Basic Acoustics

by Cliff Kaminsky

Vibro-Acoustic Sciences, Inc.

Room Acoustics

In any acoustic environment, there exist to some extent two distinct acoustic fields: the direct field and the reverberant field. Both of these fields can have the same acoustic source, but their temporal and spatial properties differ. In a room with very high absorption (short reverberation time), the direct field will dominate the acoustic response. Conversely, in a room with low absorption, the reverberant field will dominate. Real rooms tend to have a combination of direct and reverberant field characteristics.

The sound as it emanates from the source and before it strikes any surface constitutes the direct field. The most notable property of the direct field is its spatial distribution. The direct field sound pressure tends to decrease with distance from the source, due primarily to the distribution of the radiated power over a sphere of increasing surface area as the wave propagates outward from the source. A finite plane radiator will, in fact, have varying degrees of attenuation with distance due to geometric factors. At high frequencies, close to the radiating panel, the wave will act as a plane wave, with negligible attenuation with distance. Further out, the panel will act as a line source and the radiated field will decay at a rate of 3 dB per doubling of distance. Still further out, the source will appear as a point source and the decay will be 6 dB per doubling of distance. For a plane source of dimensions WxL, where W<L, the sound power is given as:

Equation 1 - behavior of the direct field

Another variable that affects the direct field sound pressure is the so-called "Q factor." The Q factor is a coefficient multiplied into the acoustic power to account for reflected energy if the source is placed near a surface or corner. For an omnidirectional source in free space, Q=1. For an omnidirectional source placed at a plane surface, Q=2 because a receiver in the room would perceive both the direct acoustic wave and the

wave immediately reflected from the wall. For a source in a two-dimensional corner (i.e. the intersection of two walls), Q=4, and in a three-dimensional corner, Q=8. Theoretically, this term applies to omnidirectional sources placed infinitesimally close to a perfectly reflecting surface so that the reflected energy is indistinguishable from the direct field.

The reverberant field is comprised of the sound that has reflected from one or more interior room surfaces. Away from the cavity walls, this component of the sound field does not vary spatially, but is primarily a function of the acoustic absorption of the room surfaces. Room acoustics theory generally uses an average room absorption coefficient that is simply an area-weighted average of the individual absorption coefficients of the individual interior room surfaces:

Equation 2 - Calculation of average absorption level

This approximation requires that the sound absorption be fairly evenly distributed over the interior room surfaces. Significant differences between interior room surfaces will cause spatial variation in the reverberant field that cannot be easily predicted. The most commonly used diffuse room theory, primarily for its simplicity, is the Sabine theory. Sabine proposed that the reverberant sound pressure in a room is inversely proportional to the acoustic absorption:

Equation 3 - Sabine mean square reverberant pressure level

This approximation however, is only applicable for low values of a because this formulation does not allow the predicted sound pressure to approach zero for a =1. In practice, Sabine theory is correct for absorption levels below about one third (33%). For higher levels of absorption, one turns to the formula put forth by Eyring:

Equation 4 - Eyring mean square reverberant pressure level

A comparison of the two methods is shown in Error! Reference source not found.. Note that in both equations the value of 4 in the numerator represents the existence of a perfectly reverberant sound field. In the case where the sound is not propagating equally in all directions, that value may decrease. In the limit when the sound is traveling in only one direction, that value will equal 1 (one).

The total acoustic field is calculated with the sum of the mean square reverberant and direct-field pressure levels:

Equation 5 - Computation of total pressure field

REFERENCES

Beranek, Leo L and Vér, István L., Noise and Vibration Control Engineering, Principles and Applications. (John Wiley & Sons; New York, 1992.)

Vibro-Acoustic Sciences, Inc., AutoSEA Theory and Quality Assurance Module, (Kent Town, Australia, 1995.)

Cremer, L., Muller, H. A., and Schultz, T. J., Principles and applications of room acoustics, (Applied Science Publishers; New York, 1982.)

Morse, P. M. and Ingard, K. U., Theoretical Acoustics, (Princeton University Press; Princeton, NJ, 1986.)

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