Sound reflection by a resonator array in a multimode cylindrical waveguide

The paper considers the problem of scattering of the mth symmetric mode by an array of Q rings of identical, closely located Helmholtz resonators joined by necks to the walls of a wide circular pipe. The distance between rings is equal to half the wavelength of this mode at frequency ??, equal or close to the eigen-frequency of the resonator ring with allowance for the connected mass and interaction of neighboring rings via inhomogeneous modes. The coefficient of reflection of the mth mode from this grating array is calculated.