Question 249: Waves

Question

Two sound waves travelling in the same direction have displacement $\mathrm{y}_1=\mathrm{a} \sin (0.2 \pi \mathrm{x}-50 \pi \mathrm{t})$ and $\mathrm{y}_2=\mathrm{a} \sin (0.15 \pi \mathrm{x}-46 \pi \mathrm{t})$.
How many times, a listener can hear sound of maximum intensity in one second?

Accepted Answer

$$ \begin{aligned} & y_1=a \sin (0.2 \pi x-50 \pi t) \ & y_2=a \sin (0.15 \pi x-46 \pi t) \end{aligned} $$
When two waves of slightly different frequencies travel through the same medium and superimpose, beats are produced. Number of times maximum intensity is heard in 1 second is equal to the beat frequency.
$$ \mathrm{f}=\left|\mathrm{f}_1-\mathrm{f}_2ight| $$
Comparing the above two equations with general form $y=A \sin (k x-2 \pi f t)$, we get $\mathrm{f}_1=25$ and $\mathrm{f}_2=23$
$ 	herefore  f=\left|f_1-f_2ight|=25-23=2 $$
$$ 	herefore $$ A listener can hear sound of maximum intensity 2 times in one second.

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Waves

Two sound waves travelling in the same direction have displacement \mathrm{y}_1=\mathrm{a} \sin (0.2 \pi \mathrm{x}-50 \pi \mathrm{t}) and \mathrm{y}_2=\mathrm{a} \sin (0.15 \pi \mathrm{x}-46 \pi \mathrm{t}). How many times, a listener can hear sound of maximum intensity in one second?

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