10.1.3WFS in a horizontal plane with a linear array of secondary sources

446  Spatial Sound

Equation (10.1.14). Substituting z′a = 0 into Equation (10.1.14), then substituting the result of integral into Equation (10.1.12), and using the far-field approximation k | r′− rS| >>1 yield

For the integral on the right side of Equation (10.1.16), the second square brackets can be regarded as the transfer function from the secondary source at r′ to the receiver position at r. After Qp(f) is multiplied, the first square brackets can be regarded as the driving signal of the secondary source. However, the transfer function here is inversely proportional to the square root of the distance |r – r′| between the secondary source and the receiver position rather than inversely proportional to |r – r′|. A secondary source with such a radiation characteristic is difficult to be realized. If this mismatched distance dependence of secondary source radiation is merged with driving signals, Equation (10.1.16) is consistent with the general formulation of multichannel sound field reconstruction given by Equation (9.2.1):

where Gfree3D r, r , f is the free-field Green’s function in a three-dimensional space expressed in Equation (10.1.2), i.e., the transfer function from a secondary point source to the receiver position; and

Equation (10.1.17) indicates that the sound field of a target point source in a horizontal plane can be approximately reconstructed by an infinite linear array of secondary monopole (point) sources. The E(r′, rS, f) in Equation (10.1.18) is the driving signal of the secondary source located at r′. Therefore, when both the target source and the receiver are restricted in the horizontal plane, WFS can be approximately implemented by an infinite linear array of secondary monopole point sources. Thus, this process is a remarkable simplification in comparison with WFS involving an infinite array of secondary sources in a vertical plane.

Equation (10.1.18) can also be written as

Spatial sound reproduction by wave field synthesis  447

The right side of Equation (10.1.19) involves a multiplication of three terms. P(r′, rS, f) is the pressure at r′ of the secondary source caused by the target point source at rS, and θsn is the angle of the target point source with respect to the horizontal-outward-normal direction at r′ of the secondary source array. Therefore, the third term cosθsn P(r′, rS, f) can be regarded as the output of a bidirectional (velocity field) microphone at r′, with the main axis of the microphone pointing to the horizontal-outward-normal direction of the secondary source array. Here, the magnitude responses of bidirectional microphones are assumed independent of the frequency for a far-field incidence plane wave.

The first term 4π jk/2π in Equation (10.1.19) is the response of a high-pass filter with an

appropriate gain. This high-pass filter can be easily obtained because it is independent from the positions of the target source, the secondary source, and the receiver. If the driving signals of secondary sources are created via simulation, all secondary sources can share a common high-pass filter.

The second term in Equation (10.1.19) aims to equalize the distance-dependent magnitude of secondary source radiation. However, this term depends on the target source position rS, the secondary source position r′, and the receiver position r. In other words, a given magnitude equalization is valid only at a special receiver position. However, this result is unreasonable. In practice, the magnitude of the reconstructed sound pressure is equalized at a given reference position rref, so Equation (10.1.19) becomes

In a receiver position deviating from the reference position, the phase of reconstructed pressure is correct, but magnitude errors occur (Sonke et al., 1998). In the approximation of a target plane wave, a double source distance of the receiver position causes a −3 dB attenuation in the level of the reconstructed sound pressure. For a target point source, a double source distance of the receiver position causes an attenuation between −3 dB and −6 dB in the level of the reconstructed sound pressure. This magnitude error in a reconstructed sound field is due to the mismatched distance dependence of secondary source radiation, i.e., using secondary point sources to replace secondary straight-line sources with an infinite length.

When the target source is distant from the secondary source array with | r′ – rS| >>1, Equation (10.1.20) is simplified as

Equation (10.1.21) is also valid for a target plane wave provided that P(r′, rS, f) on the right side is replaced by the pressure P(r′, f) = PA(f) exp (−jk r′) of the target plane wave. This condition is due to the plane wave approximation of the far-field spherical wave caused by a point source.

For a receiver position distant from the secondary source array with | x-x′| >> 1, the driving signal in Equation (10.1.19) is simplified as