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Superposition, Beats, and Harmonics

24 min Vibrations & Waves Math

Superposition and Harmonics

When two waves overlap, they add. This simple principle creates interference, beats, and the rich sound of music.

Principle of Superposition

$$y_{\text{total}} = y_1 + y_2$$

Beats

Two waves of close frequencies $f_1$ and $f_2$ produce beats at frequency:

$$f_{\text{beat}} = |f_1 - f_2|$$

Harmonics

Standing waves on a string of length $L$:

$$f_n = n \cdot \frac{v}{2L} \quad (n = 1, 2, 3, \ldots)$$

$f_1$ = fundamental, $f_2$ = 2nd harmonic, etc.

Example 1

Two tuning forks: 440 Hz and 444 Hz. Beat frequency?

$f_{\text{beat}} = |444 - 440| = 4$ Hz (4 beats per second).

Example 2

A guitar string is 0.65 m long; wave speed is 300 m/s. Find the first three harmonics.

$f_1 = 300/(2 \times 0.65) = 230.8$ Hz.

$f_2 = 2 \times 230.8 = 461.5$ Hz.

$f_3 = 3 \times 230.8 = 692.3$ Hz.

Example 3

$y_1 = \sin(6\pi t)$ and $y_2 = \sin(8\pi t)$. What is the beat frequency?

$f_1 = 6\pi/(2\pi) = 3$ Hz, $f_2 = 4$ Hz.

$f_{\text{beat}} = |3 - 4| = 1$ Hz.

1. Two forks: 256 Hz and 260 Hz. Beat frequency?

2. String: $L = 1$ m, $v = 400$ m/s. Fundamental frequency?

3. What is the 4th harmonic if $f_1 = 100$ Hz?

4. When two identical waves meet in phase, amplitude is ___?

5. When two identical waves meet 180° out of phase?

6. $y = 3\sin(2\pi t) + 3\sin(2\pi t)$. Result?

Show Answer Key

1. 4 Hz

2. $f_1 = 400/2 = 200$ Hz

3. 400 Hz

4. Doubled (constructive interference)

5. Zero (destructive interference)

6. $6\sin(2\pi t)$ — amplitude doubles