We study the interaction of noise and regular signals with a front whose steepness increases or decreases owing to nonlinear distortion of the profile of an intense pumping wave. Projective transformation is used, which is a result of one of the Burgers equation symmetries. Signal interaction with the pumping wave at its leading edge results in an increase in signal amplitude, a decrease in its time scale, an increase in the signal evolution rate, and earlier merging of discontinuities. At the trailing edge, an increase in signal amplitude, an increase in the time scale, and deceleration of the evolution rate occur. Formulas are obtained that describe the transformation of the spectrum and the correlation function of noise. Laws of the change in noise energy for both small and large Reynolds numbers are found. We study the interaction of weak noise with a nonstationary shock front in a medium with a finite viscosity. It is shown that, owing to competition between amplification at the shock front and high-frequency attenuation, the dependence on the noise intensity on distance has a nonmonotonic character, and at large distances, the intensity tends to zero, while the correlation time tends to a finite value.