A representation formula for the wave equation revisited

Abstract

We give an elementary, rigorous and self-contained derivation of a Kirchhoff representation formula for solutions of the wave equation for a bounded region in three-dimensional space for the case of non-homogeneous initial conditions. A special feature of the result is the role of spherical caps which seems not to be noticed previously. The formula is readymade for boundary integral formulations of initial/boundary-value problems for the wave operator.

Keywords:

• wave equation
• Kirchhoff formula
• fundamental solution
• Dirac delta
• mollifier
• Green's formula
• initial/boundary-value problem

AMS Subject Classifications::

• 35C15
• 35L05

Acknowledgements

We are grateful to Ivar Stakgold for advice and encouragement.