S1 and S are two identical sound sources of frequency 656 Hz. The source S1 is located at O, and S₂ moves anticlockwise with a uniform speed 4 √ 2 m/s on a circular path around O, as shown in the figure. There are three points P, Q, and R on this path such that P and R are diametrically opposite, while Q is equidistant from them. A sound detector is placed at point P. The source S₁ can move along the direction OP.

Consider both sources emitting sound. When S₂ is at R and S₁ approaches the detector with a speed of 4 m/s, the beat frequency measured by the detector is:

The correct answer is 8.2 Hz (or simply 8.2).

1. Understanding the Given Data:

• Original frequency of both sources S₁ and S₂: $f=656\text{Hz}$
• Speed of sound in air (standard value): $v=324\text{m/s}$ (or typically $320\text{m/s}$ to $330\text{m/s}$; specifically for this standard problem, $v=324\text{m/s}$).
• Position of detector D: fixed at point P.
• Motion of source S₁: moves along the direction OP towards detector P with speed $v_1=4\text{m/s}$.
• Motion of source S₂: moves anticlockwise on a circular path around O with speed $v_2=4\sqrt{2}\text{m/s}$.

2. Apparent Frequency from Source S₁:

Since S₁ is moving directly towards the stationary detector at P with speed $v_1=4\text{m/s}$, the apparent frequency $f_1$ observed at P is given by the Doppler effect formula for a moving source approaching a stationary observer:

$f_1=f\left(\frac{v}{v-v_1}\right)$

Substituting the values:

$f_1=656\times \left(\frac{324}{324-4}\right)=656\times \frac{324}{320}=664.2\text{Hz}$

3. Apparent Frequency from Source S₂:

When S₂ is at point R, its velocity vector is directed horizontally to the left (tangent to the circle at R), which is perpendicular to the line of joining R and P (the vertical diameter RP).

Since the direction of motion of S₂ at point R is perpendicular to the line connecting the source to the detector at P, there is no component of velocity along the line of sight RP. Thus, the relative motion towards or away from P is zero:

$v_{2\left|\right|}=0$

Therefore, no Doppler shift occurs for sound coming from S₂ when it is at point R:

$f_2=f=656\text{Hz}$

4. Calculating the Beat Frequency:

The beat frequency $f_{\text{beat}}$ measured by the detector at P is the absolute difference between the two apparent frequencies:

$f_{\text{beat}}=|f_1-f_2|=664.2-656=8.2\text{Hz}$