1+x)^{2}}t]^{\frac{2\beta+1}{2}}},$

so that the integral related to $F(a,b;c;z)$ [85, 15.2.E1]

$\displaystyle F(a,b;c;z)$
$\displaystyle=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}\frac{t^{b-1}\,(1-t)^{c-b-1}}{(1-zt)^{a}}\;\mathrm{d}t,$

helps to find the expressions of eqs. (34) and (35), which used
$a=\frac{2\beta}{2}$ or $a=\frac{2\beta+1}{2}$, $b=\frac{D-3}{3}+1$ or
$b=\frac{D-1}{2}+1$, and $c=\frac{D-3}{2}+b+1$. The known value $F(a,b;c;0)=1$
supporting the normalization of $\rho\,c^{2}\,w(0)=1$ without additional
factors, and symmetry-induced $\rho\,c\,I_{\mathrm{x}}(0)=0$ at $x=0$ were
used to verify scaling. The above equations also yield eq. (36) for
diffuseness $\psi$.

##  ?appendixname? D 2.5D WFS

?figurename? 17: Sound energy density (color/contours) and sound intensity
(arrows) of stationary-phase approximated 2.5D WFS for non-focused (left
$x_{0}=1,\infty$) and focused (right $x_{0}=\infty,1$) virtual source,
$0\text{\,}\mathrm{d}\mathrm{B}$ reference contour is green.

Formal background on wave field synthesis (WFS) as analyzed here was prepared
in [106]. WFS employs the Kirchhoff-Helmholtz integral, where
$r_{1}=\|\bm{x}_{\mathrm{s}}-\bm{x}_{1}\|$ and
$r_{0}=\|\bm{x}_{\mathrm{s}}-\bm{x}_{0}\|$ are: the distance $r_{1}$ between a
secondary source $\bm{x}_{\mathrm{s}}$ of the convex, closed, linear contour