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

308 12. Walls, Enclosures, and Barriersa machine) and hood maintains a fairly equilibrium condition, the space averageof the time-average energy density remains a constant value under the hood. Theentire acoustic power emitted by the source is absorbed by the hood as losses oris radiated through the walls and from thence outside the hood. That amount ofacoustic energy which passes through the hood is W 2 , which can be approximatedby Equations (12.26) and (12.28):whereW 2 = W 0(Sh α hᾱW 2 = power radiated into the room by the hoodW 0 = acoustic power of the sourceS h = area of the walls and ceiling of the hoodα h = absorption coefficient of the walls and ceilingS = total area under the hood)¯τ (12.45)ᾱ = average value of sound absorption coefficient under the hood¯τ = average value of transmission coefficient for the hood,excluding the floor.We shall now simplify Equation (12.45) by assuming that the total surface S =S h + S f , where S f is the floor area, is such that S f ≪ S h so that S ≈ S h . Withthe system in equilibrium, we can set α h = 1, which means that all of the energyimpinging on the walls of the hood becomes absorbed one way or the other.Equations (12.45) becomes¯τW 2 = W 0 (12.46)ᾱwhere ¯τ ≤ ᾱ ≤ 1. The limits are established so that when ᾱ approaches unity, thedefining expression for τ is satisfied. But when ᾱ approaches ¯τ, nearly the entireacoustic power of the source is radiated outside the hood.Setting Q 0 = Q 2 = Q in Equations (12.43) and (12.44) (i.e. the Q of the hoodsourcecombination equals that of the source alone) and substituting into Equation(12.42), we obtain( )W0IL = L w0 − L w2 = 10 log . (12.47)W 2Inserting Equation (12.46) into Equation (12.47) yields(ᾱ )IL = 10 log(12.48)¯τwhere ¯τ ≤ ᾱ ≤ 1.In actuality the average absorption coefficient ᾱ has a lower nonzero limit as theresult of air absorption inside the enclosure, viscous losses of waves near grazingincidence in the acoustical boundary layer on inside of the hood, and the changefrom adiabatic to isothermal compression in the immediate vicinity of the inner