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

12.3 Mass Control Case 283Figure 12.1. Transmission loss through a panel in the y–z plane.12.3 Mass Control CaseConsider the panel as being quite thin, i.e., its thickness is considerably smallerthan one wavelength of the sound in air; and let us also neglect the material stiffnessand damping in the panel. We can stipulate the conditions of (a) the continuity ofvelocities normal to the panel and (b) a force balance occurs inclusive of the inertialforce. From application of the first condition we express the particle velocity u atthe panel asu panel = u T cos θ = u I cos θ − u R cos θ. (12.6)When the sound pressure and the particle velocity are in-phase they are related byZ = ρ u= ρc (12.7)where Z denotes the characteristic resistance, which represents a special case ofspecific acoustic resistance.Setting x = 0 at the panel, we apply Equation (12.7) and Equation set (12.5) toEquation (12.6) to obtainP T = P I − P R . (12.8)The second condition applied over a unit surface area of the panel givesp I + p I − p T − ma panel = 0 (12.9)where m represents the mass density per unit area of the panel and a panel theacceleration normal to the surface. Through the use of Equation (12.7) we can