On Symmetry Analysis in Finding Solutions of the One Dimensional Wave Equation

In this paper we obtain the Lie invariance condition for second order partial differential equations. This condition is used to obtain the determining equations of the 1-dimensional wave equation with constant speed. The determining equations are split to obtain an overdetermined system of partial differential equations which are solved to obtain the symmetries of the wave equation. By making an appropriate transformation between the dependent and independent variable, the wave equation is reduced to an easily solvable ordinary differential equation. We solve this resulting differential equation to obtain the solutions of the wave equation. In particular, the one dimensional wave equation with unit speed has been solved.