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: \(C) \frac{v}{2}\)