  A cocoon is a by-product of a propagating jet that results from shock heating
at the jet head. Herein, considering simultaneous cocoon formation, we study
the stability of relativistic jets propagating through the uniform ambient
medium. Using a simple analytic argument, we demonstrate that independent from
the jet launching condition, the effective inertia of the jet is larger than
that of the cocoon when the fully relativistic jet oscillates radially owing to
the pressure mismatch between jet and cocoon. In such situations, it is
expected that the onset condition for the oscillation-induced Rayleigh-Taylor
instability is satisfied at the jet interface, resulting in the destabilization
of the relativistic jet during its propagation. We have quantitatively verified
and confirmed our prior expectation by performing relativistic hydrodynamic
simulations in three dimensions. The possible occurrences of the
Richtmyer-Meshkov instability, oscillation-induced centrifugal instability, and
Kelvin-Helmholtz instability are also discussed.
