  In this paper, we consider the one-dimensional isentropic compressible Euler
equations with source term $\beta(t,x)\rho|u|^{\alpha}u$ in a bounded domain,
which can be used to describe gas transmission in a nozzle.~The model is
imposed a subsonic time-periodic boundary condition.~Our main results reveal
that the time-periodic boundary can trigger an unique subsonic time-periodic
smooth solution and this unique periodic solution is stable under small
perturbations on initial and boundary data.~To get the existence of subsonic
time-periodic solution, we use the linear iterative skill and transfer the
boundary value problem into two initial value ones by using the hyperbolic
property of the system. Then the corresponding linearized system can be
decoupled.~The uniqueness is a direct by-product of the stability. There is no
small assumptions on the damping coefficient.
