Sound field, spatial hearing, and sound reproduction  3

origin or determined with the position vector Rr from the origin to the receiver position, where the subscript “r” refers to the receiver position. The receiver position can also be determined with Cartesian coordinates (Xr, Yr, and Zr) or spherical coordinates (Rr, Θr, and Φr), where 0 ≤ Rr < +∞ denotes the distance with respect to the origin, −180° < Θr ≤ 180° represents the azimuthal angle (or azimuth), and 0° ≤ Φr ≤ 180° indicates the polar angle, i.e., the angle between the position vector and the polar axis (Z-axis).

The clockwise spherical coordinate system with respect to a listener’s head center or a specific receiver position is also used in some literature and books on spatial hearing and sound reproduction. The azimuth in this coordinate system is labeled clockwise. Therefore, in the horizontal plane, azimuths −90°, 0°, 90°, and 180° represent the left, front, right, and back directions, respectively. Overall, the anticlockwise spherical coordinate system with respect to the head center is often used in studies on stereophonic and multichannel sounds, and the clockwise spherical coordinate system with respect to the head center is often utilized in the literature on spatial hearing, head-related transfer function, and virtual auditory display. However, ambiguity may arise if two or more coordinate systems are used in a book without explanation.

The anticlockwise spherical coordinate system with respect to the head center or a specific receiver position, i.e., coordinate system A, is the default in this book to be consistent with most studies on stereophonic and multichannel sounds. The results from the literature have been converted into coordinate system A with specific explanations unless otherwise stated. The default coordinate system here is different from that in the author’s other book on head-related transfer functions and virtual auditory display, which utilizes the clockwise spherical coordinate system with respect to the head center (Xie, 2008a, 2013). Moreover, for convenience in analyzing the field generated by a sound source, the coordinate system with respect to sound source, i.e., coordinate system B, is sometimes used in this book with specific explanations. The variables in coordinate systems A and B are labeled with lowercase (r, θ, ϕ) and uppercase (Rr, Θr, Φr) letters, respectively. ϕ = 0° in coordinate system A is the horizontal plane, whereas Φr = 0° in coordinate system B is the direction of the polar axis (Z-axis). Confusion in identifying the two coordinate systems can be eliminated by being careful.

1.2  SOUND FIELDS AND THEIR PHYSICAL CHARACTERISTICS

Some basic physical concepts in sound waves and sound fields are briefly reviewed in this section before they are discussed in the succeeding chapters. The detailed analyses and mathematical derivations can be found in some textbooks on fundamental acoustics (Morse and Ingrad, 1968; Piere 2019; Du et al., 2001).

1.2.1  Free-field and sound waves generated by simple sound sources

Sound waves generated by sound sources propagate in a space with a medium. A sound field is defined as a region in a medium where sound waves are being propagated. Air can be considered as a uniform and isotropic medium. A sound field in air is physically characterized by the temporal and spatial distribution of sound pressure p(r, t) in the time domain or equally characterized by the frequency and spatial distribution of sound pressure P(r, f ) in the frequency domain. In this book, vectors are denoted by boldface letters, functions in the time domain are indicated by lowercase letters, and functions in the frequency domain are represented by uppercase letters.

In a source-free region, sound pressure in air satisfies the homogeneous wave equation. In principle, sound pressure can be calculated by solving the wave equation imposed on appropriate initial and boundary conditions. Therefore, a sound field in air is determined by

4  Spatial Sound

the physical characteristics and position of a sound source and the geometrical and acoustic characteristics of a given boundary.

Free field is an important concept in acoustic analysis. It refers to a special sound field in a uniform and isotropic medium in which the influences of boundaries are completely negligible (i.e., the absence of reflections from the boundary). Cases of the free field under actual conditions are rare. Sound fields can be approximated as a free field only in an anechoic chamber or a local region, which is sufficiently high above the ground. However, the concept of the free field specifies an ideal and standard condition for acoustic analysis. Therefore, many acoustic analyses and measurements are conducted under free-field conditions. In the succeeding chapters, the reproduced sound field are often analyzed and discussed under the assumption of a free field for convenience.

