Perfectly diffuse sound fields play an important role in architectural acoustics and there are established theoretical characterizations of perfect diffuseness. Although sound fields in real rooms are diffuse to some extent, they are not perfectly diffuse, and therefore theories are required to describe pseudo-perfectly diffuse sound fields. Here, we aim to spatially characterize pseudo-perfect diffuseness via directional characterization of that, finite-degree spherical harmonic diffuseness. Our results show that finite-degree diffuse sound fields yield local spatial diffuseness, suggesting that spatial pseudo-perfect diffuseness is characterized using the effective radius of diffuseness.

## 1. Introduction

Perfectly diffuse sound fields are theoretically characterized by using their directional and spatial properties. For example, a random-wave model^{1,2} composed of an infinite number of isotropic and incoherent plane waves simulates perfectly diffuse sound fields based on the directionally random properties. The random-wave model can also be defined by using random coefficients of spherical wave functions.^{3,4} The random-wave model also enables spatial characterization of the perfect diffuseness. The spatial perfect diffuseness is defined by exploiting the correspondence of the two-point covariance of sound pressures at a pair of positions with the imaginary part of the Green's function of the field between the positions^{5} or with the sinc function^{2} (the imaginary part of the Green's function in a free sound field). The above models and characterizations establish perfect diffuseness theoretically and provide the foundations for architectural acoustics such as the measurements of absorption coefficients using reverberation rooms.^{6}

In contrast, sound fields in real environments never achieve perfect diffuseness; therefore, previous studies in room acoustics have attempted to evaluate the extent of diffuseness in a room.^{7–10} Although they are not perfectly diffuse, real sound fields are assumed to be diffuse to some extent. This therefore requires another theory describing pseudo-perfect diffuseness.

We have recently characterized pseudo-perfect diffuseness by discretizing or truncating the directional properties of perfect diffuseness.^{11,12} In these studies, discrete random-wave models composed of a finite number of random plane waves simulate pseudo-perfectly diffuse sound fields, and finite-degree spherical harmonic diffuseness, which is defined using random coefficients of truncated spherical harmonics, characterizes pseudo-perfect diffuseness. Nonetheless, spatial characterization of pseudo-perfect diffuseness remain unclear.

Here, we show that pseudo-perfect diffuseness can be defined spatially as *local* diffuseness, spatial diffuseness valid for a restricted area, via finite-degree spherical harmonic diffuseness.

Section 2 preliminarily introduces models of perfectly diffuse sound fields, defines spatially characterized perfect diffuseness, and introduces finite-degree diffuseness, directionally characterized pseudo-perfect diffuseness. Section 3 then shows that spatial diffuseness is locally valid in sound fields with finite-degree diffuseness and suggests defining spatial pseudo-perfect diffuseness as local diffuseness with an effective radius.

## 2. Preliminaries

### 2.1 Models of perfectly diffuse sound fields

There are two established methods of modeling perfectly diffuse sound fields, as shown below.

^{1}which is rigorously formulated as follows:

^{5}

^{,}

*k*[rad/m] and a position $r$. $ A ( k , \theta , \varphi )$ is a complex-valued function representing plane wave amplitudes arriving from a direction indicated by a unit vector,

^{5}for $ k \u2260 0$:

*repeatedly synthesized*sound fields), $ ( \xb7 ) \xaf$ denotes taking a complex conjugate, and $ \delta S 2 ( \xb7 )$ denotes the Dirac delta on the unit sphere in a 3-dimensional space. The last two equations respectively represent the pseudo covariance and the covariance of the distribution of plane wave amplitudes, although the Dirac-delta covariance may be difficult to physically understand as also indicated in an optical literature.

^{13}Notably, the above (integral) formulation is more rigorous than a formulation including the infinite sum of plane waves,

^{14}where the arrival directions of the plane waves are provided by words rather than equations. The latter formulation is not employed here also because the question remains whether it is equivalent to the following spherical harmonic formulation of perfectly diffuse sound fields. We hereafter assume $ k \u2260 0$ as well and omit the argument

*k*without notice.

^{3,4}

*l*th-order spherical Bessel function of the first kind, $ Y l , m ( \xb7 )$ denotes the real-valued spherical harmonics of the

*l*th degree and the

*m*th order, with $ \u2211 l , m : = \u2211 l = 0 \u221e \u2211 m = \u2212 l l$. The coefficients $ a l , m$ are assumed to have Gaussian distributions with the following statistical properties:

^{3,4,11}

### 2.2 Spatially characterized perfect diffuseness

^{2,5}

^{15}of spherical harmonics and spherical Bessel functions allow the last transformation. Notably, the sinc function corresponds to the imaginary part of a Green's function in free sound fields. In fact, Weaver and Lobkis have shown that the two-point covariance in diffuse sound fields includes the imaginary part of the Green's function between the two positions and suggested defining the spatial diffuseness by the correspondence between the covariance and the Green's function.

^{5}The sinc covariance derived above thus represents a special case of the diffuseness in a free sound field.

### 2.3 Directionally characterized pseudo-perfect diffuseness: Its definition and sound field model

Finite-degree spherical harmonic diffuseness, directionally characterized pseudo-perfect diffuseness, is defined by simply truncating the infinite-degree spherical harmonic diffuseness.^{11,12} Namely, Gaussian random coefficients $ \alpha l , m$ of $ 0 \u2264 l \u2264 L$ and $ | m | \u2264 l$ with the statistical properties in Eqs. (5a)–(5c) define the *L*th-degree spherical harmonic diffuseness.

