Question

A sound source \(S\) is moving along a straight track with speed \(v ,\) and is emitting sound of frequency \(v_{0}\) (see figure). An observer is standing at a finite distance, at the point \(O ,\) from the track. The time variation of frequency heard by the observer is best represented by : 
\(\left( t _{0}\right.\) represents the instant when the distance between the source and observer is minimum)

• A.

  

• B.

  

• C.

  

• D.

  

Answer 1

The Correct Option is D

The problem involves determining how the frequency of sound heard by an observer changes when a sound source is moving relative to the observer. This is a classic scenario involving the Doppler Effect in sound.

The Doppler Effect describes the change in frequency of a wave in relation to an observer who is moving relative to the wave source. The formula for the observed frequency \(f\) when the source is moving towards or away from the observer is given by:

\(f = \frac{v + v_{o}}{v + v_{s}} \times f_{0}\)

where:

• \(v\) is the speed of sound in the medium.
• \(v_{o}\) is the velocity of the observer relative to the medium.
• \(v_{s}\) is the velocity of the source relative to the medium.
• \(f_{0}\) is the emitted frequency of the source.

Since the observer is stationary, \(v_{o} = 0\). Therefore, the frequency \(f\) heard by the observer when the source is moving towards or away from them becomes:

\(f = \frac{v}{v + v_{s}} \times f_{0}\) (if moving away), and \(f = \frac{v}{v - v_{s}} \times f_{0}\) (if moving towards).

Based on the given description, when the source \(S\) is moving towards the observer at the shortest distance (at \( t_0 \)), the frequency is highest. Conversely, when moving away, the frequency is lowest. The frequency heard will smoothly change as the source moves from the closest point (time \( t_0 \)) and diverges.

The shape of the curve representing the frequency change should thus show maximum at \( t_0 \) and decrease symmetrically around this point as the sound source moves further away.

The correct representation of this is shown in the option with a peak in frequency at time \( t_0 \), which is depicted in the following figure:

 

 

. The frequency graph clearly demonstrates a peak at \( t_0 \) and falls off symmetrically, matching the expected temporal behavior as per the Doppler Effect.