10.2.3  General theory of two-dimensional WFS

464  Spatial Sound

Equation (10.2.31) describes the driving signals of secondary monopole straight-line source. The spatial window similar to that in Equation (10.2.26) is chosen, but in this case, n′ is the unit vector at the inward-normal direction of L′Σ. Equations (10.2.30) and (10.2.31) indicate that an array of secondary monopole straight-line sources can reconstruct the target sound field approximately in a horizontal receiver region, or more exactly, can reconstruct the vertical direction-independent sound field in a vertical cylindrical region inside the array of secondary sources, by appropriately choosing the driving signals.

For a target incident plane wave, the driving signals of secondary straight-line sources are identical to those expressed in Equation (10.2.28)

For a target straight-line source, the radiated pressure in Equation (1.2.7) is described as

Driving signals are derived by substituting Equation (10.2.33) into Equation (10.2.31) and using the derivative formula dH0(ξ)/dξ = −H1(ξ) of the zero-order Hankel function of the second kind as

w r Qli f jk2 cos snH1 k|r rS | ,

where H1(k |r′−rS|) is the first-order Hankel function of the second kind, and θsn is the angle between the target source direction and the outward-normal direction at r′ of the boundary surface.

According to the asymptotic formula of the Hankel function of the second kind, when k |r – r′| >> 1, the following equation is obtained:

As in Section 10.1.3, at the far-field, the magnitude of cylindrical wave created by a straight-line secondary source is inversely proportional to the square root of the distance between the secondary source and receiver position (−3 dB law rather than −6 dB law for a point source). In practice, WFS can be conveniently implemented using secondary point sources. Substituting Equation (9.2.35) into (9.2.5) and using Equation (10.1.2) yield

Spatial sound reproduction by wave field synthesis  465

The reconstructed pressure at r is approximated by substituting Equation (10.2.36) into Equation (10.2.30) as follows:

In Equation (10.2.37), when the target source and receiver position are restricted in the horizontal plane, a sound field can be reconstructed by a uniform and continuous array of secondary monopole point sources arranged on L′Σ instead of a straight-line source. In this case, driving signals should be equalized; that is,

L

The equalized driving signals for secondary monopole point sources are related to those for secondary monopole straight-line sources by the following equation:

Equation (10.2.38) specifies a 2.5-dimensional WFS or reproduction. Equation (10.2.39) describes 2.5-dimensional driving signals, which are obtained through frequencyand dis- tance-dependent equalization of two-dimensional driving signals. Equalization depends on the receiver position. In 2.5-dimensional reproduction, the reconstructed sound field deviates from the target sound field when either the source or receiver position deviates from the horizontal plane.

The driving signals in Equation (10.2.39) depend on the receiver position. As indicated in Section 10.1.3, the magnitude of driving signals is equalized at a given reference position rref in practice. Then, 2.5-dimensional driving signals in Equation (10.2.39) become

For a target plane wave, substituting Equation (10.2.32) into Equation (10.2.40) yields

where P(r′, f) = PA(f) exp (−jk r′) is the pressure of the target plane wave at the position of secondary sources. The physical significance of Equation (10.2.41) is similar to that of Equations (10.1.20) and (10.1.21) except for the spatial window w(r′) of secondary sources arranged in L′Σ.

466  Spatial Sound

Spors et al. (2008) suggested deriving 2.5-dimensional driving signals of secondary sources from the pressure of a three-dimensional target point source rather than a target straight-line source for a target source at the horizontal position rS because of the uneven frequencyspectral characteristics of the radiation of a target straight source in Equation (10.2.33). Substituting E2D(r′, f) in Equation (10.2.40) with Ep3D r , f in Equation (10.2.29) yields

where θsn is the angle of the target source with respect to the outward-normal direction at r′ of the secondary source array. For a horizontal linear array of secondary monopole point sources in Section 10.1.3, w(r′) = 1. Equation (10.2.43) is equivalent to Equation (10.1.21) when | r′−rS | >> 1. Therefore, a horizontal linear array of secondary monopole point sources is a special case of the discussion in this section.

If driving signals are directly derived from the pressure of a target straight-line source, the two-dimensional driving signals in Equation (10.2.34) should be equalized or multiplied with the factor jk so that the radiation of the target straight-line source has a frequency-spectral characteristic identical to that of a point source. Then, the equalized two-dimensional driving signals are converted into 2.5-dimensional driving signals by using Equation (10.2.40):

When k |r′-rS| >> 1, the asymptotic formula of the first-order Hankel function of the second kind yields

In the case of Qli(f) = Qp(f), Equation (10.2.44) becomes