Gabrielle Allen
Horst Beyer



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

Ψ(r,t) = 1 r f(r + t) + g(r t) (1)

where the functions f and g can be freely chosen.

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

f(r) = g(r) (2)

Thus if the solution at t=0 is given by ϕ(r), the general solution will be

Ψ(r,t) = 1 2r (r + t)ϕ(r + t) + (r t)ϕ(r t) (3)