A band playing music at a frequency f is moving towards a wall at a speed `v_h`. A motorist is following the band with a speed `v_m` . If v is the speed of sound, the expression for the beat frequency heard by the motorist is `(n(V+V_m)V_b)/((V^2 - V_b^2))` . Then n

Answer

To solve the problem of finding the beat frequency heard by a motorist following a band moving towards a wall, we can break down the steps as follows:

### Step 1: Understand the scenario
- A band is playing music at a frequency \( f \) and moving towards a wall at speed \( v_h \).
- A motorist is following the band at speed \( v_m \).
- The speed of sound is \( v \).

### Step 2: Determine the frequency heard by the wall
- The band is moving towards the wall, so the frequency \( f' \) heard by the wall can be calculated using the Doppler effect formula:
\[ f' = f \frac{v}{v - v_h} \]
Here, \( v \) is the speed of sound, and \( v_h \) is the speed of the band.

### Step 3: Determine the frequency reflected back to the motorist
- The wall acts as a new source of sound, reflecting the frequency \( f' \) back towards the motorist. The frequency \( f'' \) heard by the motorist from the wall is:
\[ f'' = f' \frac{v + v_m}{v} \]
Substituting \( f' \) from Step 2:
\[ f'' = \left( f \frac{v}{v - v_h} \right) \frac{v + v_m}{v} \]
Simplifying this gives:
\[ f'' = f \frac{(v + v_m)}{(v - v_h)} \]

### Step 4: Calculate the beat frequency
- The beat frequency \( f_b \) is the difference between the frequency heard by the motorist from the band and the frequency reflected back from the wall:
\[ f_b = f'' - f \]
Substituting \( f'' \):
\[ f_b = f \frac{(v + v_m)}{(v - v_h)} - f \]
Factoring out \( f \):
\[ f_b = f \left( \frac{(v + v_m)}{(v - v_h)} - 1 \right) \]
Simplifying further:
\[ f_b = f \left( \frac{(v + v_m) - (v - v_h)}{(v - v_h)} \right) \]
\[ f_b = f \left( \frac{v_m + v_h}{(v - v_h)} \right) \]

### Step 5: Relate to the given expression
- The beat frequency can be expressed in terms of \( n \) as given in the problem:
\[ f_b = \frac{n (v + v_m) v_h}{(v^2 - v_h^2)} \]
Comparing both expressions, we can deduce that:
\[ n = 2 \]

### Final Answer
Thus, the value of \( n \) is \( 2 \).

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