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A siren mounted on the roof of a firehouse emits sound at a frequency of \(900 \mathrm{Hz}\). A steady wind is blowing with a speed of \(15.0 \mathrm{m} / \mathrm{s} .\) Taking the speed of sound in calm air to be \(343 \mathrm{m} / \mathrm{s}\), find the wavelength of the sound (a) upwind of the siren and (b) downwind of the siren. Firefighters are approaching the siren from various directions at \(15.0 \mathrm{m} / \mathrm{s} .\) What frequency does a firefighter hear \((\mathrm{c})\) if he or she is approaching from an upwind position, so that he or she is moving in the direction in which the wind is blowing? (d) if he or she is approaching from a downwind position and moving against the wind?

Step by step solution

01

Calculate the speed of wind-affected sound

The raw speed of sound is given as \(343 \mathrm{m} / \mathrm{s}\), and the wind speed as \(15.0 \mathrm{m} / \mathrm{s} .\) But the speed of sound changes due to the wind. When going upwind (against the direction of wind), the effective speed decreases. So, the effective speed of sound upwind is \(343 - 15 = 328 \mathrm{m} / \mathrm{s}\). Similarly, when going downwind (in the direction of wind), the effective speed increases. Therefore, the effective speed of sound downwind is \(343 + 15 = 358 \mathrm{m} / \mathrm{s}\).

02

Find the wavelength of the sound

We know the sound source has a frequency of \(900 \mathrm{Hz}\). Using the formula \(speed = frequency \times wavelength\) to find the wavelength upwind and downwind. The wavelength upwind is \( \lambda_{upwind} = \frac{speed_{upwind}}{frequency} = \frac{328}{900} \approx 0.36 \mathrm{m}\) and the wavelength downwind is \( \lambda_{downwind} = \frac{speed_{downwind}}{frequency} = \frac{358}{900} \approx 0.40 \mathrm{m}\).

03

Apply the Doppler Effect

The frequency heard by the firefighters approaching the siren will be different due to the Doppler effect. The Doppler effect formula in this case (where both source and observer are moving) is \(f' = \frac{(v+vo)}{(v-vs)} f\), where \(f'\) is the perceived frequency, \(v\) is the speed of sound, \(vo\) is observer's velocity, \(vs\) is source's velocity, and \(f\) is source frequency. Here \(vs=0\) as siren is stationary.

04

Calculate the frequencies

For (c) the firefighter is moving in the same direction as wind, thus \(vo = 15 \mathrm{m} / \mathrm{s}\). So, the perceived frequency \(f'_{upwind} = \frac{(328+15)}{328} * 900 \approx 953 \mathrm{Hz}\). For (d) the firefighter is moving against the wind, thus \(vo = -15 \mathrm{m} / \mathrm{s}\). So, the perceived frequency \(f'_{downwind} = \frac{(358-15)}{358} * 900 \approx 846 \mathrm{Hz}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Speed

Wave speed refers to how fast a wave travels through a medium. In our case, it's the speed at which sound waves move. Typically, the wave speed of sound in calm air is about \(343 \mathrm{m/s}\). However, this speed can change if there are external factors, like wind.
When sound travels upwind against the wind's flow, its speed decreases because the wind opposes it. Conversely, when sound moves downwind in the same direction as the wind, its speed increases. Understanding these variations is crucial in scenarios where sound propagation is affected by wind, such as when using sirens in emergency situations.

Frequency

Frequency is the number of vibrations occurring per second at a certain point and is measured in Hertz (Hz). It determines the pitch of the sound we hear.
In the original problem, the frequency of the siren sound is reported as \(900 \mathrm{Hz}\). This means the siren emits 900 sound wave cycles every second. The frequency remains constant, regardless of the medium's changes or listener's motion. However, the perceived frequency can shift due to the Doppler effect, especially noticeable when the listener or source moves towards or away from each other.

Wavelength

Wavelength is the distance between two consecutive points of a wave that is in phase, such as crest to crest or trough to trough.
The formula \(\text{wave speed} = \text{frequency} \times \text{wavelength}\) illustrates the interconnectedness of wave speed, frequency, and wavelength. Given this, changing the speed of sound due to wind will affect its wavelength. In the original example, due to wind influence, wavelength varies from approximately \(0.36 \mathrm{m}\) when traveling upwind to \(0.40 \mathrm{m}\) when going downwind.

Sound Waves

Sound waves are vibrations that travel through a medium, usually air in our everyday experience. They are caused by variations in air pressure and can vary in both frequency and amplitude.
Sound waves are longitudinal in nature, meaning the air particles move parallel to the wave's direction in a push-and-pull manner. Understanding how sound waves behave, including how they interact with environmental factors like wind, is essential to predicting how sound will propagate over long distances or in different directions.

Wind Effect on Sound Propagation

Wind can significantly alter the propagation of sound. When wind is present, it can either aid or hinder the travel path of sound waves based on its direction.

• When sound travels with the wind, it gains speed, making it reach farther distances more quickly.
• When traveling against the wind, the speed of sound decreases, which can lead to shorter travel distances and reduced loudness.

This change in speed not only affects how far sound travels, but also impacts how it is heard by listeners, thanks to the Doppler effect, which shifts the perceived frequency based on movement relative to the sound source.