  $L^2$ norm error estimates of semi- and full discretisations, using
bulk--surface finite elements and Runge--Kutta methods, of wave equations with
dynamic boundary conditions are studied. The analysis resides on an abstract
formulation and error estimates, via energy techniques, within this abstract
setting. Four prototypical linear wave equations with dynamic boundary
conditions are analysed which fit into the abstract framework. For problems
with velocity terms, or with acoustic boundary conditions we prove surprising
results: for such problems the spatial convergence order is shown to be less
than two. These can also be observed in the presented numerical experiments.
