Chapter 6.2, Problem 65E

Pressure on the Eardrum See Example 6. No musical instrument can generate a true pure tone. A pure tone has a unique, constant frequency and amplitude that sounds rather dull and uninteresting. The pressures caused by pure tones on the eardrum are sinusoidal. The change in pressure P in pounds per square foot on a person’s eardrum from a pure tone at time t in seconds can be modeled using the equation

P = A sin(2πft + Φ),

where f is the frequency in cycles per second, and Φ is the phase angle. When P is positive, there is an increase in pressure and the eardrum is pushed inward. When P is negative, there is a decrease in pressure and the eardrum is pushed outward. (Source: Roederer, J., Introduction to the Physics and Psychophysics of Music, Second Edition, Springer-Verlag.)

(a) Determine algebraically the values of t for which P= 0 over [0, 0.005].

(b) From a graph and the answer in part (a), determine the interval for which P ≤ 0 over [0, 0.005].

(c) Would an eardrum hearing this tone be vibrating outward or inward when P < 0?