Exploring the Relationship Between Cost Function of Hybrid Traveling Waves and Structural Reflection Coefficient Adapted From Acoustics

Abstract The Basilar Membrane (BM) is the structural component of the mammalian cochlea that transmits auditory information as traveling structural waves, and inner hair cells transduce acoustic waves into electrical impulses in the inner ear. These waves go up towards the cochlea’s apex from its base. The primary structure at the apex of the cochlea that keeps waves from returning to the base is the helicotrema. People can hear continuous sound waves without acoustic reflection or overlap because of this property of the BM. Our research is motivated by this biological phenomenon and aims to comprehend and passively reproduce it in engineering structures. By studying the dynamics of a uniform beam linked to a spring-damper system as a passive absorber, we can use this characteristic of the inner ear to explain some of the observed phenomenological behaviors of the basilar membrane. The spring-damper system’s position separates the beam into two dynamic regions: one with standing waves and the other with non-reflecting traveling waves. This study presents the computational realization of traveling waves co-existing with standing waves in the two different zones of the structure. Moreover, this study aims to establish a correlation between two approaches to analyze the characteristics of the wave profiles: (i) the absorption coefficient approach and (ii) the cost function based on the wave envelope. The Basilar Membrane (BM) serves as the crucial structural conduit for transmitting auditory information through traveling structural waves, with inner hair cells in the inner ear transducing these waves into electrical impulses. These waves ascend from the cochlea’s base towards its apex, and the helicotrema, positioned at the cochlear apex, plays a pivotal role in preventing wave reflection and overlap, thereby facilitating the perception of continuous sound waves. The intrinsic characteristics of the Basilar Membrane (BM) inspire our research as we seek to comprehend and passively replicate this phenomenon in simplified forms. The investigation involves the exploration of the dynamics exhibited by a uniform beam connected to a spring-damper system acting as a passive absorber. This chosen system allows us to take advantage of the unique property of the inner ear, shedding light on some of the observed phenomenological behaviors of the basilar membrane. The positioning of the spring-damper system engenders two distinct dynamic regions within the beam: one characterized by standing waves and the other by non-reflecting traveling waves. The comprehensive analysis incorporates analytical and computational aspects, providing a holistic understanding of the coexistence of traveling and standing waves within these two dynamic zones.