Homogenization of the Neumann Spectral Problem in a Thin Cylinder with a Periodic Family of Disturbances

The asymptotic behavior of the eigenvalues of the Neumann problem for a second-order elliptic equation in a thin finite cylinder with a periodic family of fine resonators (an acoustic waveguide with rigid walls) is studied. The main purpose is to construct an asymptotics of the coefficient in the homogenized differential operator. Its principal term does not depend on the shape of the resonators, but the first correction term includes the polarization coefficient of an infinite cylinder with a resonator of unit size. Properties of the polarization coefficient are studied, which is defined as the constant in the expansion of a linearly growing solution of the homogeneous problem in the perturbed cylinder at infinity. The obtained asymptotic expansions are substantiated by using a refined one-dimensional model and Poincaré’s asymptotically exact inequalities.