Standing Waves & Normal Modes

When a wave travels along a string or through a column of air and meets a boundary — such as a fixed wall or an open end — it reflects back. If the reflected wave overlaps the incoming wave under exactly the right conditions, the two waves superpose to create a pattern that appears to stand still. This is a standing wave, also called a stationary wave. Unlike a travelling wave, which moves energy from place to place, a standing wave stores energy in a fixed pattern of motion, with certain points oscillating vigorously and others remaining perfectly still. The points of zero displacement in a standing wave are called nodes, and the points of maximum displacement are called antinodes. Nodes and antinodes are permanently fixed in space as long as the conditions that produced the standing wave remain unchanged. This happens because the two counter-propagating waves interfere constructively at antinodes and destructively at nodes. The distance between adjacent nodes (or adjacent antinodes) is always exactly half the wavelength of the original travelling wave. Standing waves only form at specific frequencies called natural frequencies or normal mode frequencies. Each distinct pattern — called a normal mode — has a characteristic frequency, wavelength, and number of nodes. The lowest possible frequency is the fundamental mode (or first harmonic), and higher modes are called overtones or harmonics. These concepts underpin how musical instruments produce their characteristic sounds.

Key takeaways
  • Superposition of Two Waves
  • Nodes and Antinodes
  • Boundary Conditions and Normal Modes
  • Energy in a Standing Wave

Superposition of Two Waves

A standing wave arises from the superposition (addition) of two waves with the same amplitude and frequency travelling in opposite directions. Mathematically, if one wave is y₁ = A sin(kx − ωt) and the other is y₂ = A sin(kx + ωt), their sum is y = 2A sin(kx) cos(ωt). This result is no longer a travelling wave — the spatial part sin(kx) and the time part cos(ωt) are separated, meaning every point on the wave oscillates in phase but with an amplitude that depends on its position.

Nodes and Antinodes

From the formula y = 2A sin(kx) cos(ωt), nodes occur where sin(kx) = 0, giving kx = nπ for integer n. Since k = 2π/λ, nodes are spaced λ/2 apart. Antinodes occur where |sin(kx)| = 1, also spaced λ/2 apart and sitting exactly halfway between consecutive nodes. At a node the string (or air) never moves; at an antinode the displacement oscillates between +2A and −2A, twice the amplitude of either original wave.

Boundary Conditions and Normal Modes

The boundaries of the medium impose strict rules. A fixed end must be a node (displacement is always zero there). A free or open end must be an antinode. For a string of length L fixed at both ends, the condition is that a whole number of half-wavelengths must fit exactly: L = nλ/2, so λₙ = 2L/n and the allowed frequencies are fₙ = nv/(2L) for n = 1, 2, 3 … Each value of n corresponds to one normal mode. For a pipe open at one end and closed at the other, only odd harmonics are present: fₙ = nv/(4L) for n = 1, 3, 5 …

Energy in a Standing Wave

Unlike a travelling wave, a standing wave does not transport net energy along the medium. Instead, energy is exchanged back and forth between kinetic energy (maximum at antinodes when displacement is zero) and potential (elastic) energy (maximum at nodes when the medium is most strained). The total mechanical energy remains localised between nodes. This is why a plucked guitar string vibrates in place rather than sending a pulse racing to the bridge.

The Key Equation

fn=nv2Lf_n = \frac{nv}{2L}
  • f_n — Harmonic frequency (Hz): The frequency of the nth normal mode of the standing wave.
  • n — Mode number (harmonic number) (dimensionless): A positive integer: n = 1 is the fundamental, n = 2 the second harmonic, and so on.
  • v — Wave speed (m s⁻¹): The speed at which travelling waves propagate along the medium (depends on tension and mass per unit length for a string).
  • L — Length of the string (m): The distance between the two fixed endpoints of the string or resonating medium.

Worked Example

Problem. A guitar string is 0.65 m long and the speed of transverse waves on it is 520 m s⁻¹. (a) Calculate the fundamental frequency. (b) Calculate the frequency of the third harmonic. (c) How many nodes does the third harmonic pattern have (including the endpoints)?

f1=400 Hz,f3=1200 Hz,nodes=4f_1 = 400\text{ Hz},\quad f_3 = 1200\text{ Hz},\quad \text{nodes} = 4

Answer. The fundamental frequency is 400 Hz; the third harmonic is 1200 Hz; the third harmonic pattern has 4 nodes.

Figure
Diagram
Standing Waves on a Fixed String: First Three HarmonicsNANNn=1 f₁NNNANANn=2 f₂=2f₁NNNNANANANn=3 f₃=3f₁N = Node (zero displacement)AN = Antinode (max displacement)Solid line: positive half-cycle Dashed line: negative half-cycle
Origins · 1 minMarin Mersenne and how this was discovered
MM
Marin Mersenne
1588–1648 · French

Mersenne was the first to systematically study the vibrations of stretched strings and quantify how the frequency of a vibrating string depends on its length, tension, and mass per unit length. His laws of vibrating strings (published in Harmonie Universelle, 1636) laid the mathematical foundation for understanding normal modes and harmonics centuries before the wave equation was formalised. Mersenne corresponded with virtually every leading scientist of his era — Galileo, Descartes, and Pascal — effectively acting as the scientific 'post office' of 17th-century Europe, sharing results and ideas across the continent long before scientific journals existed.

  • Superposition and Interference — Standing waves are a direct consequence of the superposition principle — understanding constructive and destructive interference is essential to explain why nodes and antinodes form.
  • Reflection of Waves at Boundaries — Standing waves require a reflected wave; the nature of the boundary (fixed or free) determines whether reflection causes a phase change and whether a node or antinode forms there.
  • Resonance — Normal modes are the resonant frequencies of a system. Driving a medium at one of its normal mode frequencies produces large-amplitude standing waves — this is resonance.
  • Sound Waves and Harmonics — Musical instruments produce sound through standing waves in strings or air columns; the harmonic series created by normal modes determines the timbre and pitch of musical notes.