Three-dimensional acoustic waves in the ear canal and their interaction with the tympanic membrane

The long and slender geometry of the ear canal supports an infinite number of cross-sectional acoustic modes. The lower mode(s) travel along the length of the ear canal, while the higher modes are trapped near the ends of the canal. Many of these waves are introduced as a result of the complex vibrational shape of the eardrum. A three-dimensional mathematical model of the ear canal is formulated that includes this acoustic interaction. The coupled system is solved using matched asymptotic expansions that take advantage of the small slenderness ratio. This solution in the ear canal is in the form of a series of modes, the first being the plane-wave solution. As an illustrative example, the analysis is applied to a geometry that partially represents the ear canal and eardrum of a cat. The results indicate that the plane-wave solution is supplemented by multidimensional trapped modes at low frequencies and by a limited number of traveling waves at high frequencies. The magnitude of these higher modes generally increases with frequency and can significantly influence the acoustic coupling of the ear.