If we have $u_t + c(x,t) u_x = 0 \; \; $ describes uni-directional wave propagation in a medium with variable wave speed.
a) Explain how to solve it by the method of charichtaristics for general $c(x,t)$ and Cauchy data $u(x,0)=f(x)$.
b) If $c=1+\epsilon \sin x$ with small parameter $\epsilon$ goes to $0$, find the explicit form of the solution including terms up to $O(\epsilon)$.