Reverberant Field Effects

Energy Density and Intensity

We have seen that, as the modal spacing gets closer and closer together, it becomes less useful to consider individual modes and we must seek other ways of describing the behavior of sound in a room. One concept is the energy density . A plane wave moves a distance c0 in one second and carries an energy per unit area equal to its intensity, I. The direct-field energy density Dd per unit volume is

Id p2

c0 p0 c0

where p2 is the rms acoustic pressure.

The energy density in a diffuse field has the same relationship to the pressure squared, which is not a vector quantity, but a different relationship to the intensity. In a diffuse field the sound energy can be coming from any direction. The intensity is defined as the power passing through an area in a given direction. In a diffuse field, half the energy is passing through the area plane in the opposite direction to the one of interest. When we integrate the energy incident on the area in the remaining half sphere, the cosine term reduces the intensity by another factor of two. Thus in a reverberant field the intensity is only a quarter of the total power passing through the area. This is shown in Fig. 8.12.

4 VP0c0

Figure 8.12 Intensity in a Reverberant Field

Norma

Energy Plow in this direction does not c-ount, so half the reverberant field energy does not contribute to the intensity

Figure 8.12 Intensity in a Reverberant Field

Norma

Half of the remaining energy is not aligned with tne normal and so it too does not contribute to the intensity

Semireverberant Fields

Occasionally we encounter a semireverberant field, where energy falls onto one side of a plane with equal probability from any direction. Most often this occurs when sound is propagating from a reverberant field through an opening in a surface of the room. Under these conditions the power passing through the plane of an opening having area Sw is given by

2 \PocoJ

Room Effect

When a sound source that emits a sound power WS is placed in a room, the energy density will rise until the energy flow is balanced between the energy being created by the source and the energy removed from the room due to absorption. After a long time the total energy in a room having a volume V due to a source having a sound power WS is

VDr = ^^ [1 + (1 - a) + (1 - a) +•••] = —^ (8.81)

co coa which has been simplified using the limit of a power series for a2 < 1

4WS V

co Sj a and the sound pressure in the room will be p

2 4W 4W

Equation 8.83 is the reverberant-field contribution to the sound pressure measured in a room and can be combined with the direct-field contribution to obtain p2 QWS 4WS

Taking the logarithm of each side we can express this equation as a level

where K is 0.1 for metric units and 10.5 for FP units. The numerical constants follow from the reasoning given in Eq. 2.67.

Equation 8.85 is based on Sabine's theory and was published in 1948 by Hopkins and Stryker. It is a useful workhorse for the calculation of the sound level in a room given the sound power level of one or more sources. It holds reasonably well where the diffuse field condition exists; that is, in relatively large rooms with adequate diffusion if we are not too A

close (usually within —) to reflecting surfaces. The increase in sound pressure level due to the reverberant field over that which we would expect from free field falloff is called the room effect.

Figure 8.13 gives the result from Eq. 8.85 for various values of the room constant. Near the source the direct-field contribution is larger than the reverberant-field contribution and the falloff behavior is that of a point source in a free field. In the far field the direct-field contribution has dropped below the reverberant-field energy, and the sound pressure level is constant throughout the space. The level in the reverberant field can be reduced only by adding more absorption to the room. According to this theory, only the total amount of absorption is important, not where it is placed in the room. In practice absorption placed where the particle velocity is the highest has the greatest effect. Thus absorption mounted in a corner, where the pressure has a maximum and the velocity a minimum, would be less effective than absorption placed in the middle of a wall or other surface. Absorption, which is hung in the center of a space, has the greatest effect but this is not a practical location.

Figure 8.13 Difference between Sound Power and Pressure Level in a Diffuse Room Due to an Omnidirectional Source

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