What is meant by beats? Derive a formula for beat frequency of two waves.
The Correct Answer and Explanation is:
What is Meant by Beats?
Beats refer to the periodic variation in the amplitude (loudness) of sound resulting from the interference of two sound waves of slightly different frequencies traveling in the same direction. When two waves of close but unequal frequencies interfere, they periodically reinforce and cancel each other, producing a fluctuating sound—this fluctuation is called a beat.
Derivation of Beat Frequency
Let two waves with slightly different frequencies f1f_1f1 and f2f_2f2 be represented as:y1=Asin(2πf1t)y2=Asin(2πf2t)y_1 = A \sin(2\pi f_1 t) \\ y_2 = A \sin(2\pi f_2 t)y1=Asin(2πf1t)y2=Asin(2πf2t)
The resultant wave yyy is given by the principle of superposition:y=y1+y2=Asin(2πf1t)+Asin(2πf2t)y = y_1 + y_2 = A \sin(2\pi f_1 t) + A \sin(2\pi f_2 t)y=y1+y2=Asin(2πf1t)+Asin(2πf2t)
Using the trigonometric identity:sina+sinb=2sin(a+b2)cos(a−b2)\sin a + \sin b = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a – b}{2}\right)sina+sinb=2sin(2a+b)cos(2a−b)
We get:y=2Acos[π(f1−f2)t]sin[2π(f1+f22)t]y = 2A \cos\left[\pi (f_1 – f_2)t \right] \sin\left[2\pi \left(\frac{f_1 + f_2}{2}\right)t \right]y=2Acos[π(f1−f2)t]sin[2π(2f1+f2)t]
This expression shows a modulated wave: a sinusoidal wave with a carrier frequency f1+f22\frac{f_1 + f_2}{2}2f1+f2, and an amplitude that varies sinusoidally with a modulation (beat) frequency:fbeat=∣f1−f2∣f_{\text{beat}} = |f_1 – f_2|fbeat=∣f1−f2∣
Explanation
Beats arise due to the interference between two sound waves of slightly different frequencies. When these waves meet, they combine according to the principle of superposition. At certain points in time, their crests align (constructive interference), resulting in a louder sound. At other times, the crest of one wave aligns with the trough of the other (destructive interference), canceling out the sound. This alternation of loud and soft sounds is perceived as “beats.”
The rate at which the loudness fluctuates is known as the beat frequency, and it is equal to the absolute difference between the frequencies of the two interfering waves:fbeat=∣f1−f2∣f_{\text{beat}} = |f_1 – f_2|fbeat=∣f1−f2∣
For example, if a tuning fork of 256 Hz is struck simultaneously with one of 260 Hz, the listener will hear 4 beats per second.
Beats are commonly used in tuning musical instruments. Musicians listen to the beat frequency and adjust their instrument until the beats disappear (i.e., when both instruments produce the same frequency, the beat frequency becomes zero).
In essence, beats are a powerful and observable result of wave interference, demonstrating how small differences in frequency can have a large impact on what we perceive in sound.
