VRIJE UNIVERSITEIT BRUSSEL Acoustics - the Dept. of ...

5.2. AIRBORNE SOUND INSULATION OF A WALL 87with p d the sound pressure at transmission side, m the mass per m 2 and atheacceleration of this wall mass. In practice one may write 2p i = ma becausep d ≪ p i . For an harmonic wave we have p i = P i cosωt. The wall accelerationis thus a = 2 m P icosωt and his velocity :v = 2mω P isinωt (5.8)Because there is no wave phenomenon in the transverse direction in thewall it follows that v = v i = v d . The velocity v d is transmitted to the air incontact with the wall at the receiving side and gives rise to a plane travellingwave with pressure given by p d = ρcv d . It then follows that :And thus :The sound insulation is given by :and consequently :p d = ρc 2mω P isinωt = P d sinωt (5.9)P i= mωP d 2ρc = π mf (5.10)ρc∆L = 10log I iI d= 20log P iP d(5.11)∆L = 20log πmfρc(5.12)This simplified law is known as the acoustic mass-frequency law. We can seethat :By doubling the wall mass the insulation is doubled (sound reductionlevel +6 dB).By doubling the frequency the insulation is doubled (sound reductionlevel +6 dB).For a brick wall (m ≈ 100 kg/m 2 ) the application of the mass-frequency lawgives ∆L40 dB at 500 Hz.In practice the increase is smaller than what the mass-frequency law predicts:Doubling the wall mass : +5 dB.Doubling the frequency : +5 dB. (frequency-law).