Engineering Acoustics. Malcolm J. Crocker

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 DI_θ,ϕ may 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.

      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 I_i, reflected intensity I_r, and transmitted intensity I_t 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 I_r and transmitted intensity I_t equals 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 ρ₁ c₁ and ρ₂ c₂ does 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.

      Example 3.12

      A plane sound wave in air is incident normally on a boundary between air and water. If each medium can be assumed to be infinite in extent, compute the energy reflection and transmission coefficients.

      Solution

      At standard temperature and atmospheric pressure we have that in water ρ = 998 kg/m³ and c = 1480 m/s. Then, the characteristic impedance of water is ρc = 1 480 000 rayls.

      Then, R = [(415–1 480 000)/(415 + 1 480 000)]² = 0.999 and

T = 4(415)(1 480 000)/(415 + 1 480 000)² = 0.001, or simply T = 1–R = 0.001. Therefore, when a sound wave passes from air to water most of its energy is reflected back because of the impedance offered by the liquid medium.

Figure 3.14 Refraction of sound in air with wind speed U(h) increasing with altitude h.