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

12.7 The Double-Panel Partition 289often falls in the range of speech frequencies, rendering some partitions nearlyuseless for providing privacy and preventing speech interference.From Equation (12.14), it is seen that the coincidence effect depends on characteristicsof the plate or panel and of the airborne sound wave. At coincidenceBω 2 sin 4 θ ∗= 1mc 4and inserting the above condition into Equation (12.14) yields the sound transmissioncoefficient for the coincidence conditionτ = τ ∗ 1=∗]mωη cos θ 2.[1 +2ρcThe corresponding transmission loss for the coincidence condition is( ) 1TL = TL ∗ ∗)fmη cos θ= 10 log = 20 log(1 + .τ ∗ ρcFrom the above equations, it would appear that τ ∗ = 1 and TL ∗ = 0 for undampedpanels (in which the loss factor η = 0). The above transmission loss equation ispremised on the theoretical behavior of an infinite plate, and the finite boundariesof actual structures such as windows and walls and the presence of damping inreal construction materials will produce a different response to sound waves.12.7 The Double-Panel PartitionA single-panel wall can exhibit resonant frequencies that fall below the rangeof speech frequencies. Most walls are constructed of two panels with an airspacebetween them, and they may yield low-frequency resonances in the speech range. Atypical interior partition consists of two gypsum board panels (ranging in thicknessfrom 1.3 cm to 1.9 cm, or from 0.5 in. to 0.75 in.) separated 9 cm (3.5 in.) by 5 cm ×10 cm (2 × 4) 1 wood or metal studs. In Figure 12.6 a double-panel configuration isshown; the two panels of mass per unit area m 1 and m 2 , respectively, are separatedby an airspace h. Because the air entrapped between the two panels behaves asa spring, a spring-mass analogy, shown in Figure 12.7, can be applied, with krepresenting the spring constant between two masses. The wall response modecan be depicted by two masses vibrating at the same frequency. A node (i.e. a“motionless” point) on the spring exists, thus effectively resolving the spring intotwo springs with spring constants (or spring rates) k 1 and k 2 , The natural frequencyof the dual-mass system is given byf n =√k1/m 1=2π√k2/m 2. (12.16)2π1 Lumber sizes are given by figures that are almost always nominal rather than representative of actualvalues. A two-by-four stud generally measures 1.75 × 3.5 in.