plane wave case.
For a point in a room at distance r from a source of power W watts, we will have a direct field intensity contribution W/4πr2 from an omnidirectional source to the mean square pressure and also a reverberant contribution.
We may define the reverberant field as the field created by waves after the first reflection of direct waves from the source. Thus the energy/second absorbed at the first reflection of waves from the source of sound power W is W
(3.76)
where p2rms is the mean‐square sound pressure contribution caused by the reverberant field.
There is also the direct field contribution to be accounted for. If the source is a broadband noise source, these two contributions: (i) the direct term p2d,rms = ρcW/4πr2 and (ii) the reverberant contribution,
(3.77)
and after dividing by p2ref, and Wref and taking 10 log, we obtain
where R is the so‐called room constant
3.15.1 Critical Distance
The critical distance rc (or sometimes called the reverberation radius) is defined as the distance from the sound source where the direct field and reverberant field contributions to p2rms are equal:
(3.79)
thus,
(3.80)
Figure 3.23 gives a plot of Eq. (3.78) (the so‐called room equation).
Figure 3.23 Sound pressure level in a room (relative to sound power level) as a function of distance r from sound source.
3.15.2 Noise Reduction
If we are situated in the reverberant field, we may show from Eq. (3.78) that the noise level reduction, ΔL, achieved by increasing the sound absorption is
(3.81)
(3.82)
Then A = S
By considering the sound energy radiated into a room by a directional broadband noise source of sound power W, we may sum together the mean squares of the sound pressure contributions caused by the direct and reverberant fields and after taking logarithms obtain the sound pressure level in the room:
where Qθ,ϕ is the directivity factor of the source (see Section 3.9) and R is the so‐called room constant:
(3.84)
A plot of the sound pressure level against distance from the source is given for various room constants in Figure 3.23. It is seen that there are several different regions. The near and far fields depend on the type of source [21] and the free field and reverberant field. The free field is the region where the direct term Qθ,ϕ /4πr2 dominates, and the reverberant field is the region where the reverberant term 4/R in Eq. (3.83) dominates. The so‐called critical distance rc = (Qθ,ϕ R/16π)1/2 occurs where the two terms are equal.
3.16 Sound Radiation From Idealized Structures
The sound radiation from plates and cylinders in bending (flexural) vibration is discussed in Refs. [9, 27] and chapter 10 in the Handbook of Acoustics [1]. There are interesting phenomena observed with free‐bending waves. Unlike sound waves, these are dispersive and travel faster at higher frequency. The bending‐wave speed is cb = (ωκcl)1/2, where κ is the radius of gyration h/(12)1/2 for a rectangular cross‐section, h is the thickness, and cL is the longitudinal wave speed {E/[ρ(1 − σ2)]}1/2, where E is Young's modulus of elasticity, ρ is the material density, and σ is Poisson's ratio. When the bending‐wave speed equals the speed of sound in air, the frequency is called the critical frequency (see Figure 3.24). The critical frequency is
