Standing Waves on a String
Pluck a guitar string, blow across the mouth of a bottle, or shake a jump rope at just the right speed, and you'll see the same phenomenon: two waves traveling in opposite directions along the same medium interfere to create a pattern that appears to stand still rather than travel. This happens whenever a wave reflects off a fixed or free end and overlaps with its own reflection — at most frequencies the result is a messy jumble, but at specific resonant frequencies the interference locks into a clean, stationary pattern of points that never move (nodes) and points that swing between maximum displacement (antinodes). Nearly every musical instrument, from strings to wind instruments to drumheads, relies on exactly this behavior to produce a stable pitch.
Nodes and antinodes
Nodes are points that stay still; antinodes oscillate with maximum amplitude. The number of segments depends on frequency and boundary conditions.
Resonance and harmonics
At resonant frequencies, integer multiples of half-wavelengths fit on the string. Each harmonic adds another antinode between fixed ends.
The Wave Equation
A wave's speed is set by the medium it travels through, like a string's tightness or the air it moves in — not by how fast you shake the source. Because the speed stays fixed, every cycle you send out must fit into that same speed, so a faster wiggle packs the cycles closer together into a shorter wavelength.
The formula
v = f · λ
- v — wave speed (m/s): how fast the wave pattern moves through the medium
- f — frequency (Hz): how many wave cycles pass a point each second
- λ — wavelength (m): the length of one full wave cycle
Worked example
A wave travels along a string, completing 5 full cycles every second. Each cycle stretches out over 2 meters. How fast does the wave travel?
- f = 5 Hz
- λ = 2 m
- v = f · λ
- v = 5 Hz × 2 m
v = 10 m/s
Test yourself
That same string still carries waves at 10 meters per second. If you shake it at 10 hertz instead of 5, what happens to the wavelength?
- It doubles to 4 m
- It stays at 2 m
- Correct answer: It halves to 1 m
Right! Speed is fixed by the string, so λ = v / f = 10 m/s ÷ 10 Hz = 1 m — doubling the frequency halves the wavelength.
A different wave has a frequency of 4 hertz and a wavelength of 3 meters. What is its speed?
- Correct answer: 12 m/s
- 7 m/s
- 0.75 m/s
Yes! v = f · λ = 4 Hz × 3 m = 12 m/s.
Where you see this
Tightening a guitar string is this equation made audible: the tighter string carries waves faster, so every vibration the player sends out returns as a higher pitch. Shaking the string faster does something different — the wave still travels at the speed the string allows, but the cycles pack closer together into a shorter wavelength. The medium, not the musician, sets the speed.
Common mistakes
The reflex error is that shaking the source faster must make the wave travel faster — the speed is fixed by the medium alone, so a faster wiggle only shortens the wavelength: rearrange v = f · λ to λ = v / f and the trade is explicit. Shake a 10 m/s string at 10 hertz instead of 5 and the wavelength drops from 2 meters to 1 meter, while the speed never moves. Frequency and wavelength are not independent knobs.
How it connects
This one equation is the spine of the whole waves unit: the Doppler effect is what happens when the source's own motion distorts those wavelengths at a fixed wave speed, and beats are two nearly-identical frequencies trading places in and out of step. It reaches across modules too — interference and diffraction in optics are governed by light's wavelength through this same relation.