The simplest sound field is the plane wave in an infinite free space. In the coordinate system with respect to a specific receiver position, an arbitrary receiver position is denoted by vector r, and the frequency-domain sound pressure of a plane wave is given by

where j is the imaginary unit; f is the frequency; k is the wave vector whose direction is the propagating direction of the plane wave; |k| = k = 2π f/c is the wave number; c = 343 m/s is the speed of sound in air; PA(f ) is the complex-valued amplitude of a harmonic plane wave with frequency f, whose modulus and phase angle are the magnitude and initial phase of the plane wave, respectively; and k r is the scalar product of two vectors. Let Ω = (θ, ϕ) denote the direction of a receiver position, and ΩS = (θS, ϕS) indicate the incident direction of the plane wave (at the origin, which is opposite to the propagating direction or the direction of a wave vector), then

where ΩS = Ω − ΩS is the angle between the direction of the receiver position and the incident direction of the plane wave.

The wavefront of a plane wave is an infinite plane, and the amplitude of a plane wave is independent of the receiver position. As such, the overall power of a plane wave in an unbounded space is infinite. Therefore, plane waves can only be approximately generated in a local region, and the plane wave in an unbounded space is unrealized.

The sound field generated by a point source (monopole source) in a free field is relatively simple and can be realized physically. Let vector rS denote the position of a point source in a coordinate system with respect to a specific receiver position. The frequency-domain sound pressure at an arbitrary receiver position r is expressed in the following equation by appropriately choosing the initial phase of a sound source:

Sound field, spatial hearing, and sound reproduction  5

where Qp(f ) is the complex-valued strength of a point source; the subscript “p” is the point source; Rr = r − rS is the vector from the point source to an arbitrary receiver position, that is, the vector of the receiver position with respect to the source; and Rr = |Rr| is the distance between the receiver position and the sound source, that is, the radial coordinate of the receiver position in the coordinate system with respect to the sound source.

Equation (1.2.3) indicates that the sound field generated by a point source is a spherical wave in which the wavefront is a spherical surface, and sound pressure is independent of the direction of the receiver position with respect to the sound source. The amplitude of sound pressure is expressed as

Therefore, the magnitude |PA(f )| of sound pressure is inversely proportional to Rr. Every double distance between the receiver position and the sound source causes a −6 dB attenuation in the sound pressure level. This attenuation is the consequence of finite and constant power provided by the point source in the free field.

In Equation (1.2.4), for a spherical wave, the relative variation in pressure amplitude with Rr is dPA(f )/PA(f ) = −dRr/Rr. |dPA(f )/PA(f )| decreases as the distance increases, i.e., Rr = | r − rS |. For a large Rr, dPA(f )/PA(f ) approaches zero. Therefore, in the local region of a far field where the distance between the receiver position and the sound source is large enough, the amplitude PA(f ) of sound pressure is approximately constant. In this case, the sound field of a point source can be approximated as a plane wave, and Equation (1.2.3) becomes

The factor exp(jk rS) in Equation (1.2.5) can be omitted by choosing an appropriate initial phase of the sound source, and the incident plane wave at the receiver position near the origin of the coordinate is expressed as

This equation is the plane wave approximation of the far field generated by a point source in which PA(f ) is independent of the sound source distance. Plane wave approximation is usually convenient for the analysis of the reproduced sound field. However, this approximation is only valid within a local region of the far field, and the spherical wave in the whole space can never be approximated as the plane wave.

The sound field generated by a straight-line source with an infinite length in the free field is relatively complicated. A straight-line source can be regarded as a composition of an infinite number of point sources, which are distributed densely and uniformly in a straight line. The sound field generated by a straight-line source with an infinite length is a cylindrical wave. In the coordinate system with respect to a specific receiver position, if the z-axis is chosen to be parallel to the straight-line source, the sound pressure is independent of the z coordinate of the receiver position. In this case, our analysis can be limited to the horizontal sound field. Here, rS is the intersect position of the straight-line source and the horizontal plane, and r is the receiver position in the horizontal plane; then, the sound pressure generated by a straightline source with an infinite length is given by