In this paper, we derive a highly accurate numerical method for the solution\nof second order hyperbolic partial differential equations in one space dimension\nusing semidiscrete approximation. The space direction is discretized by wavelet-\nGalerkin method using some special types of basis functions obtained by integrating\nDaubechies functions which are compactly supported and differentiable. The time\nvariable is discretized by using various classical Newmark schemes. Theoretical and\nnumerical results are obtained for problems with Dirichlet, mixed and Neumann boundary conditions. Computational analysis of the stability of the solutions is also done. The computed solutions are found to be highly favourable as compared to the\nexact solution.