r where Nr = number of equally spaced rays. Using Eq. 2.90 the intensity from a diffuse reflection at a receiver within the hemisphere is

4 n r22

where Sb = ——— = beam area at a distance r1 from the source

Qs = directivity of the reflecting surface = 2 for an omnidirectional hemisphere = 2 cos ft for Lambert's cosine law ft = angle between the scattering surface normal and the outgoing ray

Some programs (e.g., ODEON) use directional scattering of the diffuse energy according to Lambert's cosine law, which is introduced using the surface directivity Qs, and others (e.g., RAMSETE) use omnidirectional scattering.

Multiple Reflections

When a ray undergoes multiple reflections the effect of each successive surface's absorption and diffusion coefficients must be considered. Accordingly the surface absorption coefficient and the air absorption term for a single reflection are replaced by a product n

(1 - a) e-Y(r1 + r2) ^Yl (1 - ai) e-y(rtot + r2) (22.51)

The total diffuse scattering coefficient represents the diminution in the intensity due to a succession of diffuse reflections, each of which removes some of the energy from the specular component. It is defined as

The total diffusion coefficient increases slightly with each reflection until it approaches 1. At this point all reflections are diffuse and the sound field is totally reverberant.

A given receiver is illuminated by both specular and diffuse reflections. Figure 22.19 shows two hypothetical receivers. If a receiver lies within the specularly reflected pyramid beam, it receives both the specular and diffusely reflected intensities. If it is outside the beam it receives only the diffuse intensity, but these reflections come from every surface having a diffusion coefficient greater than zero. After each reflection the intensity of the remaining direct sound field is reduced by the amount absorbed and by the amount diffused.

Figure 22.19 Pyramid Beam Impacting a Surface (Farina, 2000)

Figure 22.19 Pyramid Beam Impacting a Surface (Farina, 2000)

When there are multiple specular reflections from a series of surfaces the received intensity after the nth reflection is (Farina, 2000)

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