Fundamental solution of wave equation

[sunrah]

Homework Statement

show that
[itex]
E(x,t):= \frac{1}{2} \left\{ \begin{array}{ll}
1 & \mbox{if $|x|<t $};\\
0 & \mbox{else}.\end{array} \right. [/itex]

is a fundamental solution of the wave equation.

Homework Equations

[itex]LE = E_{tt} - \Delta E = \delta[/itex]

The Attempt at a Solution

firstly redefined [itex]E:=\frac{1}{2}H(t-|x|)[/itex]

but the second order derivatives mean [itex]LE:=\delta'(t-|x|)[/itex]

 

[mighty2000]

This means that the function E(x,t) satisfies the wave equation, as shown by the above equation. Therefore, E(x,t) is a fundamental solution of the wave equation. This can also be seen by the fact that E(x,t) satisfies the initial conditions for the wave equation, with E(x,0) = 0 and E_t(x,0) = 0 for all x. Therefore, E(x,t) is a valid solution for the wave equation and can be used as a fundamental solution.