Optimal lower bound of the resonance widths for a Helmhotz tube-shaped   resonator
  The study of the resonances of the Helmholtz resonator has been broadly described in previous works. Here, for a simple tube-shaped two dimensional resonator, we can perform a careful analysis of the transition zone where oscillations start to appear. In that way, we obtain an optimal lower bound of the width of the resonances. 
Andre Martinez and Laurence Nedelec
math.SP