Question

What should be the speed of a source of sound moving towards a stationary listener, so that the frequency of sound heard by the listener is double the frequency of sound produced by the source? \\{Speed of sound wave is \(\mathrm{v}\\}\) (A) \(\mathrm{v}\) (B) \(2 \mathrm{v}\) (C) \(\mathrm{v} / 2\) (D) \(\mathrm{v} / 4\)

Short answer

The speed of the sound source moving towards a stationary listener, so that the frequency of the sound heard is double, should be \(\frac{v}{2}\). Therefore, the correct answer is (C) \(\frac{v}{2}\).

Step-by-step solution

Review the Doppler Effect Formula

The Doppler Effect formula, as it applies to sound waves, can be expressed as: \(f_{listener}=\frac{f_{source}(v)}{v\mp vt}\) Where \(f_{listener}\) is the frequency heard by the listener, \(f_{source}\) is the frequency produced by the source, \(v\) is the speed of sound, and \(vt\) is the speed of either the listener or the source, with a plus sign if the source is moving away and a minus sign if the source is moving towards the listener.

Define the Problem

In this problem, we want the listener to hear a frequency that is double the frequency produced by the source, which can be expressed as: \(f_{listener} = 2f_{source}\) Because the listener is stationary and the source is moving toward the listener, we can use the following equation to represent the situation: \(2f_{source} = \frac{f_{source}(v)}{v-vt}\)

Solve the Equation for the Source's Speed (\(v_{t}\))

Divide both sides of the equation by \(f_{source}\): \(2 = \frac{v}{v-vt}\) Next, isolate the denominator to solve for the speed \(vt\): \(2v-2vt = v\) \(2v = v+2vt\) Now, solve for \(vt\) (the speed of the moving source): \(v=2vt\) \(v_{t} = \frac{v}{2}\)

Choose the Correct Answer

Based on our calculations, the speed of the moving source is \(\frac{v}{2}\). Therefore, the correct answer is: \(C) \frac{v}{2}\)

Key concepts

Sound Waves

Sound waves are vibrations that travel through a medium such as air, water, or solids. They are created when an object vibrates, causing the particles around it to move and collide. This motion transfers energy from one particle to the next, forming a wave. These waves have several important properties:

• Frequency: Refers to the number of wave cycles that pass a point in one second. It's measured in Hertz (Hz).
• Wavelength: The distance between one wave crest and the next.
• Amplitude: The height of the wave crest, which relates to the volume or loudness of the sound.
• Speed: The rate at which sound waves travel through a medium, often depends on the medium itself.

Understanding these properties helps us to comprehend how sound travels from a source to a listener, and how its characteristics can be altered by movement, like in the Doppler Effect. We often encounter this effect in real-life situations, for example, the sound of a car moving towards or away from us.

Frequency

Frequency is a fundamental concept in understanding sound waves and their behavior. It denotes the number of times a wave repeats a cycle in one second. The unit of measurement is Hertz (Hz). A higher frequency means more wave cycles per second, resulting in a higher-pitched sound. In the context of the Doppler Effect, the frequency heard by an observer changes if the source of the sound is moving. For a stationary listener, this can manifest as:

• Higher frequency: When the sound source moves towards the listener. This causes the sound waves to compress, increasing the frequency and pitch.
• Lower frequency: When the source moves away, leading to stretched waves, decreasing frequency and pitch.

Understanding frequency is crucial for predicting how sound changes in different conditions, and it ties directly into the calculations involved in the given exercise.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium. In air, it is approximately 343 meters per second at room temperature. However, this speed can vary based on factors like temperature, humidity, and the medium through which the sound travels. For instance:

• Medium: Sound travels faster in solids and liquids compared to gases. In water, sound moves at about 1482 meters per second.
• Temperature: Higher temperatures generally increase the speed of sound as particles move more quickly and facilitate better energy transmission.

In the exercise, the question revolves around altering the perceived frequency by changing the sound source's speed towards a stationary listener, with the speed of sound in air serving as a constant reference.

Stationary Listener

A stationary listener is someone or something that is not moving relative to the sound source. In the context of sound and the Doppler Effect, the movement of the source while the listener remains stationary can greatly impact the frequency of the sound perceived. Here's how it typically works:

• When the source moves towards a stationary listener, the waves bunch up, resulting in a higher frequency.
• When moving away, the waves spread out, creating a lower frequency.

This concept is a crucial part of the exercise, as it focuses on how the sound's frequency doubles for a stationary listener, complicating calculations related to the speed needed for the sound source. Understanding this dynamic allows one to solve and understand problems involving the Doppler Effect more thoroughly.