A band playing music at a frequency `f` is moving towards a wall at a speed `v_(b)`. A motorist is following the band with a speed `v_(m)`. If `v` is the speed of sound, obtain an expression for the beat frequency heard by the motorist.

Options

A. `((v+v_(m))/(v+v_(b))upsilon`
B. `(v+v_(m))/(v-v_(b))upsilon`
C. `(2v_(b)(v+v_(m))/(v^(2)-v_(b)^(2))upsilon`
D. `(2v_(m)(v+v_(b))/(v^(2)-v_(m)^(2))upsilon`

Explanation

To solve the problem of finding the beat frequency heard by the motorist following a band moving towards a wall, we can break down the solution into several steps.

### Step 1: Understand the Scenario
The band is moving towards a wall, and the sound they produce travels towards the wall and reflects back. The motorist is also moving towards the band. We need to determine the apparent frequencies of the sound as heard by the motorist.

### Step 2: Calculate the Apparent Frequency Heard by the Motorist
The frequency of the sound produced by the band is `f`. When the band is moving towards the wall at speed `v_b`, the frequency heard by the wall (which reflects the sound) is given by the Doppler effect formula:
\[ f' = \frac{v + v_b}{v} \cdot f \]
Here, `v` is the speed of sound. The wall reflects this frequency back, and the motorist, moving towards the wall at speed `v_m`, hears this reflected frequency.

### Step 3: Calculate the Frequency After Reflection
The frequency heard by the motorist after the sound reflects off the wall is:
\[ f_2 = \frac{v + v_m}{v - v_b} \cdot f' \]
Substituting for `f'` from Step 2, we get:
\[ f_2 = \frac{v + v_m}{v - v_b} \cdot \left(\frac{v + v_b}{v} \cdot f\right) \]

### Step 4: Calculate the Direct Frequency Heard by the Motorist
The frequency directly heard by the motorist (without reflection) is given by:
\[ f_1 = \frac{v + v_m}{v - v_b} \cdot f \]

### Step 5: Calculate the Beat Frequency
The beat frequency is the difference between the two frequencies heard by the motorist:
\[ f_{beat} = f_2 - f_1 \]
Substituting the expressions for `f_2` and `f_1`:
\[ f_{beat} = \left(\frac{v + v_m}{v - v_b} \cdot \frac{v + v_b}{v} \cdot f\right) - \left(\frac{v + v_m}{v - v_b} \cdot f\right) \]

### Step 6: Simplify the Expression
Factoring out the common term:
\[ f_{beat} = \frac{v + v_m}{v - v_b} \cdot f \left(\frac{v + v_b}{v} - 1\right) \]
This simplifies to:
\[ f_{beat} = \frac{(v + v_m)(v_b)}{v(v - v_b)} \cdot f \]

### Final Expression
Thus, the expression for the beat frequency heard by the motorist is:
\[ f_{beat} = \frac{2 v_b (v + v_m)}{v^2 - v_b^2} \cdot f \]