*L*th-degree spherical harmonic diffuseness is achieved by using a discrete random-wave model of the

*L*th-degree. The model is composed of a finite number of Gaussian random plane waves with the following statistical properties:

^{11,12}

*q*th plane wave,

*Q*denotes the number of plane waves, and $ w q$ denotes the weight of a 2

*L*th-degree spherical quadrature rule.

*L*th exactitude degree is a set of points and weights $ { n q ( \theta q , \varphi q ) ; w q} q = 1 Q$ on the sphere accurately calculating the integrals of spherical functions, yielding the following equality:

^{16}

^{,}

*L*th degree,

*L*.

^{17}In addition, the weights of the quadrature rule are all positive: $ w q > 0$, applied $ 4 \pi $-normalization: $ \u2211 q w q = 4 \pi ,$ and the sampling points of the quadrature rule $ { n q ( \theta q , \varphi q )} q = 1 Q$ determine the arrival directions of the plane waves in the discrete random-wave model. Importantly, the following

*L*th-degree discrete orthonormality of spherical harmonics is enabled by the 2

*L*th-degree quadrature rules

^{16}

*L*th degree.

^{11,12}

Note that the above models of diffuse sound fields simulate sound fields generated by multiple sound sources emitting Gaussian noise^{5} rather than room impulse responses.

## 3. Results and discussion: Spatially characterized pseudo-perfect diffuseness

This section shows that the statistical properties of sound pressure yielded using discrete random-wave models can characterize pseudo-perfect diffuseness spatially. First, we theoretically show that the expected value and the two-point pseudo covariance of sound pressures always result in zeros regardless of the diffuseness degree, *L*. Following the definition of spatial diffuseness suggested by Weaver and Lobkis,^{5} we then show numerically that the two-point covariance locally corresponds with the imaginary part of the Green's function between the two positions and suggest defining local diffuseness, which represents spatially characterized pseudo-perfect diffuseness, by exploiting such correspondence.

### 3.1 The expected value and the two-point pseudo covariance

These results show that only the zero statistical properties derived above cannot characterize spatial pseudo-perfect diffuseness because they hold for any pair of positions, $r$ and $ r \u2032 ,$ regardless of the diffuseness degree, *L*. In addition, these zero properties are identical to those in the (continuous) random-wave model [see Eqs. (6a) and (6b)].

### 3.2 The two-point covariance

*L*th-degree discrete random-wave models under free conditions without scatterers or reflectors present in the field, as follows:

*T*= 50 000 denoting the total number of syntheses. The random amplitudes $ A q$ with the statistical properties as in Eqs. (7c) to (7b) and the arrival directions $ n q$ of the plane waves were generated by using a numerical quadrature rule of the 2

_{s}*L*th degree.

^{18}The frequency and the sound speed were respectively 880 Hz and 345 m/s, yielding $ k \u2248 16.0$ rad/m. The positions $r$ were on 64 straight lines through the origin, whose directions were selected using random points on a unit hemisphere.

The upper panel of Fig. 1 shows the real part of the two-point covariance computed using discrete random-wave models of select degrees, (a) $ L = 8$, (b) $ L = 16$, (c) $ L = 24$, and (d) $ L = 32.$ Colored lines denote the two-point covariance on randomly selected straight lines and circles denote the sinc function, $ sin ( k r ) / k r$. The two-point covariance was scaled by $ 1 / 4 \pi $. The lower panel of Fig. 1 shows the absolute error between the sinc function and the two-point covariance above.

*L*and the wavenumber

*k*as follows:

*L*th-degree directionally diffuse sound fields, spatial diffuseness can hold for any position, $r$, such that

*L*th-degree directional diffuseness. Notably, in plane wave sound fields, the effective radius of local diffuseness is constant regardless of the reference position, $ r c$; however, in sound fields not composed only of plane waves, the effective radius will depend on the diffuseness degree at the reference position.

### 3.3 Supplementary remarks

^{5}Namely, the number of sampling points in the 2

*L*th-degree quadrature rule, which composes the

*L*th-degree discrete random-wave model, can be regarded as the sufficient number of random sound sources. For example, we can numerically construct the 2

*L*th-degree spherical quadrature rules with

^{19}and positive weights,

^{18}where $ \u2308 \xb7 \u2309$ denotes a ceiling function.

Note that emerging the imaginary part of the Green's function from the two-point covariance of sound pressures in diffuse sound fields is different from reproducing a sound field based on theories such as higher-order ambisonics^{20} (HOA). For example, HOA usually deals only with interior problems (i.e., sound fields without sound sources or scatterers). Our results should therefore not be interpreted based on HOA.

## 4. Conclusion

Aiming to spatially characterize pseudo-perfect diffuseness, we presented locally diffuse sound fields with the effective radius, $ R = 2 L / k$, via the *L*th-degree spherical harmonic diffuseness. Our findings connect the directional and the spatial characterizations of pseudo-perfect diffuseness.

## Acknowledgments

This work was supported by JSPS KAKENHI (Grant Nos. JP19H04153, JP22H00523) and Grant-in-Aid for JSPS Fellows (Grant No. 22KJ1941).

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Data Availability

The data are available from the corresponding author upon reasonable request.

## References

*Fundamentals of Spherical Array Processing*

*Room Acoustics*

*Physical Acoustics*

*Theory of Partial Coherence*

*Ambisonics: A Practical 3D Audio Theory for Recording, Studio Production, Sound Reinforcement, and Virtual Reality*