Question

A source traveling to the right at a speed of \(10.00 \mathrm{~m} / \mathrm{s}\) emits a sound wave at a frequency of \(100.0 \mathrm{~Hz}\). The sound wave bounces off a reflector, which is traveling to the left at a speed of \(5.00 \mathrm{~m} / \mathrm{s}\). What is the frequency of the reflected sound wave detected by a listener back at the source?

Short answer

Answer: The frequency of the reflected sound wave detected by the listener is approximately 99.06 Hz.

Step-by-step solution

Understand the Doppler effect formula

The Doppler effect formula in the one-dimensional case for the frequency perceived by an observer relative to the frequency emitted by a source is given by: \(f_{observed} = f_{source} \frac{(v \pm v_{observer})}{(v \pm v_{source})}\) where \(f_{observed}\) is the frequency detected by the observer, \(f_{source}\) is the frequency emitted by the source, \(v\) is the speed of sound in the medium, \(v_{observer}\) is the speed of the observer with respect to the medium (positive when going towards the source), and \(v_{source}\) is the speed of the source with respect to the medium (positive when going away from the observer). We will need to use this formula twice: once for the movement of the source and reflector, and once for the movement of the reflector and listener.

Calculate the frequency detected by the reflector

First, we need to find the frequency detected by the reflector due to the movement of the source and reflector. In this case, the source is moving away from the reflector, so \(v_{source} = 10.0\mathrm{~m/s}\). The reflector (consider it as an observer) is moving towards the source, so \(v_{observer} = -5.0\mathrm{~m/s}\). Assuming the speed of sound in the medium is constant at \(v = 343\mathrm{~m/s}\), we can calculate the frequency detected by the reflector as follows: \(f_{reflector} = f_{source} \frac{(v + v_{observer})}{(v - v_{source})} = 100.0 \frac{(343 - 5.0)}{(343 + 10.0)} = 95.52\mathrm{~Hz}\)

Calculate the frequency detected by the listener

Now we need to find the frequency detected by the listener when the sound wave is reflected back to them. In this case, the reflector is acting as a new source of the sound wave, so we need to use its frequency as the new source frequency. The listener (now considered as the observer) is stationary so \(v_{observer} = 0\mathrm{~m/s}\). The reflector (now considered as the source) is moving away from the listener, so \(v_{source} = -5.0\mathrm{~m/s}\). We can calculate the frequency detected by the listener as follows: \(f_{listener} = f_{reflector} \frac{(v + v_{observer})}{(v - v_{source})} = 95.52 \frac{(343 + 0.0)}{(343 - (-5.0))} = 99.06\mathrm{~Hz}\)

Determine the frequency of the reflected sound wave detected by the listener

The frequency of the reflected sound wave detected by the listener is \( f_{listener}\). Therefore, the frequency detected by the listener is approximately \(99.06\mathrm{~Hz}\).

Key concepts

Understanding Sound Wave Frequency

Sound wave frequency is a critical concept in studying the characteristics of sound and how it behaves in various mediums. It refers to the number of pressure oscillations or sound vibrations that take place in one second, and it's measured in hertz (Hz). For example, a frequency of 100 Hz means that there are 100 cycles of sound wave oscillations occurring every second.

Frequency determines the pitch of the sound we hear—higher frequencies correlate with higher pitches, while lower frequencies result in lower pitches. In the given exercise, a sound wave emitted at 100.0 Hz by a moving source must be analyzed in the context of the Doppler effect to find how its frequency changes when reflected back by another moving object.

Exploring the Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium and it varies depending on the medium's density and temperature. In air at room temperature (around 20°C or 68°F), the speed of sound is approximately 343 meters per second (m/s). This speed becomes an essential variable when applying the Doppler effect formula, as demonstrated in the textbook solution.

It's vital to remember that sound travels faster in solids and liquids than in gases because particles in denser mediums can transfer the vibrations more quickly. Understanding this concept helps in appreciating why certain assumptions, such as the speed of sound being constant at 343 m/s in air, are made in solving physics problems.

Analyzing Motion Relative to the Sound Source

The relative motion between the sound source, listener, and medium significantly influences the perceived frequency due to the Doppler effect. The Doppler effect occurs when there is a change in frequency (and consequently pitch) of the sound as perceived by an observer because the source of sound is moving relative to the observer.

When a sound source moves towards an observer, the sound waves are compressed, leading to a higher frequency and pitch. Conversely, if the sound source moves away, the waves are stretched out, resulting in a lower frequency and pitch. The problem presented showcases a practical application of the Doppler effect, where sound reflects off a moving object, thus requiring us to consider the motion of both the source and the reflector (acting as a new source) to determine the final frequency detected back at the original source.