4.4.4 - Stationary waves and harmonics

Forming stationary waves

A stationary wave is not a progressive wave that has somehow stopped moving. It is a pattern produced when two progressive waves of the same frequency and equal amplitude travel in opposite directions through the same region and superpose.

The most common way to make this happen is reflection. A wave travels along a string, microwave path or air column, reflects from a boundary, then overlaps with the incoming wave. At the right frequencies the overlap is stable: some positions keep cancelling and some positions keep reinforcing.

OCR expects you to recognise the same idea in three physical settings:

• In a stretched string, a vibrator or oscillator sends transverse waves along the string and the waves reflect from fixed ends.
• With microwaves, a source sends waves towards a metal reflector; a detector moved between source and reflector finds positions of large and small signal.
• In an air column, sound waves reflect at the ends of the tube; at resonant frequencies a strong stationary sound wave is formed.

The word "stationary" describes the pattern, not the particles in the medium. Except at nodes, particles still oscillate. What stays fixed is the position of the nodes, antinodes and overall pattern.

The component waves must also have the same wavelength in the same medium. A sequence of snapshots makes the distinction clear: two travelling profiles move in opposite directions, while their sum changes height without shifting its nodes. At some instants the entire resultant is at zero displacement. That does not make every position a node: a node stays at zero at every instant.

Worked reasoning: forming a pattern on a string

A string is fixed at both ends and driven by a mechanical oscillator near one end. The oscillator sends waves along the string. The wave reflects from the far fixed end and travels back. At a resonant frequency, incident and reflected waves of equal frequency and equal amplitude travel in opposite directions and superpose. Fixed points of no displacement and fixed points of maximum displacement appear: the stationary wave pattern.

[DIAGRAM: asset_slug: 4_04_4_stationary_waves_and_harmonics__diagram_01; caption: Counterpropagating waves produce fixed nodes. An instant of zero displacement everywhere does not make every point a permanent node.; description: Three matched snapshots show counter-propagating sine waves and their sum. Nodes at half-wavelength intervals remain fixed; the quarter-period zero profile does not make every point a permanent node.]

Nodes, antinodes and spacing

The two named positions in a stationary wave are nodes and antinodes. They are defined using displacement.

At a node the two travelling waves always cancel. A point at a node does not oscillate, even though energy is passing in the two travelling waves that formed the pattern.

At an antinode the two travelling waves reinforce to give the largest possible oscillation of the medium. In a graphical representation, the stationary wave is often drawn as an outline shape showing the largest positive and negative displacement reached at each position.

The key geometry is:

Here $\lambda$ is the wavelength of either progressive wave that formed the stationary wave. It is not the distance from one end of the pattern to the other unless that whole length happens to contain one full wavelength.

The distance from a node to the next antinode is half as large:

Worked example: using node spacing

In a microwave stationary-wave experiment, adjacent detector minima are separated by $1.60\ \text{cm}$. The minima are electric-field nodes. Determine the wavelength of the microwaves.

Adjacent nodes are separated by $\lambda/2$, so:

If the frequency were supplied, you could then use $v=f\lambda$, the wave equation from the OCR data booklet, to determine the wave speed.

Stationary and progressive waves

Stationary and progressive waves use the same basic wave quantities, such as frequency, wavelength and amplitude, but they behave differently.

A progressive wave transfers energy from one place to another. Its crests, troughs, compressions or rarefactions move through the medium. If you take a displacement-distance graph at later times, the shape has moved along.

A stationary wave has a fixed pattern. There is no net transfer of energy along the stationary pattern from one end to the other. The amplitude depends on position: it is zero at nodes and maximum at antinodes.

The phase idea is useful but easy to overcomplicate. In one loop between two adjacent nodes, every point reaches maximum displacement in the same direction at the same time, although different points have different amplitudes. Cross a node into the next loop and the motion is in antiphase: when one loop is displaced upward, the next is displaced downward.

This is why a stationary-wave drawing often shows loops above and below the centre line. The drawing is not a path followed by one particle; it is a maximum-displacement envelope for many particles at different positions.

Strings, tubes and harmonics

Only certain stationary-wave patterns fit a given string or air column. The allowed patterns are set by the boundary conditions: what the ends of the medium are forced to do.