In this article we are interested in energy estimates for hyperbolic initial boundary value problem when surface waves occur. More precisely, we construct rigorous geometric optics expansions for so-called elliptic and mixed frequencies and we show, using those expansions, that the amplification phenomenon is greater in the case of mixed frequencies. As a consequence, this result allow us to give a partial classification of weakly well-posed hyperbolic initial boundary value problems according to the region where the uniform Kreiss Lopatinskii condition degenerates.
"Geometric optics expansions for linear hyperbolic boundary value problems and optimality of energy estimates for surface waves." Differential Integral Equations 27 (5/6) 531 - 562, May/June 2014.