The semicircular canals are biomechanical sensors responsible for detecting and encoding angular motion of the head in 3D space. Canal afferent neurons provide essential inputs to neural circuits responsible for representation of self-position/orientation in space, and to compensatory circuits including the vestibulo-ocular and vestibulo-collic reflex arcs. In this work we derive, to our knowledge, a new 1D mathematical model quantifying canal biomechanics based on the morphology, dynamics of the inner ear fluids, and membranous labyrinth deformability. The model takes the form of a dispersive wave equation and predicts canal responses to angular motion, sound, and mechanical stimulation. Numerical simulations were carried out for the morphology of the human lateral canal using known physical properties of the endolymph and perilymph in three diverse conditions: surgical plugging, rotation, and mechanical indentation. The model reproduces frequency-dependent attenuation and phase shift in cases of canal plugging. During rotation, duct deformability extends the frequency bandwidth and enhances the high frequency gain. Mechanical indentation of the membranous duct at high frequencies evokes traveling waves that move away from the location of indentation and at low frequencies compels endolymph displacement along the canal. These results demonstrate the importance of the conformal perilymph-filled bony labyrinth to pressure changes and to high frequency sound and vibration.