Chapter 5.6, Problem 44E

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.

To determine

To evaluate: The frequencies of the sound that the woman hears as the car approaches her and as it moves away.

Answer to Problem 44E

The frequency of the sound as the car approaches her is $553.9\text{Hz}$ and the frequency for the sound as the car moves away is $455.6\text{Hz}$.

Explanation of Solution

Given:

The speed of car v, sound $v_0$ is $110\text{ft/s}$, $1130\text{ft/s}$ and the perceived frequency $f_0$ is $500\text{Hz}$.

Formula used:

The relation for the perceived frequency f is related to actual frequency $f_0$ is,

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

Where,

f is the perceived frequency, $f_0$ is the actual frequency, $v_0$ is the speed of the sound, v is the car speed.

Calculation:

Calculate the frequency of the sound as the car approaches to woman.

Since the car is approaches to woman. So, take the negative sign,

The formula for frequency is,

$f=f_0\left(\frac{v_0}{v_0\pm v}\right)$ (1)

Substitute $110\text{ft/s}$ for v, $1130\text{ft/s}$ for $v_0$ and $500\text{Hz}$ for $f_0$ in equation (1),

$\begin{array}{c}f=500\left(\frac{1130}{1130-110}\right)\\ =500\times \frac{1130}{1020}\end{array}$

Further solve the equation,

$\begin{array}{c}f=500\times 1.1078\\ =553.921\end{array}$

Thus, the value of frequency of the sound that the woman hears as the car approaches to woman is $553.9\text{Hz}$.

Calculate the frequency of the sound as the car moves away from her,

Since, the car moves away from her. So, take the positive sign,

Thus, the formula for frequency is,

$f=f_0\left(\frac{v_0}{v_0+v}\right)$ (2)

Substitute $110\text{ft/s}$ for v, $1130\text{ft/s}$ for $v_0$ and $500\text{Hz}$ for $f_0$ in equation (1),

$\begin{array}{c}f=500\left(\frac{1130}{1130+110}\right)\\ =500\times \frac{1130}{1240}\end{array}$

Further solve the equation,

$\begin{array}{c}f=500\times 0.9112\\ =455.6451\end{array}$

Thus the value of frequency of the sound that the woman hears as the car moves away to woman is $455.6\text{Hz}$.

To determine

The function that shows the perceived sounds as the car approaches to woman and as it recedes.

Answer to Problem 44E

The function which shows the perceived sounds as the car approaches to woman and as it recedes is $y=A\sin 1107.8\pi t$ and $y=A\sin 911.3\pi t$.

Explanation of Solution

Given:

The frequency of the function is $553.9\text{Hz}$.

Calculation:

The general simple harmonic motion is,

$y=A\sin \omega t$ (1)

The formula for f is,

$f=\frac{\omega }{2\pi }$ (2)

Substitute $553.9\text{Hz}$ for frequency in equation (2),

$\begin{array}{c}553.9=\frac{\omega }{2\pi }\\ \omega =553.9\times 2\pi \\ =1107.8\pi \end{array}$

The value of $\omega$ is $1107.8\pi$.

Substitute $1107.8$ for $\omega$ in equation (1),

$y=A\sin 1107.8\pi t$

Thus, The function which shows the perceived sounds as the car approaches to woman is $y=A\sin 1107.8\pi t$.

Substitute $455.6\text{Hz}$ for frequency in equation (2),

$\begin{array}{c}455.6=\frac{\omega }{2\pi }\\ \omega =455.6\times 2\pi \\ =911.3\pi \end{array}$

Thus, the value of $\omega$ is $911.3\pi$.

Substitute $911.3\pi$ for $\omega$ in equation (1),

$y=A\sin 911.3\pi t$

Thus, The function which shows the perceived sounds as the car recedes to woman is $y=A\sin 911.3\pi t$.