Beats
When two sound waves of slightly different frequencies but similar amplitudes are produced simultaneously, the loudness increases and decreases periodically. This phenomenon is called the beat. The periodic variation in the intensity of sound at a point, due to the superposition of two sound waves of slightly different frequencies travelling in the same direction, is called beats. The time interval in which one beat occurs is called beat period while the number of beat per second is called beat frequency.
Mathematical Derivation for Beat Frequency
Suppose two waves of frequencies f_{1} and f_{2} and each of amplitude ‘a’ are travelling in a medium in the same direction. The equations of the waves are
\begin{align*} y_1 = a\sin (\omega _1 t – k_1x) \dots (i) \\ y_2 = a\sin (\omega _2 t – k_2x) \dots (ii) \\ \text {where,} \: \omega _1 = 2\pi f_1 \: \text {and} \: \omega _2 = 2\pi f_2.\end{align*}
When two waves superpose
\begin{align*} \text{at x} = o, \: \text {then } \\ y_1 = a\sin \omega _1 t \dots (iii) \\ y_1 = a\sin \omega _2 t \dots (vi) \\ \end{align*}
And from the superposition principle the resultant displacement at that point is given by
\begin{align*}\\y = y_1 + y_2 \\ = a\sin \omega _1 t + a\sin \omega _2 t \\ =2a\sin \left (\frac {\omega _1 + \omega _2}{2}\right ) t \cos \left (\frac {\omega _1  \omega _2}{2}\right )t \\ =2a\cos 2\pi \left (\frac {f_1 – f_2}{2} \right ) t \sin 2\pi \left (\frac {f_1 + f_2}{2} \right ) t \\ \text {or,} \: y = A\sin 2\pi \left (\frac {f_1 + f_2}{2} \right ) t \\ \text {where A} \: = \: 2a\cos 2\pi \left (\frac {f_1 – f_2}{2} \right ) t. \: \\\end{align*} The above equation shows that resultant wave has an amplitude A that depends on time t and varies with a frequency of f \begin{align*}= \frac {f_1 – f_2}{2}. \end{align*}
Condition for Maxima
The resultant amplitude A will be maximum when
\begin{align*} \: \cos 2\pi \left ( \frac {f_1 – f_2}{2}\right ).t \: \text {is maximum. That is,} \\ \cos 2\pi \left ( \frac {f_1 – f_2}{2}\right ).t = \pm 1 \\ \text {or,} \: \cos 2\pi \left ( \frac {f_1 – f_2}{2}\right ).t = \cos n\pi \\ \text {where n} = 0, 1, 2, 3, 4, \dots \\ \text {So,} \: 2\pi\left ( \frac {f_1 – f_2}{2}\right ).t = n\pi \\ \text {or,} \: t = \frac {n}{f_1 – f_2} \\ \text {or,} \: t = 0, \frac {1}{f_1 – f_2}, \frac {2}{f_1 – f_2}, \frac {3}{f_1 – f_2}, \dots \\ \end{align*}
The time interval between two consecutive maxima is the period and it is given by
\begin{align*} \\ T = \frac {1}{f_1 – f_2} – 0\: \text {or,} \:\frac {2}{f_1 – f_2}  \frac {3}{f_1 – f_2} \: \text {or,} \dots = \frac {1}{f_1 – f_2} \\ \therefore T = \frac {1}{f_1  f_2} \\ \text {Hence frequency of maxima} \: = \frac 1T = f_1 – f_2 \\ \end{align*}
Condition for Minima
The resultant amplitude A will be minimum when
\begin{align*} \: \cos 2\pi \left (\frac {f_1 – f_2}{2}\right )t \: \text {is miniimum. } \\ \text {That is,} \\ \cos 2\pi \frac {f_1 – f_2}{2}.t = 0 \\ \text {or,} \: \cos 2\pi \left (\frac {f_1 – f_2}{2}\right ). t = \cos (2n + 1)\pi/2 \\ \text {where n} = 0, 1, 2, 3, 4, \dots \\ \text {So,} \: 2\pi\left ( \frac {f_1 – f_2}{2}\right ).t = (2n + 1)\pi/2 \\ \text {or,} \: t = \frac {2n + 1}{2(f_1 – f_2)} \\ \text {or,} \: t = \frac {1}{2(f_1 – f_2)}, \frac {3}{2(f_1 – f_2)}, \frac {5}{2(f_1 – f_2)}, \dots \\ \end{align*}
So, the time interval between two consecutive minima i.e. beat period is given as,
\begin{align*} \\ T = \frac {3}{2(f_1 – f_2)}  \frac {1}{2(f_1 – f_2)} \: \text {or,} \: \frac {5}{2(f_1 – f_2)}  \frac {3}{2(f_1 – f_2)} \\ = \frac {1}{f_1 – f_2} \\ \text {Hence, frequency of minima} = \frac 1T = f_1 – f_2, \text {same as the frequency of maxima. } \end{align*}.So, the beat frequency is equal to the difference of frequencies of two sound waves.
References
Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.
S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.
When two sound waves of slightly different frequencies but similar amplitudes are produced simultaneously, the loudness increases and decreases periodically which is called the beat.
The periodic variation in the intensity of sound at a point, due to the superposition of two sound waves of slightly different frequencies travelling in the same direction, is called beats.
The time interval in which one beat occurs is called beat period while the number of beat per second is called beat frequency.

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