The dynamic response of a massless rigid strip foundation of semi-circular cross-section embedded in a uniform or layered viscoelastic half-space is considered. The foundation is subjected to external harmonic forces and moments and to plane homogeneous P-, SV- and SH-waves. Numerical results are obtained by two indirect boundary integral equation methods: a reciprocity-based approach and a version of the least squares approach. The results for a uniform half-space are tested against the exact solution for longitudinal vibrations and against earlier results for the in-plane impedance functions and for the in-plane response to plane waves. The deffects of Poisson's ratio, material attenuation and layering on the impedance functions and on the scattering coefficients (response to waves) are described in some detail. The appearance of fictitious eigen-frequencies at which the solution may fail is also discussed.