Analytical aspects of an artificial damping layer for the wave equation

The use of damping layers is one way of absorbing water waves travelling on the free surface. In the mathematical formulation these layers are introduced by adding certain lower-order terms in the free surface equations. In this paper some analytical aspects of such an artificial damping layer are studied by considering the addition of comparable lower-order terms in the one-dimensional wave equation. The wave equation is closely connected with the formulation of a linear free-surface model describing gravity waves travelling over a potential flow. It is shown that the addition of such lower-order terms does not harm the well-posedness of a given Cauchy problem. Furthermore, reflection properties of the absorbing layer are presented and analytical solutions are given for layers in which the damping coefficient has simple lower order polynomial behaviour. It is shown that the form of the damping coefficient can affect the reflection properties considerably.