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200 4QO &OO FREQUENCY, Hz

IfeOO 32QQ

200 4QO &OO FREQUENCY, Hz

IfeOO 32QQ

of incidence. When two planes of reflectors are employed, for a given separation distance there is a relationship between the angle of incidence and the frequency of cancellation of the reflected sound. This effect was used by Bragg to study the crystal structure of materials with x-rays. When sound is scattered from two reflecting planes, certain frequencies are missing in the reflected sound. This was one cause of the problems in Philharmonic Hall in New York (Beranek, 1996).

An illustration of this phenomenon, known as Bragg imaging, is shown in Fig. 7.13. When two rows of reflecting panels are placed one above the other, there is destructive interference between reflected sound waves when the combined path-length difference has the relationship

Figure 7.13 Geometry of Bragg Scattering from Rows of Parallel Reflectors

The path length difference between the lower and upper reflectors is 2 d cos 0.

Figure 7.13 Geometry of Bragg Scattering from Rows of Parallel Reflectors

The path length difference between the lower and upper reflectors is 2 d cos 0.

where X = wavelength of the incident sound (m or ft)

d = perpendicular spacing between rows of reflectors (m or ft) n = positive integer 1, 2, 3, ... etc.

9 = angle of incidence and reflection with respect to the normal (rad or deg) The use of slightly convex reflectors can help diffuse the sound energy and smooth out the interferences; however, stacked planes of reflecting panels can produce a loss of bass energy in localized areas of the audience.

Scattering from Curved Surfaces

When sound is scattered from a curved surface, the curvature induces diffusion of the reflected energy when the surface is convex, or focusing when it is concave. The attenuation associated with the curvature can be calculated using the geometry shown in Fig. 7.14. If we consider a rigid cylinder having a radius R, the loss in intensity is proportional to the ratio of the incident-to-reflected beam areas (Rindel, 1986). At the receiver, M, the sound energy is proportional to the width of the reflected beam tube (a + a2) dfi. If there were no curvature the beam width would be (a1 + a2) d^1 at the image point M1.

Accordingly the attenuation due to the curvature is a t i0i (a + a2)10l (a + a2X^/^) ALcurv = —10 log-- 2 =—10 log---—— (7.37)

Using Fig. 7.14 we see that a d^ = a1 d^1 = R cos 9, and that d^ = d^1 + 2 d0, from which it follows that

d^1 Rcos 9

and plugging this into Eq. 7.37 yields

a"

R cos 9

where a* is given in Eq. 7.24. For concave surfaces the same equation can be used with a negative value for R. Figure 7.15 shows the results for both convex and concave surfaces.

Figure 7.14 Geometry of the Reflection from a Curved Surface (Rindel, 1986)

Figure 7.14 Geometry of the Reflection from a Curved Surface (Rindel, 1986)

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