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In this paper, we consider a method for solving a system of Maxwell's equations in the case of time-dependent boundary conditions at the end of a waveguide with superconducting walls. An explicit analytical solution is obtained for a quasi-harmonic signal whose pulse width in the frequency domain is much smaller than the carrier frequency. Numerical examples are calculated in the case of a Gaussian pulse as a superposition of modes propagating in a circular hollow metal waveguide. The calculation of dynamic invariants of short pulses propagating in a waveguide with an arbitrarily-shaped conducting shell is considered. A procedure for quantizing an electromagnetic field in a waveguide with superconducting walls is described.