  In this paper a mathematical model is given for the scattering of an incident
wave from a surface covered with microscopic small Helmholtz resonators, which
are cavities with small openings. More precisely, the surface is built upon a
finite number of Helmholtz resonators in a unit cell and that unit cell is
repeated periodically. To solve the scattering problem, the mathematical
framework elaborated in [Ammari et al., Asympt. Anal., 2019] is used. The main
result is an approximate formula for the scattered wave in terms of the lengths
of the openings. Our framework provides analytic expressions for the scattering
wave vector and angle and the phase-shift. It justifies the apparent
absorption. Moreover, it shows that at specific lengths for the openings and a
specific frequency there is an abrupt shift of the phase of the scattered wave
due to the subwavelength resonances of the Helmholtz resonators. A numerically
fast implementation is given to identify a region of those specific values of
the openings and the frequencies.
