This paper is concerned with an interior penalty discontinuous Galerkin (IPDG) method
based on a flexible type of non-polynomial local approximation space
for the Helmholtz equation with varying wavenumber.
The local approximation space consists of multiple polynomial-modulated
phase functions which can be chosen according to the phase information of the solution.
We obtain some {approximation} properties for this space
and \textit{a prior} $L^2$ error estimates
for the \textit{h}-convergence of the IPDG method
using duality argument.
We also provide ample numerical examples to show that,
building phase information into the local spaces
often gives more accurate results comparing to using the
standard polynomial spaces.