We investigate nonlinear plane-wave solutions of the classical Minkowskian Yang-Mills (YM) equations of motion. By imposing a suitable ansatz which solves Gauss' law for the SU(3) theory, we derive solutions which consist of Jacobi elliptic functions depending on an enumerable set of elliptic modulus values. The solutions represent periodic anharmonic plane waves which possess arbitrary nonzero mass and are exact extrema of the nonlinear YM action. Among them, a unique harmonic plane wave with a nontrivial pattern in phase, spin, and color is identified. Similar solutions are present in the SU(4) case, while they are absent from the SU(2) theory.
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