An ambhulance blowing a siren of frequency 700Hz is travelling slowly towards vertical reflectoing wall with a speed 2m/s. The speed of sound is 350m/s. How many beats ar heard per sec to the driver of het ambluance?

Answer

To solve the problem step by step, we need to determine the frequency of the sound heard by the driver of the ambulance after it reflects off the wall and then calculate the beat frequency.

### Step 1: Calculate the frequency of the sound heard by the wall (f')
The formula for the frequency heard by a stationary observer (the wall) when the source (the ambulance) is moving towards it is given by:

\[ f' = \frac{v + v_o}{v - v_s} f_0 \]

Where:
- \( f_0 = 700 \, \text{Hz} \) (frequency of the siren)
- \( v = 350 \, \text{m/s} \) (speed of sound)
- \( v_o = 0 \, \text{m/s} \) (speed of the wall, since it's stationary)
- \( v_s = 2 \, \text{m/s} \) (speed of the ambulance)

Substituting the values:

\[ f' = \frac{350 + 0}{350 - 2} \cdot 700 \]

\[ f' = \frac{350}{348} \cdot 700 \]

Calculating this gives:

\[ f' \approx 707.17 \, \text{Hz} \]

### Step 2: Calculate the frequency of the sound heard by the ambulance after reflection (f'')
Now, the wall acts as a source of sound, and the ambulance (now moving towards the wall) acts as the observer. The formula for the frequency heard by the moving observer is:

\[ f'' = \frac{v + v_o}{v - v_s} f' \]

Where:
- \( v_o = 2 \, \text{m/s} \) (speed of the ambulance)
- \( v_s = 0 \, \text{m/s} \) (speed of the wall)

Substituting the values:

\[ f'' = \frac{350 + 2}{350 - 0} \cdot 707.17 \]

\[ f'' = \frac{352}{350} \cdot 707.17 \]

Calculating this gives:

\[ f'' \approx 712.17 \, \text{Hz} \]

### Step 3: Calculate the beat frequency
The beat frequency is the absolute difference between the frequency of the siren and the frequency heard after reflection:

\[ f_{\text{beat}} = |f'' - f_0| \]

Substituting the values:

\[ f_{\text{beat}} = |712.17 - 700| \]

Calculating this gives:

\[ f_{\text{beat}} \approx 12.17 \, \text{Hz} \]

### Final Answer
The number of beats heard per second by the driver of the ambulance is approximately **12.17 beats per second**.

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