Solution to one-dimensional Wave Equation with Method of Characteristics

I'm not sure I understand exactly what your specific problem is, but for a general description of the method of characteristics for the one-dimensional wave equation $$u_t+c(u)u_x=0$$ (which includes the linear case with $c(u)=c_0$ constant), I can recommend Chapter 2 of Whitham's Linear and Nonlinear Waves.

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Real Analysis

Wave Equation

Partial Differential Equations

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