An observer rides with a sound source of frequency `f` and moving with velocity `v` towards a large vertical wall. Considering the velocity of sound waves as `c`, find :
(i) The number of waves striking the surface of wall per second
(ii) The wavelength of the reflected wave
(iii) The frequency of reflected wave as observed by observer.
(iv) Beat frequency heard by the observer.

Answer

To solve the problem step by step, we will break down each part of the question:

### Given:
- Frequency of sound source = \( f \)
- Velocity of observer (and sound source) = \( v \)
- Velocity of sound = \( c \)

### (i) The number of waves striking the surface of the wall per second

1. **Calculate the frequency of the sound waves as perceived by the wall**:
The frequency \( f_1 \) of the sound waves emitted by the source is given by:
\[ f_1 = \frac{c}{c - v} f \]
Here, \( c - v \) is the effective speed of sound relative to the wall since the observer is moving towards it.

2. **Number of waves striking the wall per second**:
The number of waves striking the wall per second is equal to the frequency \( f_1 \):
\[ N = f_1 = \frac{c}{c - v} f \]

### (ii) The wavelength of the reflected wave

1. **Calculate the wavelength of the reflected wave**:
The wavelength \( \lambda \) of the reflected wave can be found using the relationship:
\[ \lambda = \frac{c}{f_1} \]
Substituting \( f_1 \) from above:
\[ \lambda = \frac{c}{\frac{c}{c - v} f} = \frac{(c - v)}{f} \]

### (iii) The frequency of the reflected wave as observed by the observer

1. **Calculate the frequency of the reflected wave**:
The frequency of the reflected wave \( f_2 \) as perceived by the observer moving towards the wall is given by:
\[ f_2 = \frac{c + v}{c} f_1 \]
Substituting \( f_1 \):
\[ f_2 = \frac{c + v}{c} \cdot \frac{c}{c - v} f = \frac{(c + v)}{(c - v)} f \]

### (iv) Beat frequency heard by the observer

1. **Calculate the beat frequency**:
The beat frequency \( f_b \) is the difference between the frequency of the reflected wave and the original frequency:
\[ f_b = f_2 - f = \left(\frac{c + v}{c - v} f\right) - f \]
Simplifying this:
\[ f_b = \left(\frac{c + v}{c - v} - 1\right) f = \left(\frac{(c + v) - (c - v)}{c - v}\right) f = \frac{2v}{c - v} f \]

### Summary of Results:
1. Number of waves striking the wall per second: \( N = \frac{c}{c - v} f \)
2. Wavelength of the reflected wave: \( \lambda = \frac{(c - v)}{f} \)
3. Frequency of the reflected wave: \( f_2 = \frac{(c + v)}{(c - v)} f \)
4. Beat frequency heard by the observer: \( f_b = \frac{2v}{c - v} f \)