  In the present work we analize the behavior of 5-dimensional gravitational
waves propagating on a Kaluza-Klein background and we face separately the two
cases in which respectively the waves are generated before and after the
process of dimensional compactification. We show that if the waves are
originated on a 5-d space-time which fulfills the principle of general
relativity, then the process of compactification can not reduce the dynamics to
the pure 4-dimensional scalar, vector and tensor degrees of freedom. In
particular, while the electromagnetic waves evolve independently, the scalar
and tensor fields couple to each other; this feature appears because, when the
gauge conditions are splitted, the presence of the scalar ripple prevents that
the 4-d gravitational waves are traceless. The phenomenological issue of this
scheme consists of an anomalous relative amplitude of the two independent
polarizations which characterize the 4-d gravitational waves. Such profile of
polarization amplitudes, if detected, would outline the extra-dimension in a
very reliable way, because a wave with non-zero trace can not arise from
ordinary matter sources. We discuss the above mentioned phenomenon either in
the case of a unit constant value of the background scalar component (when the
geodesic deviation is treated with precise outputs), and assuming such
background field as a dynamical degree (only qualitative conclusion are
provided here, because the details of the polarization amplitudes depend on the
choice of specific metric forms). Finally we perturb a real Kaluza-Klein theory
showing that in this context, while the electromagnetic waves propagate
independently, the 4-d gravitational waves preserve their ordinary structure,
while the scalar plays for them the role of source.
