17-6 Beats

17-6 Beats#

Prompts

  • Two sound waves with frequencies 440 Hz and 444 Hz are played together. What beat frequency do you hear? What is the perceived pitch?

  • Starting from \(s_1 = s_m \cos(\omega_1 t)\) and \(s_2 = s_m \cos(\omega_2 t)\), use the identity \(\cos a + \cos b = 2\cos\frac{a-b}{2}\cos\frac{a+b}{2}\) to find the resultant. Identify the fast oscillation and the slow amplitude modulation.

  • Why does the beat frequency equal \(|f_1 - f_2|\)? (Hint: how many times per second does the amplitude reach a maximum?)

  • How do musicians use beats to tune an instrument? What happens when the instrument is perfectly in tune?

Lecture Notes#

Overview#

  • Beats are the slow, periodic variation in loudness when two sound waves of slightly different frequencies are heard together.

  • The beat frequency is \(f_{\text{beat}} = |f_1 - f_2|\)—the rate at which the combined sound waxes and wanes.

  • The perceived pitch is the average of the two frequencies.

  • Beats are used to tune instruments: match a tone to a reference until beats disappear.

Superposition of two nearly equal frequencies#

Two waves of equal amplitude \(s_m\) and slightly different angular frequencies \(\omega_1\), \(\omega_2\):

(133)#\[ s_1 = s_m \cos(\omega_1 t), \qquad s_2 = s_m \cos(\omega_2 t) \]

Using \(\cos a + \cos b = 2\cos\frac{a-b}{2}\cos\frac{a+b}{2}\), the resultant is

(134)#\[ s = s_1 + s_2 = \left[2s_m \cos(\omega' t)\right] \cos(\omega t) \]

where

(135)#\[ \omega' = \frac{1}{2}(\omega_1 - \omega_2), \qquad \omega = \frac{1}{2}(\omega_1 + \omega_2) \]

  • \(\cos(\omega t)\): fast oscillation at the average angular frequency.

  • \(2s_m \cos(\omega' t)\): slowly varying amplitude (envelope).

Beat frequency#

The amplitude \(A(t) = |2s_m \cos(\omega' t)|\) reaches a maximum when \(\cos(\omega' t) = \pm 1\)—twice per cycle of \(\cos(\omega' t)\). So the beat angular frequency is \(\omega_{\text{beat}} = 2\omega' = \omega_1 - \omega_2\), and

(136)#\[ f_{\text{beat}} = |f_1 - f_2| \]

  • Perceived pitch: \(f_{\text{avg}} = \frac{1}{2}(f_1 + f_2)\).

  • Example: 440 Hz and 444 Hz → beat frequency 4 Hz (4 loudness cycles per second), perceived pitch ~442 Hz.

When are beats audible?

Beats are heard only when \(f_{\text{beat}}\) is small (typically \(\lesssim 12\) Hz). If the frequencies differ too much, the ear hears two separate tones instead of a single tone with varying loudness.

Poll: Beat frequency from envelope

Two pairs of waves produce beats. Pair 1 has a slower loudness variation than Pair 2. For which pair is \(|f_1 - f_2|\) greater?

[FIGURE: Two beat traces — one with slow envelope, one with fast envelope; time axis labeled]

  • (A) Pair 1

  • (B) Pair 2

  • (C) Same for both

  • (D) Cannot tell

Tuning with beats#

Musicians tune by sounding the instrument against a reference (e.g., concert A at 440 Hz). If beats are heard, the instrument is out of tune. Adjust the instrument until the beats disappear—then the frequencies match.

Poll: Beat frequency

A tuning fork at 256 Hz and a piano string produce 4 beats per second. What are the possible frequencies of the piano string?

  • (A) 252 Hz or 260 Hz

  • (B) 254 Hz or 258 Hz

  • (C) 260 Hz only

Summary#

  • Beats: slow loudness variation when two nearly equal frequencies combine.

  • \(f_{\text{beat}} = |f_1 - f_2|\)—beat frequency equals the difference.

  • Perceived pitch ≈ \(\frac{1}{2}(f_1 + f_2)\).

  • Tuning: match to reference until beats vanish.