This paper introduces a low-thickness sandwich waveguide structure comprising silicon nitride -sapphire -silicon nitride layers. By exploiting its dispersion wave radiation effect and mid-infrared phase matching condition, combined with the waveguide pulse transmission model, this study examines the impact of different physical sizes of the sandwich waveguide on the phase matching point and spectral broadening. Through numerical simulation, a supercontinuum spectrum ranging from 0.5 to 4 μm is generated, producing a farther mid-infrared dispersion wave at a -40 dB level. Moreover, this model provides an in-depth mechanism for nonlinear waveguide pulse transmission. Theoretical analysis reveals that modifying the physical size of the silicon nitride and sapphire interlayer and altering the phase-matching conditions can regulate the position of the dispersion wave across a broader wavelength range.