  We show that the 1D planar ultrarelativistic shock-tube problem with an
ultrarelativistic polytropic equation of state can be solved analytically for
the case of a working surface, i.e. for the case when an initial discontinuity
on the hydrodynamical quantities of the problem form two shock waves separating
from a contact discontinuity. The procedure is based on the extensive use of
the Taub jump conditions for relativistic shock waves, the Taub adiabatic and
performing Lorentz transformations to present the solution in a system of
reference adequate for an external observer at rest. The solutions are found
using a set of very useful theorems related to the Lorentz factors when
transforming between systems of reference. The energy dissipated inside the
working surface is relevant for studies of light curves observed in
relativistic astrophysical jets and so, we provide a full analytical solution
for this phenomenon assuming an ultrarelativistic periodic velocity injected at
the base of the jet.
