Human hearing occurs when a sound wave traveling through the air enters the ear and strikes against the eardrum, or tympanic membrane, causing it to vibrate. The eardrum in turn passes the vibration to the cochlea via the ossicles (ear bones) as part of the complex process that converts physical vibration into an electrical signal which the brain registers as auditory perception. In the moments after a sound wave makes contact with the eardrum, the vibrating system of the eardrum and ossicles can be modeled as a simple harmonic oscillator, where the tension of the membrane provides the restoring force. (In an upcoming lecture, we will learn how this system could be better modeled as a damped oscillator due to the stiffening muscles of the inner ear, but simple harmonic oscillation is good enough for short time scales). Auricle External acoustic meatus Elastic cartilages Auditory ossicles Semicircular canals Tympanis moribrarie Tympanic cavity Vestibule Vestibulocochlear nerve Auditory tube Oval window -Cochlea Round window (a) In an experiment, researchers find that a certain frequency sound wave causes the patient's eardrum to displace 2 m backward from equilibrium every 2 ms. Write equations for the position r(t), velocity v(t), and acceleration a(t) of the eardrum over time. (b) If the oscillating system of the eardrum and ossicles has a combined mass of 65 mg, what is the spring constant of the patient's eardrum? How does this rank compared to a slinky (k = 0.15 N/m) or a car tire (k = 190,000 N/m)? (c) Damage to the eardrum can occur when it displaces more than 1 mm. What sound intensity level (SIL) would pose an immediate safety threat to this patient's hearing? Sound intensity level, L1, measured in decibels (dB) is given by the equation L = 10 log10 (), where p is sound pressure and Po is the reference sound pressure (typically po = 20 10-6 J/m). Sound pressure, likewise, can be approximated as p =, or force over volume. The dimensions of the ear drum are about 0.1 mm in thickness and 10 mm in diameter.