To solve the problem of determining the number of beats per second heard by a man running towards a stationary wall while holding a tuning fork, we can follow these steps:
### Step 1: Understand the Doppler Effect
The Doppler effect describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave. In this case, the man is the source of sound (tuning fork) and the wall acts as a stationary observer.
### Step 2: Identify Given Data
- Frequency of the tuning fork, \( f = 90 \, \text{Hz} \)
- Speed of sound in air, \( V = 340 \, \text{m/s} \)
- Velocity of the man (source), \( V_s = 3.7 \, \text{m/s} \)
### Step 3: Calculate the Frequency Heard by the Wall
When the man runs towards the wall, the frequency heard by the wall can be calculated using the Doppler effect formula:
\[
f' = f \frac{V}{V - V_s}
\]
Substituting the values:
\[
f' = 90 \, \text{Hz} \cdot \frac{340 \, \text{m/s}}{340 \, \text{m/s} - 3.7 \, \text{m/s}}
\]
Calculating the denominator:
\[
340 - 3.7 = 336.3 \, \text{m/s}
\]
Now substituting back:
\[
f' = 90 \cdot \frac{340}{336.3}
\]
Calculating \( \frac{340}{336.3} \):
\[
\frac{340}{336.3} \approx 1.011
\]
Now calculating \( f' \):
\[
f' \approx 90 \cdot 1.011 \approx 91.99 \, \text{Hz} \approx 92 \, \text{Hz}
\]
### Step 4: Calculate the Frequency Heard by the Man
When the sound reflects off the wall, the wall acts as a source of sound with frequency \( f' \). The man, now moving towards the wall, hears this frequency as follows:
\[
f'' = f' \frac{V + V_s}{V}
\]
Substituting the values:
\[
f'' = 92 \, \text{Hz} \cdot \frac{340 + 3.7}{340}
\]
Calculating the numerator:
\[
340 + 3.7 = 343.7 \, \text{m/s}
\]
Now substituting back:
\[
f'' = 92 \cdot \frac{343.7}{340}
\]
Calculating \( \frac{343.7}{340} \):
\[
\frac{343.7}{340} \approx 1.011
\]
Now calculating \( f'' \):
\[
f'' \approx 92 \cdot 1.011 \approx 93.092 \, \text{Hz}
\]
### Step 5: Calculate the Beat Frequency
The beat frequency is the difference between the frequency of the tuning fork and the frequency heard by the man:
\[
\text{Beat Frequency} = f'' - f = 93.092 - 90 = 3.092 \, \text{Hz} \approx 3 \, \text{Hz}
\]
### Final Answer
The number of beats per second heard by the man is approximately **3 beats per second**.
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