We present analytic solutions to the scalar and vector wave equation with variable coefficients in one spatial dimension using Laplace transform methods. These solutions are used in convergence tests of finite difference methods satisfying a summation-by-parts rule with weak enforcement of boundary conditions through the simultaneous approximation term with order of accuracy 2, 4, 6 and 8. We explore linearly varying wave speeds that contain singularities. Constant and smooth, linearly varying wave speeds yield standard convergence results, whereas convergence rates for piecewise linear, non-smooth wave speeds drop to 2 in all cases, and drop to 1 for wave speeds with a jump discontinuity. Plots of discrete and continuous spectra suggest strict stability for constant and linearly varying wave speeds, which is lost with non-smooth coefficients.