Acoustic Radiation from Plates into a Wedge-Shaped Fluid Region: Application to the Free Plate Problem

The statistical energy analysis (SEA) method, applied to interacting systems, requires estimates for certain coupling coefficients that characterize the flow of acoustic energy between adjacent subsystems. These are calculated with reference to appropriate canonical problems that have similar local geometry. One such case, considered here, is that of two plates that are joined together along a common straight edge, with compressible fluid in the wedge region 0 < θ < θ ₀ = π /q. One of the bounding surfaces is rigid and the other (given by x > 0, -∞ < y < ∞ , z = 0) has outward normal velocity Re{u(x, y) exp(-iω t)} that depends on the governing plate equation and edge conditions. For a thin elastic plate the mode shapes have the form u(x,y) = v₀ sin k₁x sin k₂y at some distance from x = 0, with local edge behaviour that depends on the precise boundary conditions there. When k₁² + k₂² < k², where k = ω /c is the acoustic wavenumber at radian frequency ω , the radiated acoustic power is proportional to the area of the whole vibrating surface. When k₁² + k₂² > k² and k₂ < k, the radiated power is identified as an edge effect; it is proportional to the length of the edge and depends on the velocity u near the apex. Explicit formulae are given for R_q, defined as the power per unit span divided by the average value of u² with respect to space and time, for a variety of plate edge conditions. A key result is that, for general edge constraints, R_q = qR₁ + o(α ^-2), as α = k₁/k -> ∞ . The approximation is likely to provide an adequate approximation for R_q, for the purpose of SEA estimates, for all wavenumbers such that k₁² + k₂² > k², k₂ < k. A related problem is that of a `free plate' at x > 0, z = 0, corresponding to the wedge of angle θ ₀ = 2π but with the prescribed velocity u(x,y) on both sides z = ± 0. The solution is expressed as a combination of the above wedge potentials with q = 1/2 and q = 1. It is found that the radiation resistance R_free is asymptotically small compared with the reference value for the basic value for a simply supported baffled plate.