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Chapter 14: Problem 66

The apparent frequency is \(f_{1}\) when a source of sound approches a stationary observer with a speed \(u\) and \(f_{2}\) when the observer approches the stationary source with same speed. If \(v\) is the velocity of sound, then: (a) \(f_{1}=f_{2}\) (b) \(f_{1}>f_{2}\) if \(uf_{1}\) if \(u

Short Answer

Expert verified
(b) \(f_{1} > f_{2}\) if \(u < v\).

Step by step solution

01

Understand the Situation with the Source Approaching the Observer

When the source of sound approaches the stationary observer with a speed \(u\), the apparent frequency \(f_1\) heard by the observer can be computed using the Doppler Effect formula given by \(f_1 = f \frac{v}{v-u}\), where \(f\) is the actual frequency of the source and \(v\) is the velocity of sound.
02

Analyze the Case with the Observer Approaching the Source

When the observer moves towards a stationary source with a speed \(u\), the apparent frequency \(f_2\) is calculated as \(f_2 = f \frac{v+u}{v}\). This formula also results from the Doppler Effect, considering the motion of the observer.
03

Compare the Expressions for \(f_1\) and \(f_2\)

Now we compare the two expressions: \(f_1 = f \frac{v}{v-u}\) and \(f_2 = f \frac{v+u}{v}\). To determine the relationship between \(f_1\) and \(f_2\), let's focus on the factors influencing them: in \(f_1\) the denominator \(v-u\) makes the frequency higher, whereas in \(f_2\), the numerator \(v+u\) increases it.
04

Derive and Interpret the Comparison

For \(f_1 > f_2\), then \(\frac{v}{v-u} > \frac{v+u}{v}\). Simplifying gives \(v^2 > (v-u)(v+u)\), or \(v^2 > v^2 - u^2\), leading to \(u^2 > 0\). Since \(u > 0\), when \(u < v\), it indeed holds that \(f_1 > f_2\).
05

Conclusion

The analysis shows that for \(u < v\), the apparent frequency when the source approaches is greater than when the observer approaches, thus \(f_1 > f_2\). Therefore, the correct answer is (b): \(f_1 > f_2\) if \(u < v\).

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Key Concepts

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

Apparent Frequency
The concept of apparent frequency is crucial to understanding how sound waves are perceived in motion. Apparent frequency refers to the frequency of a sound as perceived by an observer moving relative to the sound source. It differs from the actual frequency emitted by the source.

This difference occurs due to the Doppler Effect:
  • When the sound source moves towards the observer, the sound waves are compressed, increasing the perceived frequency. This is known as the "approaching effect."
  • Conversely, when the observer moves towards a stationary sound source, the rate at which they encounter sound waves increases, also leading to a higher perceived frequency.
  • The perceived pitch of the sound changes based on relative movements, which can be calculated through specific Doppler Effect equations.
Both situations modify how we hear the sound, creating an apparent frequency that changes dynamically with motion. Understanding this helps in various applications, including radar technology and medical imaging.
Motion of Sound Source
The motion of a sound source directly affects the way sound waves propagate towards an observer. In the scenario where the sound source is moving towards a stationary observer, the Doppler Effect comes into play.

Here's how it works:
  • As the source approaches, the waves are bunched closer together, effectively shortening the wavelength.
  • A shorter wavelength means that more waves reach the observer each second, which increases the frequency鈥攖his is why the sound may seem higher in pitch.
  • This effect can be quantified mathematically by the formula: \(f_1 = f \frac{v}{v-u}\), where \(f_1\) is the apparent frequency, \(f\) is the actual frequency, \(v\) is the speed of sound, and \(u\) is the speed of the source.
Understanding the motion of the sound source is vital in physics because it influences how and why we hear differences in pitch in everyday phenomena like cars passing by or trains honking while approaching.
Sound Wave Velocity
The velocity of sound waves, usually denoted as \(v\), is a fundamental aspect in calculating the Doppler Effect and apparent frequencies. Sound velocity depends on the medium through which the sound travels.

Key details about sound velocity include:
  • In air at room temperature, the speed of sound is approximately 343 meters per second. This speed can vary based on temperature, humidity, and altitude.
  • When calculating apparent frequency, the velocity of sound acts as a constant against which the motion of either the observer or the source is measured.
  • In scenarios like the one described in the exercise, sound wave velocity \(v\) is crucial in both formulas: \(f_1 = f \frac{v}{v-u}\) and \(f_2 = f \frac{v+u}{v}\), representing the medium's role in propagation.
By understanding sound wave velocity, we appreciate its influence on how sound behaves as it travels, impacting perceived frequency and pitch. This knowledge is essential in diverse fields, such as acoustics engineering and atmospheric science.

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Most popular questions from this chapter

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