  We investigate in detail gravitational waves in an Schwarzschild-anti-de
Sitter bulk spacetime surrounded by an Einstein static brane with generic
matter content. Such a model provides a useful analogy to braneworld cosmology
at various stages of its evolution, and generalizes our previous work
[gr-qc/0504023] on pure tension Einstein-static branes. We find that the
behaviour of tensor-mode perturbations is completely dominated by quasi-normal
modes, and we use a variety of numeric and analytic techniques to find the
frequencies and lifetimes of these excitations. The parameter space governing
the model yields a rich variety of resonant phenomena, which we thoroughly
explore. We find that certain configurations can support a number of lightly
damped `quasi-bound states'. A zero-mode which reproduces 4-dimensional general
relativity is recovered on infinitely large branes. We also examine the problem
in the time domain using Green's function techniques in addition to direct
numeric integration. We conclude by discussing how the quasi-normal resonances
we find here can impact on braneworld cosmology.
