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In this paper a mathematical model is given for the scattering of an incidentwave from a surface covered with microscopic small Helmholtz resonators, whichare cavities with small openings. More precisely, the surface is built upon afinite number of Helmholtz resonators in a unit cell and that unit cell isrepeated periodically. To solve the scattering problem, the mathematicalframework elaborated in [Ammari et al., Asympt. Anal., 2019] is used. The mainresult is an approximate formula for the scattered wave in terms of the lengthsof the openings. Our framework provides analytic expressions for the scatteringwave vector and angle and the phase-shift. It justifies the apparentabsorption. Moreover, it shows that at specific lengths for the openings and aspecific frequency there is an abrupt shift of the phase of the scattered wavedue to the subwavelength resonances of the Helmholtz resonators. A numericallyfast implementation is given to identify a region of those specific values ofthe openings and the frequencies.

A mathematical and numerical framework for gradient meta-surfaces built upon periodically repeating arrays of Helmholtz resonators