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We investigate the dispersive behavior of shear horizontal (SH) surface waves propagating in a layered structure consisting of a piezoelectric layer and an elastic half-space, in which the top and bottom of the layer are electrically shorted. The interface between the layer and the half-space is assumed to be imperfect bonding. The degree of imperfection of the interface is described by the so-called shear-lag model. The dispersion equations are expressed in an explicit closed form. The phase velocities are calculated to show the influences of the interfacial imperfection and the material properties of piezoelectric layers on the dispersive characteristics.