BeatsJEE Main

Concept

When two sound waves of nearly equal frequency overlap, the loudness rises and falls periodically — beats. The ear hears a tone at the average frequency that throbs at the beat frequency equal to the difference of the two.

Key formula

Derivation

Add $y_1 = A\sin(2\pi f_1 t)$ and $y_2 = A\sin(2\pi f_2 t)$ using the sum-to-product identity.

This gives a fast oscillation at the mean frequency $\tfrac{f_1+f_2}{2}$ multiplied by a slow envelope $2A\cos(2\pi\tfrac{f_1-f_2}{2}t)$.

Loudness peaks each time the envelope reaches $\pm 2A$, which happens twice per envelope cycle, so the beat frequency is $|f_1 - f_2|$, not half of it.

Scenarios to explore

• Beats — Two close frequencies producing a beat envelope.

Real-world applications

• Tuning musical instruments against a reference until the beats vanish.
• Measuring small frequency differences precisely.
• Heterodyne radio receivers.

JEE exam tips

• If a tuning fork of known frequency gives n beats/s, the unknown is $f \pm n$ — load it to resolve the sign.
• Adding wax to a fork lowers its frequency; use this to decide the sign of the difference.

Common mistakes

• Taking the beat frequency as $\tfrac{|f_1-f_2|}{2}$ — the envelope's amplitude maxima occur twice per cycle.
• Confusing the heard pitch (average) with the throb rate (difference).
• Expecting beats when the two frequencies are far apart (they merge into separate tones).

Exam traps to avoid

• Beats require the frequencies to be close; otherwise the sensation disappears.
• The number of beats heard per second equals the magnitude of the frequency difference.