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We present a homogenization method to treat the problem of the reflection of waves at the free boundary of an elastic body, the edge being structured periodically at the subwavelength scale. The problem is considered for shear waves and the wave equation in the time domain is considered. In the homogenized problem, a boundary condition at an equivalent flat edge is obtained, which links the normal stress to its derivatives, instead of the usual traction free condition. The problem of the position of the equivalent flat boundary with respect to the real roughnesses is addressed and this is done considering the equation of energy conservation in the homogenized problem and considering the accuracy of the homogenized solution when compared to the real one.

Effective boundary condition for the reflection of shear waves at the periodic rough boundary of an elastic body