We study standing-wave solutions of Born-Infeld electrodynamics, with nonzero electromagnetic field in a region between two parallel conducting plates. We consider the simplest case which occurs when the vector potential describing the electromagnetic field has only one nonzero component depending on time and on the coordinate perpendicular to the plates. The problem then reduces to solving the scalar Born-Infeld equation, a nonlinear partial differential equation in 1+1 dimensions. We apply two alternative methods to obtain standing-wave solutions to the Born-Infeld equation: an iterative method, and a ``minimal surface method. We also study standing wave solutions in a uniform constant magnetic field background.
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