Plasma-wave behavior in multiple dimensions is studied using two- and three-dimensional particle-in-cell simulations. We find that large-amplitude waves with kλ(D)≳0.2, where k is the wave number of the wave and λ(D) is the Debye length, localize in the transverse direction around their axis due to nonlinear, local damping caused by transiting particles. The center of the wave behaves like a plane wave in which trapped particles maintain a quasisteady state at approximately constant amplitude, while the transverse edges damp away.