The present work investigates the problem of finite amplitude longitudinal oscillations in a straight duct with initially present temperature gradient. The one-dimensional gasdynamics equations: continuity, momentum and energy; with appropriate boundary conditions specified at both the ends are solved using Galerkin method. The formulation leads to a system of coupled nonlinear ordinary differential equations in time. These equations are solved numerically from initial state till the limit cycle. An example of linear temperature gradient is considered. The temperature gradient is increased form zero to a finite value, and in the each case, the limit cycle waveform is obtained. In case of zero temperature gradient, the pressure waveform has shock that decays over a saw tooth profile. This is same as obtained in the previous studies; both experimental and numerical. As the temperature gradient is increased, the pressure waveform changes its shape. At high temperature gradients, the shock disappears and the wave form becomes smoother. The nonlinear oscillations in temperature gradient show softening behavior.