Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide

In this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height $H$. The perforations are periodically placed along the ordinate axis at a distance $O(\epsilon )$ between them, where $\epsilon$ is a parameter that converges towards zero. Another parameter $\eta$, the Floquet-parameter, ranges in the interval $[-\pi ,\pi ]$. The boundary conditions are quasi-periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters $\epsilon$ and $\eta$ and strongly depend on $H$. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves.