Pseudospectra for the wave equation with an absorbing boundary

For systems which can be described by u(t) = Au, with a highly nonnormal matrix or operator A, the spectrum of A may describe the behavior of the system poorly. One such operator arises from the one-dimensional wave equation on a finite interval with a homogeneous Dirichlet condition at one end and a linear damping condition at the other. In this paper the pseudospectra (norm of the resolvent) of this operator are computed in an energy norm, using analytical techniques and computations with discrete approximations. When the damping condition is perfectly absorbing, the pseudospectra are half-planes parallel to the imaginary axis, and in other cases they are periodic in the imaginary direction and approximate strips of finite thickness. The nonnormality of the operator is related to the behavior of the system and the limitations of spectral analysis.