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The shear horizontal wave is solved using the high-order waveguide modal theory.The convergence and correctness are proved.The band structure caused by geometry-induced mismatched impedance is revealed.The rainbow trapping of shear horizontal waves is achieved.The high-order waveguide modal theory, usually used in electromagnetics and acoustics, is adopted to investigate the propagation properties of shear horizontal waves in a periodic stubbed plate. Beyond the sub-wavelength regime, higher-order modes are included to calculate the exact band structures caused by the stubs. Theoretical solutions are obtained in a closed form, in which both the dynamic governing equations and the boundary conditions are strictly satisfied. It is shown that the proposed modelling approach exhibits good convergence and accuracy, in agreement with results obtained from the finite element method. After a systematic investigation on the influence of the stub on the evolution of the band structures, the so-called rainbow trapping phenomenon of SH waves is revealed and explored in a graded stubbed plate with monotonously increasing height or width of the stubs, featuring an obvious reduction of the group velocity and blocking of the wave propagation at different locations for SH waves of different frequencies. The proposed model is expected to provide a useful theoretical tool for the physical mechanism exploration, structural design and eventually system optimization to guide various engineering applications of SH waves.