Physics › 🌊 Waves & Sound

Standing Waves & Resonance

Two waves collide head-on and, instead of chaos, you get a pattern that just sits there and pulses — that stillness is the whole trick.

High schoolAP Physics 1 level
💡
The big idea: A standing wave isn't a new kind of wave — it's what you see when two identical waves travel through each other in opposite directions, usually because a wave keeps reflecting back off a fixed boundary. The overlap creates spots that never move (nodes) and spots that swing between maximum extremes (antinodes). Because a string of length \(L\) can only fit certain wave shapes between its fixed ends, only certain wavelengths — and therefore only certain frequencies — are allowed. Those allowed frequencies are the harmonics, and driving a system at one of them is resonance.
🎯 By the end, you'll be able to
  • Explain a standing wave as the superposition of two traveling waves moving in opposite directions.
  • Identify nodes and antinodes on a vibrating string and state their spacing in terms of wavelength.
  • Use \(\lambda_n = 2L/n\) and \(v = f\lambda\) to find the wavelength and frequency of any harmonic on a fixed string.
  • Explain resonance as driving a system at one of its natural (harmonic) frequencies.
📎 You should already know
  • Wave basics: amplitude, wavelength, frequency, and period
  • Wave speed equation \(v = f\lambda\)
  • Superposition and interference of waves

Two waves, one still pattern

Pluck a guitar string and send a wave pulse down its length. It hits the fixed end, flips over, and travels straight back toward you. Now you've got two waves — the one you sent and its reflection — moving through the exact same string in opposite directions at the same time.

Most of the time, overlapping waves make a messy, shifting jumble. But if the timing lines up just right, something strange happens: the jumble locks into a fixed shape. Certain points never move at all. Other points swing back and forth with the biggest amplitude the string ever shows. Nothing in the pattern appears to travel left or right anymore — it just pulses in place. That's a standing wave.

🔑 Nodes, antinodes, and why they exist

A node is a point that stays motionless the entire time — at that exact spot, the crest of the rightward wave always arrives at the same instant as the trough of the leftward wave, so they cancel completely, forever. An antinode is a point where the two waves' crests (and later, troughs) always arrive together, adding up to double amplitude.

Adjacent nodes are always spaced half a wavelength apart, and an antinode sits exactly halfway between each pair of nodes. A string fixed at both ends must have a node at each end — the string can't move where it's clamped — and that one geometric fact is what restricts which wavelengths are allowed to fit.

\[ v = f\lambda \]
The wave speed equation still holds for standing waves — it just relates the frequency and wavelength of whichever harmonic you're looking at.
\[ \lambda_n = \dfrac{2L}{n}, \quad n = 1, 2, 3, \ldots \]
For a string of length \(L\) fixed at both ends, only these wavelengths fit — n is the harmonic number, equal to the number of antinodes (and also the number of half-wavelength 'loops') along the string.
\[ f_n = \dfrac{nv}{2L} \]
Combining the two equations above gives the allowed (harmonic) frequencies directly: the fundamental is \(f_1 = v/2L\), and every higher harmonic is a whole-number multiple of it.
🎮 Interactive: Build harmonics on a string LIVE
Drag the harmonic number to see how the node/antinode pattern reshapes itself — watch how the wavelength shrinks and the frequency climbs as n increases, while the string length stays fixed.

Resonance: pushing at exactly the right rhythm

Every physical string, air column, or bridge has its own set of natural (harmonic) frequencies, set entirely by its length and wave speed. If you drive the system with a periodic push — a bow, a speaker, a gust of wind — at a frequency that doesn't match any harmonic, the energy you put in mostly cancels itself out and the response stays small.

But if you drive it at (or very near) one of those harmonic frequencies, each new push arrives in step with the motion already there, and the energy keeps adding up. The amplitude grows dramatically even though each individual push is small. That's resonance — it's why singers can shatter glasses, why marching soldiers break step crossing a bridge, and why a swing goes higher when you pump it at the right moments rather than randomly.

📝 Worked example: A string is 2.0 m long and waves travel along it at 40 m/s. Find the wavelength and frequency of the first three harmonics.
  1. Use \(\lambda_n = 2L/n\) with \(L = 2.0\text{ m}\).
  2. n = 1 (fundamental): \(\lambda_1 = 2(2.0)/1 = 4.0\text{ m}\), so \(f_1 = v/\lambda_1 = 40/4.0 = 10\text{ Hz}\).
  3. n = 2 (2nd harmonic): \(\lambda_2 = 2(2.0)/2 = 2.0\text{ m}\), so \(f_2 = 40/2.0 = 20\text{ Hz}\).
  4. n = 3 (3rd harmonic): \(\lambda_3 = 2(2.0)/3 \approx 1.33\text{ m}\), so \(f_3 = 40/1.33 \approx 30\text{ Hz}\).
  5. Notice each harmonic frequency is just n times the fundamental: 10, 20, 30 Hz — that pattern always holds.
✓ f₁ = 10 Hz, f₂ = 20 Hz, f₃ = 30 Hz (each a whole-number multiple of the 10 Hz fundamental).
📝 Worked example: A guitar string 0.65 m long vibrates at its fundamental frequency of 110 Hz. What is the wave speed on the string?
  1. The fundamental corresponds to n = 1, where \(\lambda_1 = 2L = 2(0.65) = 1.3\text{ m}\).
  2. Use \(v = f\lambda\): \(v = (110\text{ Hz})(1.3\text{ m})\).
  3. \(v = 143\text{ m/s}\).
✓ v ≈ 143 m/s.
⚠️ Common trap: don't confuse 'number of humps' with wavelength

It's tempting to think the n-th harmonic pattern shows n full wavelengths on the string — it doesn't. It shows n half-wavelengths (n antinodes, n+1 nodes counting both ends). That's exactly why \(\lambda_n = 2L/n\) has that factor of 2: a full wavelength needs a full crest-to-crest cycle, but each 'loop' you see between two nodes is only half of one.

Also remember this specific formula assumes both ends are fixed (nodes at both ends), like a guitar or violin string. A tube open at one end, or a string free at one end, has a different pattern — node at the fixed end, antinode at the free end — and a different formula.

Check your understanding

1. What is a node on a standing wave?
A node is where the two overlapping traveling waves always cancel exactly, so that point never moves, regardless of time.
2. A string 3.0 m long carries waves at 60 m/s. What is the frequency of its 2nd harmonic?
λ₂ = 2L/n = 2(3.0)/2 = 3.0 m. Then f = v/λ = 60/3.0 = 20 Hz.
3. Why does a system respond with unusually large amplitude when driven at one of its harmonic frequencies?
Resonance happens when the timing of the driving force matches the system's natural rhythm, so energy adds constructively push after push instead of partially canceling.
4. For a string fixed at both ends, how many nodes does the 3rd harmonic (n = 3) have, including the two fixed ends?
The n-th harmonic has n antinodes and n+1 nodes total. For n = 3, that's 4 nodes — the two fixed ends plus 2 more in between.
✅ Key takeaways
  • A standing wave forms when two identical waves travel in opposite directions through the same medium, typically from a wave reflecting off a fixed boundary.
  • Nodes (always still) and antinodes (maximum swing) alternate along the pattern, spaced half a wavelength apart.
  • A string fixed at both ends only allows wavelengths given by \(\lambda_n = 2L/n\), producing harmonic frequencies \(f_n = nv/2L\) that are whole-number multiples of the fundamental.
  • Resonance is what happens when a driving force matches one of these natural harmonic frequencies, letting small pushes build into large amplitude.