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 variables separation, have been found. The real valueness and the properties of the solutions produced by computer algebra programs are not always manifest and in this work we provide ready to use solutions. We discuss especially the potential $(m_1 + m_2 sinh x)cosh^{–2} x$. Such potential approximates the Schwarzschild black hole potential and its use for determining black holes quasi-normal modes is hinted to.
Solutions of all one-dimensional wave equations with time independent potential and separable variables
FERRARIS, Marco;
2004-01-01
Abstract
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 variables separation, have been found. The real valueness and the properties of the solutions produced by computer algebra programs are not always manifest and in this work we provide ready to use solutions. We discuss especially the potential $(m_1 + m_2 sinh x)cosh^{–2} x$. Such potential approximates the Schwarzschild black hole potential and its use for determining black holes quasi-normal modes is hinted to.
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