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A left-handed rectangular waveguide periodically filled with tangentially magnetized ferrite medium has been proposed, and backward wave characteristics in the structure have been confirmed numerically by using a mode-matching method. The operational frequency is selected to be below the cutoff frequency for a dominant TE10 mode in a rectangular waveguide where the effective permittivity is negative. The obtained results show that there exist backward-wave modes in the negative permeability frequency region where magnetostatic surface wave modes can be supported, and the field profiles have a sinusoid-like form composed of a series of exponentials in each section. In addition, three-dimensional full-wave simulation is demonstrated, which shows that unexpected excitations of magnetostatic surface wave mode transversely propagating along a surface of the ferrite layer near the input port plays a role of a transformer from a guided TE10 mode to the backward wave mode and vice versa near the output port.