  As an application of the Cartan invariants obtained using the Karlhede
algorithm, we study a simple subclass of the PP-wave spacetimes, the
gravitational plane waves. We provide an invariant classification of these
spacetimes and then study a few notable subcases: the linearly polarized plane
waves, the weak-field circularly polarized waves, and another class of plane
waves found by imposing conditions on the set of invariants. As we study these
spacetimes we relate the invariant structure (i.e., Cartan scalars) to the
physical description of these spacetimes using the geodesic deviation equations
relative to timelike geodesic observers.
