in [@Marcuvitz_1991; @Littlejohn_1993; @GCT15].

In this paper, we will use the WDF approach inside finite domains which may be regularly or irregularly shaped. Extending the treatment in [@GCT15], we will consider here multiple reflections and interference effects explicitly. We derive the ray-tracing limit in this case and discuss various limits leading to deviations from a pure ray-tracing approach. The ray-tracing component itself can be evaluated efficiently when combined with fast phase-space propagation methods such the [*Dynamical Energy Analysis*]{} (DEA) developed in the context of vibro-acoustics [@Tanner2009] and mesh-based implementation tools such as [*Discrete Flow Mapping*]{} (DFM) techniques [@Chappell2013].

We will focus here on linear and stationary (frequency domain) scalar wave problems using as reference point the wavenumber $k$ or the frequency $\omega$. When considering quantum systems, we may identify the scale $1/k$ with $\hbar$; extensions to vector wave equations is straightforward [@GG15]. In Sec. \[sec:trans\], we introduce the concept of transfer operators and write correlation functions of stationary wave fields in terms of these operators. We derive relations between correlation functions and phase space densities and verify the results with the help of a simple quantum map, the kicked Harper map [@Art_Scholar]. In Sec. \[sec:var\], we extend the results to 