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We derive the weak limit of a linear viscoacoustic model in an acoustic linerthat is a chamber connected to a periodic repetition of elongated chambers --the Helmholtz resonators. As model we consider the time-harmonic and linearizedcompressible Navier-Stokes equations for the acoustic velocity and pressure.Following the approach in Schmidt et al., J. Math. Ind 8:15, 2018 for theviscoacoustic transmission problem of multiperforated plates the viscosity isscaled as $\delta^{4}$ with the period $\delta$ of the array of chambers andthe size of the necks as well as the wall thickness like $\delta^2$ such thatthe viscous boundary layers are of the order of the size of the necks. Applyingthe method of two-scale convergence we obtain with a stability assumption inthe limit $\delta \to 0$ that the acoustic pressure fulfills the Helmholzequation with impedance boundary conditions. These boundary conditions dependon the frequency, the length of the resonators and through the effectiveRayleigh conductivity -- that can be computed numerically -- on the shape oftheir necks. We compare the limit model to semi-analytical models in theliterature.

Surface homogenization of an array of Helmholtz resonators for a viscoacoustic model using two-scale convergence