Question

A person in a car blows a trumpet sounding at $438\mathrm{Hz}$. The car is moving toward a wall at $19.3\mathrm{m}/\mathrm{s}.$ Calculate $(a)$ the frequency of the sound as received at the wall and $(b)$ the frequency of the reflected sound arriving back at the source.

Short answer

After calculations, you'll find: (a) the frequency of the sound as received at the wall is higher than 438 Hz due to the Doppler Effect. And (b) the frequency of the reflected sound arriving back at the source is lower than the frequency observed at the wall but still higher than the original frequency of 438 Hz.

Step-by-step solution

Calculate the Observed Frequency

Using the formula $f^{\prime }=f\frac{v+v_0}{v}$ where $f^{\prime }$ is the observed frequency, $f=438$ Hz is the source frequency, $v=343$ m/s (the speed of sound in air), and $v_0=19.3$ m/s (the speed of the car). Substituting these values in the formula, we get $f^{\prime }=438\mathrm{Hz}\cdot \frac{343\mathrm{m}/\mathrm{s}+19.3\mathrm{m}/\mathrm{s}}{343\mathrm{m}/\mathrm{s}}$.

Calculate the Reflected Frequency

We use the formula $f^{″}=f\frac{v}{v+v_s}$ where $f^{″}$ is the reflected frequency for part (b). We've found in the previous step that $f^{\prime }$ is the frequency as perceived by the wall, so we'll use this as the new source frequency. $v_s=19.3$ m/s is still the relative speed of the observer (now the car in case of the reflected sound). We substitute these values as well as the speed of sound in the formula to find the reflected frequency: $f^{″}=f^{\prime }\frac{343\mathrm{m}/\mathrm{s}}{343\mathrm{m}/\mathrm{s}+19.3\mathrm{m}/\mathrm{s}}$.

Final Calculation

Calculate $f^{\prime }$ from the first step and then use this value to calculate $f^{″}$ in step 2. This will give us the observed frequency at the wall and the frequency of the reflected sound.

Key concepts

Sound Frequency

Sound frequency is a fundamental characteristic of sound waves; it represents the number of sound wave cycles that pass a point in one second and is measured in Hertz (Hz). A higher frequency means a higher pitch of sound, and a lower frequency results in a lower pitch. In the exercise, the trumpet in the car emits sound at a frequency of 438 Hz, which is the pitch as originally produced and heard in still air by someone inside the car.

Understanding the frequency of the source is crucial for solving problems related to the Doppler effect, as it is the original sound frequency that gets affected by the movement of the source or the observer. In daily life, the frequency of a sound affects not only musical instruments but also the perception of voices, environmental noises, and electronic devices.

Speed of Sound in Air

The speed of sound in air is the velocity at which sound waves travel through the air. It is affected by various factors such as temperature, humidity, and air pressure but is typically around 343 meters per second (m/s) at room temperature (20°C or 68°F).

This value is considered a constant in many physics problems, including those involving the Doppler effect. When solving the exercise, it is important to use the correct value for the speed of sound as it directly affects the observed and reflected frequencies. Knowing that sound travels faster in warmer air and slower in colder air might help you understand its behavior in different conditions.

Observed Frequency Calculation

The observed frequency calculation involves determining how the sound frequency is perceived by an observer when there is relative motion between the source of sound and the observer. This is described by the Doppler effect formula:\
$$f^{\prime }=f\frac{v+v_o}{v}$$
where $f^{\prime }$ is the observed frequency, $f$ is the source frequency, $v$ is the speed of sound in the medium, and $v_o$ is the speed of the observer towards the source. If the observer is moving away from the source, $v_o$ would be negative.

In our case, the observer is the wall, and it is stationary relative to the air. However, it 'perceives' an increase in frequency as the car (and thereby the sound source) approaches it. By substituting the given numbers into the formula, students can find how the frequency appears to change due to the relative motion.

Reflected Sound Frequency

The frequency of the reflected sound is the frequency at which the sound wave, after hitting a surface like a wall, is reflected back and then observed by the source or another observer. The Doppler effect comes into play again when we calculate the reflected sound frequency:\
$$f^{″}=f^{\prime }\frac{v}{v+v_s}$$
where $f^{″}$ becomes the new observed frequency of the reflected sound, $f^{\prime }$ is the frequency that hits the wall, $v$ is the speed of sound, and $v_s$ is the relative speed of the source regarding the reflected sound. In this problem, after the sound reflects off the wall, the car is now the 'observer' of its own sound, and the frequency it hears will be lower than the one that was emitted due to it moving away from the reflected sound.

This concept is experienced in real life when hearing the siren of an ambulance as it passes by: the pitch seems to drop the moment the vehicle starts moving away from the observer. For students, mastering reflected sound frequency calculations is essential in understanding real-world phenomena related to sound reflection.