  This paper concerns the existence of transonic shock solutions to the 2-D
steady compressible Euler system in an almost flat finite nozzle ( in the sense
that it is a generic small perturbation of a flat one ), under physical
boundary conditions proposed by Courant-Friedrichs in
\cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at
the exit of the nozzle. In the resulting free boundary problem, the location of
the shock-front is one of the most desirable information one would like to
determine. However, the location of the normal shock-front in a flat nozzle can
be anywhere in the nozzle so that it provides little information on the
possible location of the shock-front when the nozzle's boundary is perturbed.
So one of the key difficulties in looking for transonic shock solutions is to
determine the shock-front. To this end, a free boundary problem for the
linearized Euler system will be proposed, whose solution will be taken as an
initial approximation for the transonic shock solution. In this paper, a
sufficient condition in terms of the geometry of the nozzle and the given exit
pressure is derived which yields the existence of the solutions to the proposed
free boundary problem. Once an initial approximation is obtained, a further
nonlinear iteration could be constructed and proved to lead to a transonic
shock solution.
