Malcolm J. Crocker - Engineering Acoustics

(3.53)

Figure 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 p 2 rms( 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 p 2 rmsis a constant, independent of angles θ and ϕ .

We may therefore write:

(3.56)

and

(3.57)

where is the space‐averaged mean‐square sound pressure.

We define the directivity factor Q as

(3.58)

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.

The directivity index DI is just a logarithmic version of the directivity factor Q . It is expressed in decibels.

A directivity index DI θ,ϕmay be defined, where

(3.59)

(3.60)

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.

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?

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

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)

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 I i, reflected intensity I r, and transmitted intensity I tin 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 I rand transmitted intensity I tequals the incident intensity I i:

(3.64)

and dividing throughout by I i,

(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):

(3.66)

and

(3.67)

Some interesting facts can be deduced from Eqs. (3.66)and (3.67). Both the reflection and transmission coefficients are independent of the direction of the wave since interchanging ρ 1 c 1and ρ 2 c 2does not affect the values of R and T . For example, for sound waves traveling from air to water or water to air, almost complete reflection occurs, independent of direction; the reflection coefficients are the same and the transmission coefficients are the same for the two different directions.