Musical instruments ring because vibrating parts can store and exchange energy in regular patterns. A plucked guitar string, bowed violin string, air column, drumhead, or metal bar vibrates most strongly at certain natural frequencies. These preferred frequencies shape the pitch and tone that we hear.
Resonance makes some vibrations grow louder while others fade quickly.
Key Facts
- Wave speed on a stretched string: v = sqrt(T/mu), where T is tension and mu is mass per unit length.
- Allowed standing-wave wavelengths on a string fixed at both ends: lambda_n = 2L/n.
- Harmonic frequencies for a fixed string: f_n = n v/(2L), where n = 1, 2, 3, ...
- The fundamental frequency is the lowest allowed frequency: f_1 = v/(2L).
- Nodes have zero displacement, while antinodes have maximum displacement.
- Resonance occurs when a driving frequency matches a natural frequency, causing a larger vibration amplitude.
Vocabulary
- Resonance
- Resonance is the large response of a system when it is driven at or near one of its natural frequencies.
- Standing wave
- A standing wave is a wave pattern that appears to stay in place because two waves of the same frequency travel in opposite directions and interfere.
- Node
- A node is a point in a standing wave that remains still because destructive interference always occurs there.
- Antinode
- An antinode is a point in a standing wave where the vibration has maximum amplitude.
- Harmonic
- A harmonic is a resonant frequency that is an integer multiple of the fundamental frequency in an ideal string or open air column.
Common Mistakes to Avoid
- Confusing loudness with pitch is wrong because loudness depends mainly on amplitude, while pitch depends mainly on frequency.
- Putting antinodes at the fixed ends of a string is wrong because fixed ends cannot move, so they must be nodes.
- Using lambda = L for the fundamental on a string fixed at both ends is wrong because the fundamental has half of a wavelength on the string, so lambda = 2L.
- Assuming every object resonates at only one frequency is wrong because most instruments have many natural frequencies called harmonics or modes.
Practice Questions
- 1 A guitar string is 0.65 m long and has a wave speed of 260 m/s. What is its fundamental frequency?
- 2 A violin string fixed at both ends is 0.33 m long and vibrates in its third harmonic at 990 Hz. What is the wave speed on the string?
- 3 A player lightly touches a guitar string at its exact midpoint while plucking it, preventing that point from moving. Which harmonics are favored or suppressed, and why?