Class 11 > Unit # 12:Acoustics > Beats

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Beats, Production & Analytical Derivation of Beat Frequency

Video Lecture: Beats and Beat Frequency

Watch the complete step-by-step video lecture explaining the definition of beats, their production, and the analytical derivation of beat frequency:

Definition of Beats

“The periodic alternation of sound between maximum and minimum loudness caused by the superposition of two sound waves of slightly different frequencies are called beats.”

Production of Beats

Consider two tuning forks, $A$ and $B$, with slightly different frequencies—say $32\text{ Hz}$ and $30\text{ Hz}$ respectively—placed at equal distances from an observer's ear:

• At $t = 0$: Both forks are in phase, with their right-hand prongs sending out compressions simultaneously. These compressions arrive at the ear together, producing a loud sound (constructive interference).
• After $\frac{1}{4}\text{ second}$: Fork $A$ completes $8$ vibrations while fork $B$ completes $7.5$ vibrations. Fork $A$ emits a compression while fork $B$ emits a rarefaction. They cancel each other out (destructive interference), resulting in silence.
• After $1\text{ second}$: Fork $A$ completes $32$ vibrations and fork $B$ completes $30$ vibrations. Both forks are once again in phase, sending out compressions together to produce another loud sound.

In this example, $2$ beats are produced per second, which corresponds exactly to the difference between their frequencies ($32 - 30 = 2$). Note that the human ear can typically detect a maximum of about $7$ beats per second; if the beat frequency exceeds $7\text{ Hz}$, individual beats blur together and cannot be distinguished clearly.

Analytical Treatment of Beats

Consider two sound waves of equal amplitude $A_0$ traveling through a medium in the same direction with slightly different frequencies $f_1$ and $f_2$. Their instantaneous displacements are represented by:

$y_1 = A_0 \cos(2\pi f_1 t) \quad \text{--- (1)}$

$y_2 = A_0 \cos(2\pi f_2 t) \quad \text{--- (2)}$

According to the superposition principle, the net displacement $y$ of the resultant wave is the sum of the individual displacements:

$y = y_1 + y_2 = A_0 \left[ \cos(2\pi f_1 t) + \cos(2\pi f_2 t) \right]$

Using the trigonometric sum-to-product identity $\cos A + \cos B = 2 \cos\left(\frac{A - B}{2}\right) \cos\left(\frac{A + B}{2}\right)$:

$y = 2 A_0 \cos\left[2\pi \left(\frac{f_1 - f_2}{2}\right) t\right] \cos\left[2\pi \left(\frac{f_1 + f_2}{2}\right) t\right] \quad \text{--- (3)}$

Equation (3) represents a wave oscillating with an effective carrier frequency equal to the average frequency $\left(\frac{f_1 + f_2}{2}\right)$, modulated by a time-varying amplitude $A$:

$A = 2 A_0 \cos\left[2\pi \left(\frac{f_1 - f_2}{2}\right) t\right] \quad \text{--- (4)}$

A maximum in loudness (loud sound) occurs whenever the modulating cosine term equals $\pm 1$. Since loudness peaks twice in each complete cosine cycle, the number of beats per second (the beat frequency $f_b$) is twice the modulation frequency:

$f_b = f_1 - f_2 \quad \text{--- (5)}$

Thus, the beat frequency is equal to the absolute difference between the frequencies of the two interfering waves.

Fig: Amplitude modulation and waveform of acoustic beats resulting from superposition.

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