Apparent Frequency Calculation Using Doppler Effect in Sound

To find the value of $n$, we apply the Doppler Effect formula for sound waves.

Let $f_0$ be the original frequency of the sound source, $V$ be the speed of sound in the medium, and $v$ be the speed of both the source and the observer relative to the ground.

Case 1: Source and observer move towards each other When both the source and the observer move towards each other with speed $v$, the observed frequency $f_1$ is given by: $f_1 = f_0 \left( \frac{V + v}{V - v} \right)$

Given:

• $f_0 = 240 \text{ Hz}$
• $f_1 = 288 \text{ Hz}$

Substituting the given values into the formula: $288 = 240 \left( \frac{V + v}{V - v} \right)$

Simplifying the ratio of frequencies: $\frac{V + v}{V - v} = \frac{288}{240} = \frac{6}{5}$

From this, we can find the relation between $V$ and $v$: $5(V + v) = 6(V - v)$ $5V + 5v = 6V - 6v$ $11v = V \implies \frac{v}{V} = \frac{1}{11}$

Case 2: Source and observer move away from each other When both the source and the observer move away from each other with speed $v$, the observed frequency $f_2 = n \text{ Hz}$ is given by: $f_2 = f_0 \left( \frac{V - v}{V + v} \right)$

Since $\frac{V + v}{V - v} = \frac{6}{5}$, its reciprocal is: $\frac{V - v}{V + v} = \frac{5}{6}$

Substituting $f_0 = 240 \text{ Hz}$ into the equation for $f_2$: $n = 240 \times \left( \frac{5}{6} \right)$ $n = 40 \times 5 = 200$

Thus, the value of $n$ is 200.