Question

You are traveling in a car toward a hill at a speed of \(40.0 \mathrm{mph} .\) The car's horn emits sound waves of frequency \(250 \mathrm{~Hz},\) which move with a speed of \(340 \mathrm{~m} / \mathrm{s}\) a) Determine the frequency with which the waves strike the hill. b) What is the frequency of the reflected sound waves you hear? c) What is the beat frequency produced by the direct and the reflected sounds at your ears?

Short answer

Answer: The beat frequency produced by the direct and the reflected sounds at our ears is 37.3 Hz.

Step-by-step solution

Convert the car's speed to meters per second

In this exercise, we are given the speed of the car in miles per hour (mph), but the speed of the sound waves is in meters per second (m/s). To find the frequency with which the waves strike the hill (the Doppler effect), we need to convert the car's speed to meters per second. To convert the speed of the car from miles per hour to meters per second, we can use the formula: \( Speed (m/s) = Speed (mph) * \frac{1609 meters}{1 mile} * \frac{1 hour}{3600 seconds} \) Given the car's speed is 40.0 mph: \(Speed (m/s) = 40.0 * \frac{1609}{1} * \frac{1}{3600} = 17.88 m/s\)

Determine the frequency with which the waves strike the hill

To determine the frequency with which the waves strike the hill considering the Doppler effect, we can use the following formula: \( f' = f * \frac{v + v_o}{v - v_s} \) Where: f' is the frequency of the wave when it strikes the hill f is the original frequency emitted by the car horn (250 Hz) v is the speed of the sound waves (340 m/s) v_o is the velocity of the observer (0 m/s, since the hill is stationary) v_s is the velocity of the source (17.88 m/s, calculated in Step 1) Plugging in the values: \( f' = 250 * \frac{340 + 0}{340 - 17.88} = 267.57 Hz \)

Determine the frequency of the reflected sound waves we hear

To find the frequency of the reflected sound waves we hear, we take into account the Doppler effect once again, as the sound waves are now moving towards the moving car (the listener). This time, the hill is considered the source, and the car is considered the observer. We use the same formula as in Step 2, modifying the variables: \( f'' = f' * \frac{v - v_o}{v + v_s} \) Where: f'' is the frequency of the reflected sound waves we hear f' is the frequency of the wave when it strikes the hill (267.57 Hz, calculated in Step 2) Plugging in the values: \( f'' = 267.57 * \frac{340 - 17.88}{340 + 17.88} = 287.30 Hz \)

Calculate the beat frequency produced by the direct and the reflected sounds

The beat frequency is the difference between frequencies of the direct and reflected sounds we hear. In this case, the direct sound has a frequency of 250 Hz, and the reflected sound has a frequency of 287.30 Hz (calculated in Step 3). To find the beat frequency, we subtract the direct sound frequency from the reflected sound frequency: \( Beat \ Frequency = f_{reflected} - f_{direct} \) \( Beat \ Frequency = 287.30 - 250 = 37.3 Hz \)

Answers

a) The frequency with which the waves strike the hill is 267.57 Hz. b) The frequency of the reflected sound waves we hear is 287.30 Hz. c) The beat frequency produced by the direct and the reflected sounds at our ears is 37.3 Hz.

Key concepts

Frequency Conversion in the Doppler Effect

In physics, frequency conversion is a core aspect when studying the Doppler effect, particularly in scenarios involving motion between a source of sound and an observer. The Doppler effect describes the change in frequency—or pitch—of a sound wave for an observer moving relative to the source of the sound.

When an object emitting sound is approaching an observer, the sound waves compress and the frequency increases, leading to a higher-pitched sound. Conversely, as the sound-emitting object moves away, the sound waves stretch, resulting in a lower frequency and a lower-pitched sound. This phenomenon is widely observed, from the siren of a passing ambulance to the changing pitch of a racing car as it zooms by.

Applying the Doppler Formula

Using the formula for Doppler effect, \[ f' = f \times \frac{v + v_o}{v - v_s} \],one can calculate the perceived frequency (\( f' \)) by considering the original frequency (\( f \)), speed of sound (\( v \)), and velocities of the observer (\( v_o \)) and source (\( v_s \)). This equation is key to solving problems related to frequency conversion due to the Doppler effect.

In the given textbook problem, the car horn’s frequency appears higher as it moves toward the hill, and then appears even higher when the sound waves, now reflected, travel back towards the moving car.

Sound Wave Propagation

Sound wave propagation describes how sound travels through a medium from the point of origin to the listener’s ears. Sound is a mechanical wave of pressure and displacement, requiring a medium like air, water, or solid for its transmission. It propagates outward from the source in all directions as a spherical wave. For sound waves, the speed of propagation is influenced by factors such as the medium's density, elasticity, and temperature.

Generally, sound travels fastest in solids and slowest in gases due to the differing molecular spacing and the forces between molecules. For instance, the speed of sound in air at standard temperature and pressure is about \( 340 \; \text{m/s} \), and it is this value that is used to solve related physics problems, including those related to the Doppler effect.

Beat Frequency

Beat frequency is perceived when two sound waves of similar, but not identical, frequencies interfere with each other. When these waves overlap, they produce a phenomenon known as interference, resulting in a new sound wave whose amplitude oscillates at a rate equal to the difference between the frequencies of the two waves. This oscillation is the beat frequency, which can be detected as a throbbing or pulsing effect in the sound heard.

Calculating Beat Frequency

The formula for determining the beat frequency (\( f_{beat} \)) is:\[ f_{beat} = |f_{high}-f_{low}| \],where \( f_{high} \) and \( f_{low} \) are the higher and lower frequencies involved, respectively.

In the provided exercise, the direct sound from the car horn and the reflected sound from the hill combine at the listener’s ears, creating a beat frequency. This is calculated by finding the difference between the car horn's emitted frequency and the perceived frequency of the reflected sound, showcasing the interference pattern that arises from the superposition of two similar frequencies.