Fundamental Solution of the Wave Equation in One Dimension

Question

I am trying to prove that $$E(x,t)=\frac{1}{2}H(t)H({t}^{2}−{x}^{2}),$$ where $H(\cdot)$ is the Heaviside function, is the fundamental solution for the operator $L={∂}_{tt}−{∂}_{xx}$. Note that the fundamental solution is a distribution which satisfies $LE=δ$, where $δ$ is the Dirac distribution.

Answer 1

It will make your life a ton easier if you first define the coordinate transform $u=t+x$ and $v=t-x$ and factor the wave operator as $\partial_u\partial_v$. Then, forget about the current answer and assume $E(u,v)=A(u)B(v)$. Show then that $\partial_u A(u)=\delta_{u=0}$ and similarly for $B$. This implies $A(u)=H(u)+a$ and $B(v)=H(v)+b$. Now show that $a=b=0$ because you're considering the future-time solution ($t\geq 0$).

Conclude that in the original coordinates, with proper restrictions on $t,x$:

$E(t,x):=(1/2)H(t-x)H(t+x)=(1/2)H(t)H(t^2-x^2)$