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A hybrid numerical-analytical method for the characterization of the propagation of the transverse-electric surface waves along a high impedance surface (HIS) is presented. The HIS is obtained by a 2-D periodic repetition of a rectangular unit cell on a Cartesian grid. The unit cell consists of a microstrip line with variable width immersed in a grounded multilayer dielectric substrate. Inside the unit cell, the effective value of the dielectric is sinusoidally modulated along the direction of the microstrip line, and the dimension of the unit cell is equal to the period of the modulation. On the basis of the modulation parameters, limits of the stop/pass bands are determined analytically employing Mathieu's functions. The synthesis of the shape of the microstrip line that guarantees the desired modulation requires the numerical determination of the phase velocity - line width relationship which is not known in closed form for the considered unbounded open structure. The limits of the pass/stop bands determined analytically are in very good agreement with those found numerically and measured values for different modulation parameters.