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

88 CHAPTER 5. SOUND INSULATIONAbovemass-frequency lawwasderived assumingnormallyincident waves.One can show the following relation for sound with oblique incidence :∆L = 20logπmf cosθρc(5.13)The more oblique the incident wave, the lower the insulation value of thewall. But starting from a certain angle reflection will occur. Above model isonly valid for 0 < θ < 78 degrees. In practice sound will be incident fromdifferent directions simultaneously (e.g. in the case of a diffuse field there isa omnidirectional incidence). One can show that in this case :∆L = 20log πmfρc−5 dB (5.14)The insulation value of the wall is thus 5 dB less than for a normal incidence.For air this formula can be rewritten as :∆L = 20logmf −47.4 dB (5.15)The mass frequency law is an engineering models which attempt to give acoarse prediction of the sound insulation behavior. It does not give an exactrepresentationofthevibroacousticbehavioroftheair-wallinteraction. Inthefollowing sections we will introduce several extensions of the mass-frequencylaw.5.2.2 Effect of the wall stiffnessThe theoretical mass-frequency law showed that the mass per unit area playsan important role in sound insulation. We assumed that the wall was characterizedby its mass only and the elastic properties were ignored. If the latterproperties are considered, we notice that for the acoustic insulation propertiesthis has some negative consequences : resonance phenomena can occurat which the wall is transparant for the sound wave. This will be shown inwhat follows.Consider p − en p + the sound pressures at respectively the left and rightside of the wall, given by :p − = 2P i cosk 1 xexpiωt−iωXρ 1 c 1 expiωt+k 1 xp + = iωXρ 2 c 2 expiωt−k 2 x = P d expiωt−k 2 xThe equation of motion can now be written :mẍ+dẋ+kx = p − (0)−p + (0) (5.16)