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A rigorous solution is derived for the propagation characteristics and field distributions of waves guided by a plane surface which possesses a surface reactance modulated sinusoidally in the propagation direction. The explicit field amplitudes and the determinantal equation for the propagation wavenumber are expressed in a continued fraction form which is rapidly convergent for all values of modulation. Numerical results are obtained for both surface wave (modal) and leaky wave (nonmodal) solutions. The relevance of these studies to high-gain modulated surface-wave antennas is discussed.