  Dirac-Born-Infeld scalar field theories which appear in the context of
inflation in string theory in general have a field dependent speed of sound. It
is however possible to write down DBI models which possess exact solutions
characterized by a constant speed of sound different from unity. This requires
that the potential and the effective D-brane tension appearing in a DBI action
have to be related in a specific way. This paper describes such models in
general and presents some examples with a constant speed of sound $c_s<1$ for
which the spectrum of scalar perturbations can be found analytically without
resorting to the slow roll approximation.
