What is a standing wave?

A standing wave is also called a stationary wave, the name used in UK A-level Physics. It looks like a string swelling and shrinking in loops on the spot, rather than ripples moving along. For waves in general, see our guide to what a wave is.

How a standing wave forms

Shake one end of a rope tied to a wall. The wave travels to the wall, reflects and comes back. Now the outgoing and returning waves overlap and add together, which is called superposition. At some points, the two waves always cancel: one pushes up exactly as the other pushes down. These are the nodes. At others, they always add up, making the biggest vibration. These are the antinodes.

A steady pattern only appears at certain frequencies, when a whole number of half-wavelengths fits exactly between the two ends. At other frequencies, the reflections pile up out of step and the pattern is a jumble. This is resonance: the system vibrates strongly at its own natural frequencies.

Here x is the position along a 6 m string in metres and y is the displacement at one moment. The string is fixed at both ends, which are always nodes, and there are nodes at 2 m and 4 m too. Half a cycle later, every loop is flipped: up becomes down. The nodes never move, so the pattern stands still. Nodes are half a wavelength apart, so here the wavelength is 4 m.

Standing wave vs travelling wave

Harmonics on a string

A string fixed at both ends must have a node at each end, so it can only hold patterns with a whole number of loops. Each loop is half a wavelength. These patterns are the harmonics.

The lowest, the fundamental, sets the note's pitch. The higher harmonics, sounding at the same time, give an instrument its character, which is why a guitar and a violin playing the same note sound different. Pressing a guitar string against a fret shortens L, which raises the frequency.

Standing waves in pipes

In wind instruments, the standing wave is in the air. A closed end is a node, because the air there can't move, and an open end is an antinode.

• Open at both ends (a flute): antinodes at both ends, so the harmonics follow the string pattern, f = nv ÷ 2L, with all whole numbers n.
• Closed at one end (a clarinet, roughly, or blowing across a bottle): the air can't move at the closed end but swings freely at the open one, so the fundamental is a quarter wavelength, f = v ÷ 4L, and only odd harmonics occur: 1st, 3rd, 5th and so on.

Worked examples

Example 1: a guitar string

A guitar string vibrates over a length of 0.65 m, and waves travel along it at 286 m/s. What is its fundamental frequency? f₁ = v ÷ 2L = 286 ÷ (2 × 0.65) = 286 ÷ 1.3 = 220 Hz. The second harmonic is 2 × 220 = 440 Hz.

Example 2: a pipe closed at one end

A pipe 0.30 m long is closed at one end. Sound travels at 343 m/s. The fundamental is f = v ÷ 4L = 343 ÷ 1.2 = 285.8 Hz. The next harmonic it can play is the third: 3 × 285.8 = 857.5 Hz. If the same pipe were open at both ends, its fundamental would be 343 ÷ 0.6 = 571.7 Hz, twice as high.

Example 3: measuring the speed of light in a microwave oven

Microwaves bouncing around inside an oven form standing waves, which is why food heats unevenly and most ovens have a turntable. Take the turntable out and heat a plate of chocolate briefly: it melts first at the antinodes. Suppose the melted spots are 6.1 cm apart. Antinodes are half a wavelength apart, so λ = 12.2 cm. With the oven's frequency of 2.45 GHz, the wave speed is c = fλ = 2.45 × 10⁹ × 0.122 = 2.99 × 10⁸ m/s, very close to the speed of light.

Standing waves in exams

In the UK, stationary waves are an A-level topic. The AQA specification covers how they form, nodes and antinodes, harmonics on a string, and a required practical investigating how the frequency of a vibrating string depends on its length, tension and mass per unit length. US high school physics covers the same ideas through strings and pipes. You can explore wave overlap in the Wave Interference Simulator and our Light and Energy course.

Common mistakes

• Saying nodes are a whole wavelength apart. Adjacent nodes, and adjacent antinodes, are half a wavelength apart.
• Saying standing waves carry energy along. Energy is stored in the pattern, not transferred along it.
• Putting antinodes at the fixed ends of a string. Fixed ends must be nodes.
• Giving a closed pipe even harmonics. A pipe closed at one end only has odd harmonics.
• Mixing up a standing wave and interference in general. A standing wave needs two matching waves moving in opposite directions.

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