Formation of stationary waves

A stationary wave forms when two waves of the same frequency travel in opposite directions and overlap. At each position, their displacements add to give the resultant displacement. This addition is called superposition.

When the two waves have equal amplitudes, their interference creates a pattern with fixed positions of zero and maximum oscillation amplitude:

• A node is a position where the resultant displacement is always zero.
• An antinode is a position where the oscillation amplitude is greatest.

At a node, the two waves always have equal and opposite displacements. At an antinode, their displacements reinforce each other as they oscillate.

Constructing the stationary pattern

To construct a displacement profile, draw the two travelling waves at the same instant. At each position, add their displacements, including their signs. Plot the resultant displacement at each position and join the points. Repeating this at successive times shows how the resultant profile changes.

For example, choose an instant when the two equal-amplitude travelling waves have identical displacement profiles. Their displacements add to give a profile with twice the amplitude of either wave. A quarter of a period later, their displacements are opposite at every position, so the resultant profile is momentarily zero everywhere. After another quarter period, the resultant profile has the same shape as initially but is inverted.

The medium therefore oscillates, but the node and antinode positions remain fixed. A point that is momentarily at zero displacement is not necessarily a node: a node remains at zero displacement at all times.

Spacing and phase

If the wavelength of each travelling wave is $$\lambda$$, adjacent nodes are separated by $$\lambda/2$$. A node and its neighbouring antinode are separated by $$\lambda/4$$.

Points between the same pair of adjacent nodes oscillate in phase: they reach corresponding parts of their oscillations at the same time. Points in neighbouring sections, on opposite sides of a node, oscillate in antiphase: when one section is displaced in one direction, the other is displaced in the opposite direction.

The two equal-amplitude waves carry energy in opposite directions, so the stationary pattern transfers no net energy along the wave.

Stationary waves formed by reflection

A stationary wave can form when an incident wave overlaps with a wave reflected back along the same path. Equal incident and reflected amplitudes produce the ideal pattern of zero-amplitude nodes and maximum-amplitude antinodes.

On a string, nodes and antinodes are positions of minimum and maximum transverse displacement. For microwaves or sound, a detector moved through the region can reveal fixed minima and maxima in the detected signal.

For sound, displacement and pressure variations have opposite node–antinode positions: a displacement node is a pressure antinode, while a displacement antinode is a pressure node.

Harmonics on a fixed string

A string fixed at both ends can form a stationary wave when waves travelling along it in opposite directions combine to produce a stable pattern. The fixed ends cannot move, so they must be nodes: points with zero displacement. Between nodes are antinodes, where the oscillation has maximum amplitude.

Only patterns that fit nodes at both ends are permitted. These patterns are called harmonics.

Harmonics and wavelength

The distance between neighbouring nodes is half a wavelength. Therefore, the string length $$l$$ must contain a whole number of half-wavelengths. For harmonic number $$n = 1, 2, 3, \ldots$$, with wavelength $$\lambda_n$$:

The first harmonic contains one half-wavelength: it has one antinode between the two fixed ends. The second harmonic contains two half-wavelengths, with two antinodes and one additional node between the ends. The third contains three half-wavelengths, with three antinodes and two additional nodes.

In general, harmonic $$n$$ has $$n$$ antinodes and $$n-1$$ nodes between the fixed ends. As the harmonic number increases, the permitted wavelength decreases.

Harmonic frequencies

For a string under tension $$T$$, with mass per unit length $$\mu$$, the frequency of the first harmonic is:

The frequency is inversely proportional to length: doubling $$l$$ halves $$f_1$$. It increases with the square root of the tension and decreases with the square root of the mass per unit length. For example, quadrupling $$T$$ doubles $$f_1$$, while quadrupling $$\mu$$ halves it, provided the other quantities stay unchanged.

The higher harmonic frequencies are whole-number multiples of the first-harmonic frequency:

Thus the second harmonic has frequency $$2f_1$$ and the third has frequency $$3f_1$$. For the same string, increasing the harmonic number makes the wavelength shorter and the frequency proportionally higher.