Chapter 5.6, Problem 56E

Doppler Effect When a car with its horn blowing drives by an observer, the pitch of the horn seems higher as it approaches and lower as it recedes (see the figure below). This phenomenon is called the Doppler effect. If the sound source is moving at speed v relative to the observer and if the speed of sound is v₀, then the perceived frequency f is related to the actual frequency f₀ as follows.

$f=f_0\left(\frac{v_0}{v_0\pm v}\right)$

We choose the minus sign if the source is moving toward the observer and the plus sign if it is moving away.

Suppose that a car drives at 110 ft/s past a woman standing on the shoulder of a highway, blowing its horn, which has a frequency of 500 Hz. Assume that the speed of sound is 1130 ft/s. (This is the speed in dry air at 70°F.)

1. (a) What are the frequencies of the sounds that the woman hears as the car approaches her and as it moves away from her?
2. (b) Let A be the amplitude of the sound. Find functions of the form

$y=A\sin \omega t$

that model the perceived sound as the car approaches the woman and as it recedes.