need to construct a function satisfying wave equation.

Question

I need an example of a function that satisfies wave equation and that vanishes beyond certain range. I mean if $f(x,t)$ is a function of space and time, then $f(x, t) = 0, $ for $x < a(t) $ and $x>b(t)$ where $a(t)$ and $b(t)$ be any function of time such that $b(t)<a(t)$ for all time $t$. How to construct such function?

Answer 1

You can construct such things for some functions $a,b$. For example, if $a(t)=a_0+ct$ and $b(t)=b_0+ct$, then you can let $\varphi$ be a smooth function with compact support contained in $(b_0,a_0)$, and define $$f(x,t) = \varphi(x-ct)$$ This $f$ satisfies $f_{tt} = c^2 f_{xx}$.

On the other hand, if $a(t) = t^2+1$ and $b(t)=t^2$, then there is no such $f$, because the wave equation exhibits finite propagation speed.