Question
What are beats? Prove that the number of beats produced per second by the two sound sources is equal to the difference between their frequencies.
✓
Answer
Beats: The phenomenon of alternate variation in the intensity of sound with time at a particular position, when two sound waves of nearly same frequencies and amplitudes superimpose on each other is called beats. If $v_1$ and $v_2$ are the two frequencies, $n = |v_1 - v_2|$, Beats are heard only when $|v_1 - v_2| < 10$, since the sound persist in our ear for $\frac{1}{10}\text{th}$ of a second.
Let us consider two wave trains of equal amplitude ‘A’ and slightly different frequencies $v_1$ and $v_2$ travelling in a medium in the same direction. Displacement of waves are $\text{y}_1=\text{A}\sin2\pi\text{v}_1\text{t}$ and $\text{y}_2=\text{A}\sin2\pi\text{v}_2\text{t},$ on superposition, resultant displacement of waves are:
$\text{y}=\text{y}_1+\text{y}_2=\text{A}\sin2\pi\text{v}_1\text{t}+\text{A}\sin2\pi\text{v}_2\text{t}$
$=\text{A}[\sin2\pi\text{v}_1\text{t}+\sin2\pi\text{v}_2\text{t}]$
$=2\text{A}\cos2\pi\Big(\frac{\text{v}_1-\text{v}_2}{2}\Big)\text{t}\sin2\pi\Big(\frac{\text{v}_1+\text{v}_2}{2}\Big)$
$\Big[\text{Using}\sin\text{A}+\sin\text{B}=2 \cos\frac{\text{A}-\text{B}}{2}\sin\frac{\text{A}+\text{B}}{2}\Big]$
Amplitude $=2\text{A}\cos2\pi\Big(\frac{\text{v}_1\text{v}_2}{2}\Big)\text{t}$ becomes maximum,
when $2\pi\text{t}\Big(\frac{\text{v}_1-\text{v}_2}{2}\Big)=0,\pi,2\pi,...=\text{N}\pi$
i.e. $\text{v}_1\text{v}_2=\frac{2\text{N}\pi}{2\pi\text{t}}=\frac{\text{N}}{\text{t}}=\text{n}$
Where n is the beat frequency – the number of times the maxima and minima is repeated in one second. Thus number of beats produced by two superimposing waves is equal to difference of their frequency.
Let us consider two wave trains of equal amplitude ‘A’ and slightly different frequencies $v_1$ and $v_2$ travelling in a medium in the same direction. Displacement of waves are $\text{y}_1=\text{A}\sin2\pi\text{v}_1\text{t}$ and $\text{y}_2=\text{A}\sin2\pi\text{v}_2\text{t},$ on superposition, resultant displacement of waves are:
$\text{y}=\text{y}_1+\text{y}_2=\text{A}\sin2\pi\text{v}_1\text{t}+\text{A}\sin2\pi\text{v}_2\text{t}$
$=\text{A}[\sin2\pi\text{v}_1\text{t}+\sin2\pi\text{v}_2\text{t}]$
$=2\text{A}\cos2\pi\Big(\frac{\text{v}_1-\text{v}_2}{2}\Big)\text{t}\sin2\pi\Big(\frac{\text{v}_1+\text{v}_2}{2}\Big)$
$\Big[\text{Using}\sin\text{A}+\sin\text{B}=2 \cos\frac{\text{A}-\text{B}}{2}\sin\frac{\text{A}+\text{B}}{2}\Big]$
Amplitude $=2\text{A}\cos2\pi\Big(\frac{\text{v}_1\text{v}_2}{2}\Big)\text{t}$ becomes maximum,
when $2\pi\text{t}\Big(\frac{\text{v}_1-\text{v}_2}{2}\Big)=0,\pi,2\pi,...=\text{N}\pi$
i.e. $\text{v}_1\text{v}_2=\frac{2\text{N}\pi}{2\pi\text{t}}=\frac{\text{N}}{\text{t}}=\text{n}$
Where n is the beat frequency – the number of times the maxima and minima is repeated in one second. Thus number of beats produced by two superimposing waves is equal to difference of their frequency.
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