Simulation · Physics · Class 11

Beats: tuning one string against another

From the lesson Beats in Waves. Change the values and watch what happens.

The idea behind it

NCERT §14.7

Two sounds of nearly equal frequency heard together give a sound at their average frequency whose loudness swells and fades regularly. This waxing and waning is called beats.
For two waves of equal amplitude, s = s₁ + s₂ = [2a cos ω_b t] cos ω_a t, with ω_b = (ω₁ − ω₂)/2 and ω_a = (ω₁ + ω₂)/2.
When |ω₁ − ω₂| is small compared with ω₁ + ω₂, the term cos ω_a t is a fast oscillation and 2a cos ω_b t is a slowly changing amplitude around it.
The loudness peaks whenever cos ω_b t reaches +1 or −1, which happens twice in each period of cos ω_b t. So the beat frequency is ν_beat = ν₁ − ν₂.
Example: waves of 11 Hz and 9 Hz give a resultant at 10 Hz that swells and fades 2 times a second.
Musicians use beats to tune instruments: they adjust one against a reference until the beats slow down and vanish.
Worked example (NCERT): sitar string A plays Dha at 427 Hz, and 5 beats per second are heard with string B, so B is 422 Hz or 432 Hz. Tightening B raises its frequency and the beats drop to 3 per second; only 422 Hz moves towards 427 Hz, so ν_B = 422 Hz.
Musical pillars: the Nellaiappar temple in Tamil Nadu, built by the Pandyan dynasty in the 7th century, has stone pillars that give musical notes (Sa Re Ga Ma Pa Dha Ni Sa) when tapped, grouped as Shruti, Gana Thoongal and Laya Thoongal pillars. Such pillars are also found at Hampi, Kanyakumari and Thiruvananthapuram.

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