THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

11.5 Sound Absorption Coefficients 249in incremental volume dV is EdV. The surface area of the sphere of radius rencompassing dV is 4πr 2 . The projected area of dS on the sphere is cos θ dS. Theportion of the total energy contained in dV is given by the ratio dS cos θ/4πr 2 . Theenergy from dV that strikes dS directly becomesdE = EdVdScos θ . (11.1)4π r 2Now consider the volume element dV as being part of a hemisphere shall ofradius rand thickness dr. The acoustic energy rendered to S by the complete shellis found by assuming a circular zone of radius r sin θ (with θ treated as a constant)in Figure 11.5 and integrating over the entire surface of the shell. The volume ofthe resultant element is 2πr sin θ rdrdθ. From θ = 0toθ = π/2, and Equation(11.1) yieldsE = EdSdV2∫ π/20sin θ cos θ dθ = EdSdr4This energy arrives during time interval t = dr/c. Hence, the rate of acoustic energyimpinging dS from all directions isEt= EcdS4or Ec/4 per unit area, which is therefore the intensity I of the diffused sound atthe walls. This is also equal to one fourth of a plane wave of energy intensity Iincident at a normal angle onto a plane. The intensity I of the diffuse sound at thewall becomesI = Ec(11.2)411.5 Sound Absorption CoefficientsAll materials constituting the boundaries of an enclosure will absorb and reflectsound. A fraction α of the incident energy is absorbed and the balance (1 – α) isreflected. Reflection is indicated by the reflection coefficient r defined asamplitude of reflected waver =amplitude of incident waveBecause the energy in a sound wave is proportional to the square of the amplitude,the sound absorption coefficient α and the reflection coefficient are related byα = 1 − r 2The value of the sound absorption coefficient α will vary with the frequency of theincident ray and the angle of incidence. Materials comprising room surfaces aresubject to sound waves that impinge upon them from many different angles as aresultof multiple reflections. Hence, published data for absorption coefficients