For a beat to be heard, the detector must receive two slightly different frequencies coming from the two sources. Hence we first calculate the frequency heard from each source.

Step 1 : Frequency heard from $$S_1$$
  • The observer (detector) is at rest at point P.
  • Source $$S_1$$ moves towards the detector along the line OP with speed $$v_{s1}=4\text{ m s}^{-1}$$.
  • For a moving source and a stationary observer, the Doppler-shifted frequency is
    $$f_1'=\frac{v}{\,v-v_{s1}\,}\,f\quad -(1)$$
    where $$v=324\text{ m s}^{-1}$$ is the speed of sound and $$f=656\text{ Hz}$$ is the emitted frequency.

Substituting in $$(1)$$:
$$f_1'=\frac{324}{\,324-4\,}\times656 =\frac{324}{320}\times656 =1.0125\times656 =664.2\text{ Hz}$$

Step 2 : Frequency heard from $$S_2$$
  • When $$S_2$$ is at point R, P and R are diametrically opposite.
  • The velocity of $$S_2$$ is tangential to the circular path. Thus, at R its velocity is perpendicular to the line RP that joins the source to the detector.
  • Only the component of velocity along the line of sight contributes to the Doppler effect; here that component is zero.
  • Therefore, no Doppler shift occurs and the detector hears the emitted frequency itself:
$$f_2' = f = 656\text{ Hz}$$

Step 3 : Beat frequency at the detector
The beat frequency equals the absolute difference of the two received frequencies:
$$f_\text{beat}=|f_1'-f_2'| =|664.2-656| =8.2\text{ Hz}$$

Hence, the beat frequency measured by the detector is 8.2 Hz.