## IDScalarWave

Date

Abstract

Initial Data for the 3D Scalar Wave Equation

### 1 Purpose

This thorn provides diﬀerent initial data for the 3D scalar wave equation.

### 2 Spherically Symmetric Solutions

The general spherically symmetric solution can be written

 $\Psi \left(r,t\right)=\frac{1}{r}\left(f\left(r+t\right)+g\left(r-t\right)\right)$ (1)

where the functions $f$ and $g$ can be freely chosen.

Making the additional requirement of time symmetry at $t=0$, forces

 $f\left(r\right)=g\left(r\right)$ (2)

Thus if the solution at t=0 is given by $\varphi \left(r\right)$, the general solution will be

 $\Psi \left(r,t\right)=\frac{1}{2r}\left(\left(r+t\right)\varphi \left(r+t\right)+\left(r-t\right)\varphi \left(r-t\right)\right)$ (3)