Analysis of Quiet Zones in Diffuse Fields

Abstract

The wave model of pure tone and broad-band diffuse sound fieldsGarcia-Bonito used the wave model for a pure tone diffuse field, which is comprised of large number of propagating waves arriving from various directions [Garcia et al., 1997]. However, a complete mathematical derivation of this model, which was taken from Jacobson [Jacobsen, 1979], was not found and the wave model of broadband diffuse sound fields has not been derived. For completeness, this mathematical derivation for pure tone and broadband diffuse fields is given below. When a source produces sound in an enclosure in a room, the sound field is composed of two fields. One is the sound field radiated directly from the source called the direct sound www.intechopen.com Noise Control, Reduction and Cancellation Solutions in Engineering 48field. The other is reflection of sound waves from surfaces of the room, which contributes to the overall sound field, this contribution being known as the reverberant field. Therefore at any point in the room, the sound field is a function of direct and reverberant sound fields. The sound field in a reverberant space can be divided into two frequency ranges. In the low frequency range, the room response is dominated by standing waves at certain frequencies. In the high frequency range, the resonances become so numerous that they are difficult to distinguish from one another. For excitation frequencies greater than the Schroeder frequency, for which M () = 3, where M() is the modal overlap [Garcia et al., 1997], the resulting sound field is essentially diffuse and may be described in statistical terms or in terms of its average properties. The diffuse sound field model can be derived as follows. In the model described below, the diffuse field is comprised of many propagating waves with random phases, arriving from uniformly distributed directions. Although the waves occupy a three-dimensional space, the quiet zone analysis is performed, for simplicity, over a two-dimensional area. Consider a single incident plane wave travelling along line r with its wave front parallel to lines A and B as shown in Figure 1. We assume that the plane wave has some phase when approaching line A, and has some phase shift due to the time delay when approaching line B both on the x-y plane. We next find the phase of the plane wave at (x 0 ,y 0 ) on line B. We now consider the plane perpendicular to lines A and B and parallel to line r, as illustrated in Figure 2. This incident plane wave has phase shift when approaching point (x 0 ,y 0 ) on line B. The pressure at this point can therefore be expressed as P (x 0 ,y 0, k) = (a+jb) exp (-jkd) ( 1) where a+jb account for the amplitude and phase of this incident plane wave when approaching line A, k is the wave number and d is the additional distance travelled by the plane wave when approaching point (x 0 ,y 0 ) on line B as shown in Figure 2. The equation of line A on the x-y plane can be written asThe equation of line B on x-y plane can also be written aswhere m is the distance between li...