The method of weighted residuals can efficiently enforce time-periodic solutions of flexible structures experiencing unilateral contact. The Harmonic Balance Method (HBM) based on Fourier expansion of the sought solution is a common formulation, though wavelet bases that can sparsely define nonsmooth solutions may be superior. This hypothesis is investigated using an axially vibrating rod with unilateral contact conditions. A distributional formulation in time is introduced allowing L^2(S^1)^N trial functions to approximate the second-order equations. The mixed wavelet Petrov-Galerkin solutions are found to yield consistent or better results than HBM, with similar convergence rates and seemingly more accurate contact force prediction.