Dealing with characteristics of First-Order Wave Equation with initial condition of $w(0,t)$

Question

Solve $$({\partial w}/{\partial t})+4{\partial w}/{\partial x}=0$$ with $w(0,t)=\sin(3t)$.

I know how to do this with $t=0$ and just $x$ but how do you approach this? I know that $x=4t+x_0$ but I get stumped as you have $f(t)$ not $f(x)$.

Answer 1

You already found that $w$ is constant on every line of the form $x-4t=C$, for any constant $C$. Therefore, $w$ can be written as $g(x-4t)$ where $g$ is some function to be determined. (Why? Because this form says precisely that as soon as we know which characteristic line the point $(x,t)$ is on, we know $w$.)

It remains to plug in the given condition: when $x=0$, $w=\sin 3t$. So, $$g(-4t)=\sin 3t$$ I'll leave for you to write down the formula for $g$, and formulate the answer $w(x,t)=g(x-4t)$.