Beat Frequency Measured by Sound Detector from Two Moving Sources

To find the beat frequency measured by the sound detector placed at point $P$, we need to calculate the apparent frequencies of the sound waves received from both sources $S_1$ and $S_2$.

1. Frequency received from Source $S_2$:

• The detector is placed at point $P$, which is at the bottom of the circular path.
• Source $S_2$ is at point $R$, which is diametrically opposite to $P$ (at the top of the circular path).
• $S_2$ moves anti-clockwise in a circle. At point $R$, the velocity vector of $S_2$ is directed horizontally to the left.
• The line joining the source $S_2$ at $R$ and the detector at $P$ is along the vertical diameter $RP$.
• Therefore, the velocity vector of $S_2$ is perpendicular to the line of sight joining $S_2$ and $P$.
• The component of velocity of $S_2$ towards the detector at $P$ is: $v_{2,\parallel} = 0$

Hence, there is no Doppler shift for the sound emitted by $S_2$ and received at $P$: $f_2 = f = 656\text{ Hz}$

---

2. Frequency received from Source $S_1$:

• Source $S_1$ moves along the line $OP$ directly approaching the detector at $P$ with a speed of $v_1 = 4\text{ m s}^{-1}$.
• The speed of sound in air is $v = 324\text{ m s}^{-1}$.
• Using the Doppler effect formula for a moving source approaching a stationary observer: $f_1 = f \left( \frac{v}{v - v_1} \right)$

Substitute the given values into the equation: $f_1 = 656 \times \left( \frac{324}{324 - 4} \right)$ $f_1 = 656 \times \frac{324}{320}$ $f_1 = 656 \times 1.0125 = 664.2\text{ Hz}$

---

3. Calculation of Beat Frequency:

The beat frequency $f_{\text{beat}}$ detected at point $P$ is the absolute difference between the two received frequencies: $f_{\text{beat}} = |f_1 - f_2| = |664.2 - 656| = 8.2\text{ Hz}$

The beat frequency measured by the detector is 8.2 (or 8.2 Hz).