  We study the transversal wave equation on a compact Riemannian foliated
manifold. As applications, we get an Egorov's type theorem for transversally
elliptic operators, state a relationship between the singularities of the
Fourier transform of the spectrum distribution function of a transversally
elliptic operator and the corresponding transversal bicharacteristic flow and
give a description of the noncommutative geodesic flow in transversal geometry
of Riemannian foliations.
