Solutions of all one-dimensional wave equations with time independent potential and separable variables

TL;DR

This paper derives exact, explicit solutions for all one-dimensional wave equations with specific time-independent potentials using variable separation, including a potential relevant to black hole physics.

Contribution

It provides explicit solutions for all wave equations with eight types of potentials, including a potential modeling black hole behavior, filling a gap in available analytical solutions.

Findings

Explicit solutions for eight classes of potentials.

Solutions include a potential approximating Schwarzschild black hole.

Ready-to-use solutions from computer algebra outputs.

Abstract

Exact solutions, in terms of special functions, of all wave equations $% u_{xx} - u_{tt} = V(x) u(t,x)$, characterised by eight inequivalent time independent potentials and by variable separation, have been found. The real valueness of the solutions from computer algebra programs is not always manifest and in this work we provide ready to use solutions. We discussed especially the potential $\cosh^{-2}x (m_1 + m_2 \sinh x)$. Such potential approximates the Schwarzschild black hole potential for even parity.