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Question

A source emitting sound of frequency $f_0$ moves with a constant velocity $v_s$ towards a stationary observer. The wind is blowing with a velocity $v_w$ in the same direction as the source. The apparent frequency heard by the observer is $f$. If the speed of sound in still air is $v$, which expression is correct for $f$?

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: Doppler Effect with moving source and wind
 **Related Topics**: Wave motion, sound propagation
 **Prerequisites**: Understanding of the Doppler effect, vector addition of velocities
## SOLUTION PATHWAY
**Given Information**:
  * Source frequency: $f_0$
  * Source velocity (towards observer): $v_s$
  * Wind velocity (in the same direction as the source): $v_w$
  * Speed of sound in still air: $v$
  * Apparent frequency: $f$
  * Observer is stationary
**Method & Steps**:
 1. Consider the effect of the wind on the speed of sound.
  Reasoning: The wind adds to the speed of sound in the direction of the wind and subtracts from the speed of sound in the opposite direction.
  Formula/Rule:  The effective speed of sound towards the observer is $v + v_w$.
 2. Apply the Doppler effect formula for a moving source.
  Working: The general formula for the Doppler effect with a moving source is given by:
  $f = f_0 \frac{v \pm v_o}{v \mp v_s}$
  Where:
  * $f$ is the observed frequency
  * $f_0$ is the source frequency
  * $v$ is the speed of sound in the medium
  * $v_o$ is the velocity of the observer (positive if moving towards the source)
  * $v_s$ is the velocity of the source (positive if moving towards the observer)
  In our case, the observer is stationary ($v_o = 0$), the source moves towards the observer ($v_s$), and the effective speed of sound is $v + v_w$. Therefore, the formula becomes:
  $f = f_0 \frac{v + v_w}{v + v_w - v_s}$
 3. Final Result:
  Result: $f = f_0 \frac{v + v_w}{v + v_w - v_s}$
  Verification: The denominator represents the relative velocity between the sound wave and the observer, which is reduced by the source's velocity and increased by the wind's velocity in this case.
 ## OPTION ANALYSIS
A ✓: $f = f_0 \frac{v + v_w}{v + v_w - v_s}$ (Correct - accounts for both wind and source movement)
B :x: $f = f_0 \frac{v - v_w}{v - v_w - v_s}$ (Incorrect - subtracts wind velocity from the speed of sound, which is wrong since the wind is in the direction of the sound)
C :x: $f = f_0 \frac{v + v_w}{v - v_w - v_s}$ (Incorrect - incorrectly subtracts wind velocity from the denominator)
D :x: $f = f_0 \frac{v - v_w}{v + v_w - v_s}$ (Incorrect - subtracts wind velocity from numerator and adds it to the denominator)
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Incorrectly handling the wind velocity:  Forgetting to add or subtract the wind velocity from the speed of sound depending on its direction relative to the sound wave.
  2.Confusing the signs in the Doppler effect formula:  Incorrectly assigning positive or negative signs to the observer and source velocities.
 **Quick Tips**:
  1.Draw a diagram to visualize the direction of sound waves, source movement, and wind.
  2.Remember that the wind affects the speed of sound, not the source or observer velocity.

SUBHANI · Answer

The correct Option is "$f = f_0 \frac{v + v_w}{v + v_w - v_s}$"

SUBHANI · Answer

$f = f_0 \frac{v - v_w}{v - v_w - v_s}$

SUBHANI · Answer

$f = f_0 \frac{v + v_w}{v - v_w - v_s}$

SUBHANI · Answer

$f = f_0 \frac{v - v_w}{v + v_w - v_s}$

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