  The purpose of this article is to develop a Hadamard-Riesz type parametrix
for the wave propagator in bounded planar domains with smooth, strictly convex
boundary. This parametrix then allows us to rederive an oscillatory integral
representation for the wave trace appearing in \cite{MaMe82} and compute its
principal symbol explicitly in terms of geometric data associated to the
billiard map. This results in new formulas for the wave invariants. The order
of the principal symbol, which appears to be inconsistent in the works of
\cite{MaMe82} and \cite{Popov1994}, is also corrected. In those papers, the
principal symbol was never actually computed and to our knowledge, this paper
contains the first explicit formulas for the principal symbol of the wave
trace. The wave trace formulas we provide are localized near both simple
lengths corresponding to nondegenerate periodic orbits and degenerate lengths
associated to one parameter families of periodic orbits tangent to a single
rational caustic. Existence of a Hadamard-Riesz type parametrix with explicit
symbol and phase calculations in the interior appears to be new in the
literature, with the exception of the author's previous work \cite{Vig18} in
the special case of elliptical domains. This allows us to circumvent the symbol
calculus in \cite{DuGu75} and \cite{HeZe12} when computing trace formulas,
which are instead derived from integrating our explicit parametrix over the
diagonal.
