  A new space-time discontinuous Galerkin (dG) method utilising special Trefftz
polynomial basis functions is proposed and fully analysed for the scalar wave
equation in a second order formulation. The dG method considered is motivated
by the class of interior penalty dG methods, as well as by the classical work
of Hughes and Hulbert. The choice of the penalty terms included in the bilinear
form is essential for both the theoretical analysis and for the practical
behaviour of the method for the case of lowest order basis functions. A best
approximation result is proven for this new space-time dG method with
Trefftz-type basis functions. Rates of convergence are proved in any dimension
and verified numerically in spatial dimensions $d = 1$ and $d = 2$. Numerical
experiments highlight the effectiveness of the Trefftz method in problems with
energy at high frequencies.
