Physics / Unit 7: Waves and Sound
7.6 Standing Waves and Resonance
Notes
A standing wave forms when a wave reflects and interferes with itself, making a pattern that seems to stand still.
- Nodes: points that do not move. Antinodes: points of largest motion.
- On a string fixed at both ends, the allowed wavelengths are $\displaystyle \lambda_n = \frac{2L}{n}$ for $n = 1, 2, 3, \dots$, so $f_n = n f_1$.
- The lowest frequency $f_1$ is the fundamental; the others are harmonics.
- Resonance: a system vibrates with large amplitude when pushed at its natural frequency (a singer shattering a glass, pushing a swing in time).
- Example: a 0.6 m guitar string with wave speed $240 \text{ m/s}$: $\lambda_1 = 1.2 \text{ m}$, $\displaystyle f_1 = \frac{240}{1.2} = 200 \text{ Hz}$.
Practice
A string 0.5 m long is fixed at both ends. What is the wavelength of its fundamental?
$\lambda_1 = 2L = 2(0.5) = 1.0 \text{ m}$.
A string's fundamental frequency is 150 Hz. What is the frequency of its third harmonic?
$f_3 = 3f_1 = 3 \times 150 = 450 \text{ Hz}$.
A child on a swing goes higher when pushed at just the right moment each swing. This is an example of:
Pushes at the natural frequency build up a large amplitude.
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