  We present a general method which can be used for geometrical and physical
interpretation of an arbitrary spacetime in four or any higher number of
dimensions. It is based on the systematic analysis of relative motion of free
test particles. We demonstrate that local effect of the gravitational field on
particles, as described by equation of geodesic deviation with respect to a
natural orthonormal frame, can always be decomposed into a canonical set of
transverse, longitudinal and Newton-Coulomb-type components, isotropic
influence of a cosmological constant, and contributions arising from specific
matter content of the universe. In particular, exact gravitational waves in
Einstein's theory always exhibit themselves via purely transverse effects with
D(D-3)/2 independent polarization states. To illustrate the utility of this
approach we study the family of pp-wave spacetimes in higher dimensions and
discuss specific measurable effects on a detector located in four spacetime
dimensions. For example, the corresponding deformations caused by a generic
higher-dimensional gravitational waves observed in such physical subspace, need
not be tracefree.
