Question 2: Waves

Question

Two sources of sound S$$_1$$ and S$$_2$$ are moving
towards and away from a stationary observer
with the same speed respectively. Observer
detects 3 beats per second. Find speed of
sources (approximately).
Given, $$f_1=f_2=500$$ Hz, speed of sound in air = 330 m/s

Accepted Answer

Given, frequency of sound produced source $$S_1$$ and $$S_2$$ i.e, $$f_1=f_2=500 \mathrm{~Hz}$$
speed of sound in air, $$v=330 \mathrm{~m} / \mathrm{s}$$
Beats $$=3 \mathrm{~Hz}$$
If $$v_s$$ be the speed of source, when first source is moving towards the observer and $$f_1{ }^{\prime}$$ be apparent frequency heard by the observer, then,
$$f_1{ }^{\prime}=\frac{v}{v-v_s} 	imes f_1=\frac{330}{330-v_s} 	imes 500$$
When second source is moving away from the apparent frequency heard by the observer, then
$$f_2^{\prime}=\frac{v}{v+v_s} 	imes f_2=\frac{330}{330+v_s} 	imes 500$$
$$\begin{aligned}
& 	herefore 	ext { Beats }=f_1{ }^{\prime}-f_2{ }^{\prime}=3 \
& \Rightarrow \frac{330}{330-v_s} 	imes 500-\frac{330}{330+v_s} 	imes 500=3 \
& \Rightarrow \frac{500 	imes 2 v_s 	imes 330}{330^2-v_s^2}=3
\end{aligned}$$
Since, $$v_s^2<<330^2$$
$$\Rightarrow \frac{500 	imes 2 v_s 	imes 330}{330^2}=3 \Rightarrow v_s=0.99 \approx 1 \mathrm{~m} / \mathrm{s}$$

Answer

1 m/s

Answer

2 m/s

Answer

3 m/s

Answer

4 m/s

Waves

Two sources of sound S$_1 and S_2 are moving towards and away from a stationary observer with the same speed respectively. Observer detects 3 beats per second. Find speed of sources (approximately). Given, f_1=f_2=500$ Hz, speed of sound in air = 330 m/s

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