Failure of exterior observability at equal bulk and surface wave speeds in an annulus

We consider a wave equation in a planar annulus, with a Dirichlet condition on the outer circle and a dynamic wave equation on the inner circle. The bulk and surface propagation speeds are both one. For every finite observation time, we construct smooth real solutions with unit initial energy whose exterior normal-derivative observation tends to zero. The construction uses exact separated solutions with angular order \(m\) and frequency \(\lambda_m=m+O(m^{-1/3})\). Large-order Bessel asymptotics at the inner turning point give a lower bound for the surface energy, while an exact Wronskian identity fixes the outer flux. Thus exterior observability fails at equality of the two speeds in this annulus.