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

286 12. Walls, Enclosures, and BarriersFigure 12.2. Semi-log plot of transmission loss versus product of frequency and surfacemass.Ver and Holmer (1971) develop the sound transmission coefficient equation forpanels manifesting significant bending stiffness and damping, which is given asfollows:τ =[1 + η( mω cos θ2ρc1)( Bω 2 sin 4 θmc 4 )] 2+[( mω cos θ2ρc)(1 − Bω2 sin 4 )] 2.θmc 4(12.14)The panel thickness is assumed to be small compared with the wavelength of theincident sound, B = panel bending stiffness (N m), η = composite loss factor(dimensionless), and m = panel surface density (kg/m 2 ).12.6 Coincidence Effect and Critical FrequencyIn propagating though panels and other structural elements, sound can occur aslongitudinal, transverse and bending waves. Bending waves give rise to the coincidenceeffect. In Figure 12.3 a panel is shown with an airborne sound wave ofwavelength λ incident at angle θ. A bending wave of wavelength λ b is excited in thepanel. The propagation velocity of bending waves depends on the frequency, withhigher velocities occurring at higher frequencies. The coincidence effect occurswhenθ = θ ∗ = arcsin(λ/λ b ) (12.15)