Question

With what velocity should an observer approach a stationary sound source, so that the apparent frequency of sound should appear double the actual frequency?

• A. \( \frac{v}{2} \)
• B. \( 3v \)
• C. \( 2v \)
• D. \( v \)

Hint:

When the observer moves towards a stationary sound source, the apparent frequency increases. To double the actual frequency, the observer must approach the source with a velocity equal to the speed of sound.

Answer 1

The Correct Option is D

To determine the velocity with which an observer should approach a stationary sound source so that the apparent frequency becomes double the actual frequency, we can use the Doppler effect formula for sound. The formula for the apparent frequency (\(f'\)) perceived by an observer approaching a stationary sound source is given by:

\[f' = \left(\frac{v + v_o}{v}\right)f\]

Where:

• \(f\) = actual frequency of the sound source
• \(v\) = speed of sound in air
• \(v_o\) = velocity of the observer

Since we want the apparent frequency to be double the actual frequency, we set up the equation:

\[2f = \left(\frac{v + v_o}{v}\right)f\]

Dividing both sides by \(f\), we get:

\[2 = \frac{v + v_o}{v}\]

Multiplying both sides by \(v\) results in:

\[2v = v + v_o\]

Solving for v_o, we subtract \(v\) from both sides:

\[v_o = 2v - v\]

\[v_o = v\]

Therefore, the observer should approach the stationary sound source with a velocity equal to the speed of sound (\(v\)) to perceive the frequency as double the actual frequency.

The correct answer is: \(\mathbf{v}\)

Answer 2

The Doppler effect for sound gives the apparent frequency \( f' \) as: \[ f' = f \left( \frac{v + v_o}{v} \right) \] where:
- \( f' \) is the apparent frequency,
- \( f \) is the actual frequency,
- \( v \) is the speed of sound in air,
- \( v_o \) is the velocity of the observer. In this case, the observer is moving towards the stationary source, and we want the apparent frequency \( f' \) to be double the actual frequency \( f \). Hence: \[ 2f = f \left( \frac{v + v_o}{v} \right) \] Canceling \( f \) from both sides: \[ 2 = \frac{v + v_o}{v} \] Solving for \( v_o \): \[ 2v = v + v_o \quad \Rightarrow \quad v_o = v \]
Thus, the observer must approach the sound source with a velocity equal to the speed of sound \( v \). Therefore, the correct answer is: \[ \text{(4) } v \]