3.12 Geometry used in derivation of directivity factor.
For a directional source, the mean square sound pressure measured at distance r and angles θ and ϕ is p2rms (θ,ϕ).
In the far field of this source (r ≫ λ), then
(3.54)
But if the source were omnidirectional of the same power W, then
(3.55)
where p2rms is a constant, independent of angles θ and ϕ.
We may therefore write:
(3.56)
and
(3.57)
where
We define the directivity factor Q as
the ratio of the mean‐square sound pressure at distance r to the space‐averaged mean‐square pressure at r, or equivalently the directivity Q may be defined as the ratio of the mean‐square sound pressure at r divided by the mean‐square sound pressure at r for an omnidirectional sound source of the same sound power W, watts.
3.9.2 Directivity Index
The directivity index DI is just a logarithmic version of the directivity factor Q. It is expressed in decibels.
A directivity index DIθ,ϕ may be defined, where
(3.59)
Note if the source power remains the same when it is put on a hard rigid infinite surface Q(θ, ϕ) = 2 and DI(θ, ϕ) = 3 dB.
Example 3.11
1 If a constant‐volume velocity source of sound power level 120 dB (which is equivalent to 1 acoustic watt) radiates to whole space and it has a directivity factor of 12 at 50 m, what is the sound pressure level in that direction?
2 If this constant‐volume velocity source is put very near a hard reflecting floor, what will its sound pressure level be in the same direction?
Solution
1 We have that I = 1/4π(50)2 = 1/104 π (W/m2), then But for the directional source Lp(θ, ϕ) = 〈Lp〉S + DI(θ, ϕ), then assuming ρ c = 400 rayls, Lp(θ, ϕ) = 75 + 10 log 12 = 75 + 10 + 10 log 1.2 = 85.8 dB.
2 If the direction is away from the floor, then
3.10 Line Sources
Sometimes noise sources are distributed more like idealized line sources. Examples include the sound radiated from a long pipe containing fluid flow or the sound radiated by a stream of vehicles on a highway.
If sound sources are distributed continuously along a straight line and the sources are radiating sound independently, so that the sound power/unit length is W′ watts/metre, then assuming cylindrical spreading (and we are located in the far acoustic field again and ρc = 400 rayls):
(3.61)
so,
then
(3.62)
and for half‐space radiation (such as a line source on a hard surface, such as a road)
(3.63)
3.11 Reflection, Refraction, Scattering, and Diffraction
For a homogeneous plane sound wave at normal incidence on a fluid medium of different characteristic impedance ρc, both reflected and transmitted waves are formed (see Figure 3.13).
Figure 3.13 Incident intensity Ii, reflected intensity Ir, and transmitted intensity It in a homogeneous plane sound wave at normal incidence on a plane boundary between two fluid media of different characteristic impedances.
From energy considerations (provided no losses occur at the boundary) the sum of the reflected intensity Ir and transmitted intensity It equals the incident intensity Ii:
(3.64)
and dividing throughout by Ii,
(3.65)
where R is the energy reflection coefficient and T is the transmission coefficient. For plane waves at normal incidence on a plane boundary between two fluids (see Figure 3.13):
