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Concept Questions The table shows three situations in which the Doppler effect may arise. The first two columns indicate the velocities of the sound source and the observer, where the length of each arrow is proportional to the speed. For each situation, fill in the empty columns by deciding whether the wavelength of the sound and the frequency heard by the observer increase, decrease, or remain the same compared to the case when there is no Doppler effect. Provide a reason for each answer. $$ \begin{array}{|l|c|c|c|c|} \hline & \begin{array}{c} \text { Velocity of Sound } \\ \text { Source (Toward the } \\ \text { Observer) } \end{array} & \begin{array}{c} \text { Velocity of } \\ \text { Observer (Toward } \\ \text { the Source) } \end{array} & \text { Wavelength } & \begin{array}{c} \text { Frequency Heard by } \\ \text { Observer } \end{array} \\ \hline \text { (a) } & 0 \mathrm{~m} / \mathrm{s} & 0 \mathrm{~m} / \mathrm{s} & & \\ \hline \text { (b) } & \rightarrow & 0 \mathrm{~m} / \mathrm{s} & & \\ \hline \text { (c) } & \rightarrow & \leftarrow & & \\ \hline \end{array} $$ Problem The siren on an ambulance is emitting a sound whose frequency is \(2450 \mathrm{~Hz}\). The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). (a) If the ambulance is stationary and you (the "observer") are sitting in a parked car, what is the wavelength of the sound and the frequency heard by you? (b) Suppose the ambulance is moving toward you at a speed of \(26.8 \mathrm{~m} / \mathrm{s}\). Determine the wavelength of the sound and the frequency heard by you. (c) If the ambulance is moving toward you at a speed of \(26.8 \mathrm{~m} / \mathrm{s}\) and you are moving toward it at a speed of \(14.0 \mathrm{~m} / \mathrm{s}\), find the wavelength of the sound and the frequency that you hear. Be sure that your answers are consistent with your answers to the Concept Questions.

Step by step solution

01

Analyze Conceptual Situation (a)

For situation (a), both the sound source and the observer are stationary. Hence, there is no Doppler effect affecting the wavelength or frequency. Therefore, the wavelength remains constant at its normal value, and the frequency heard by the observer remains at the emitted frequency.

02

Analyze Conceptual Situation (b)

In situation (b), the sound source moves towards the observer while the observer remains stationary. The wavelength decreases because the sound waves are compressed. This leads to an increase in the frequency heard by the observer.

03

Analyze Conceptual Situation (c)

For situation (c), both the sound source and the observer are moving towards each other. The relative velocity between them increases the effect observed in (b), resulting in an even shorter wavelength and a higher frequency heard by the observer.

04

Solve Problem (a) - Stationary Conditions

With the ambulance stationary, the frequency heard is the same as the emitted frequency, 2450 Hz. The wavelength \(\lambda\) is calculated by \(\lambda = \frac{v}{f} = \frac{343}{2450} \approx 0.140 \text{ m}\).

05

Solve Problem (b) - Source Moving Toward Observer

When the ambulance moves toward you at 26.8 m/s, only the source is moving. The observed frequency \(f'\) is given by \(f' = f \frac{v}{v - v_s} = 2450 \frac{343}{343 - 26.8} \approx 2675 \text{ Hz}\). The new shorter wavelength \(\lambda'\) is \(\lambda' = \frac{v}{f'} \approx 0.129 \text{ m}\).

06

Solve Problem (c) - Both Moving Toward Each Other

With both the ambulance and observer moving towards each other, the relative velocity is the sum, \(v_{rel} = 26.8 + 14.0 = 40.8 \text{ m/s}\). The frequency heard is \(f' = f \frac{v + v_o}{v - v_s} = 2450 \frac{343 + 14.0}{343 - 26.8} \approx 2834 \text{ Hz}\). The wavelength \(\lambda'\) becomes \(\lambda' = \frac{v}{f'} \approx 0.121 \text{ m}\).

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Key Concepts

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

Sound Waves

Sound waves are vibrations that travel through the air or another medium. These waves move by creating areas of high and low pressure. When something like a bell rings, it causes the air molecules around it to vibrate.

These vibrations create waves that move outward from the source. Sound waves are longitudinal, meaning the movement of the molecules is parallel to the direction of the wave. Think of it like a slinky toy, where compressions and rarefactions move along the length of the toy.

One key thing to know about sound waves is that they carry energy, which is why you can hear the sound from far away. In summary, sound waves are how sound travels from the source to your ears.

Frequency

Frequency refers to the number of times a wave vibrates per second. In the context of sound, it is measured in hertz (Hz), where one hertz equals one vibration per second.

Frequency determines the pitch of the sound. A higher frequency means a higher-pitched sound, like that of a whistle or a bird chirping. On the other hand, a lower frequency produces a lower-pitched sound, like a drum or a cello.

In our case of the ambulance siren, the frequency tells us how high or low the sound of the siren is when it reaches the observer. Understanding frequency is crucial when dealing with concepts like the Doppler effect, as moving sources or observers can alter the frequency, hence bringing changes in the sound we hear.

Wavelength

Wavelength is the distance between consecutive points of a wave, such as from one crest to the next. In sound waves, it's the distance between compressions in the air.

The wavelength affects the sound's frequency. A shorter wavelength results in a higher frequency, while a longer wavelength results in a lower frequency.

For an ambulance siren, as it approaches a listener, the sound waves get compressed, leading to a shorter wavelength and an increased frequency heard by the observer. Conversely, when the ambulance moves away, the wavelengths stretch out, resulting in a lower frequency. This is a visual and auditory demonstration of the Doppler effect in action.

Ambulance Siren

An ambulance siren is an excellent real-life example of the Doppler effect. Normally, the siren emits sound waves at a specific frequency. But when the ambulance moves, it affects the sound waves and, consequently, how we perceive them.

As the ambulance approaches, sound waves are compressed, making them hit our ears more frequently, increasing the frequency of the sound we interpret. When the ambulance moves away, the opposite happens; the sound waves stretch, resulting in a lower frequency. This is why the pitch of an ambulance siren seems to change as it passes by.

This Doppler effect is crucial for understanding the change in sound frequency that occurs due to the motion of the source or the observer.

Velocity of Sound

The velocity of sound is the speed at which sound waves travel through a medium. In air, this speed is typically around 343 meters per second, but this can vary depending on factors like temperature and humidity.

Knowing the velocity of sound is critical for solving problems involving the Doppler effect. It helps us calculate how the frequency and wavelength of sound change as the source or observer moves.

For instance, in the case of a moving ambulance, understanding the velocity of sound allows us to determine the new frequency and wavelength when the Doppler effect comes into play. The velocity acts as a reference against which other velocities (like that of the source or the observer) are compared in these calculations.