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Superposition overview

Topic 2 of 9

Build a standing wave

Equal-amplitude waves of the same frequency travelling in opposite directions can form a standing wave. Its fixed pattern of oscillation amplitude differs from the travelling profiles that produce it.

Take two waves in the same region and oscillation direction, with the same wavelength and speed but opposite propagation directions. At every position and instant, add their signed displacements. The result is also called a stationary wave.

Add the profiles through a cycle

Each component in this supplied model has amplitude 2.0 mm, wavelength 0.80 m and period T = 0.20 s. The view covers x = 0 to 0.80 m.

Two travelling profiles form one standing pattern

Component 1 travels right; component 2 travels left. Each has amplitude 2.0 mm. Brown long dashes and green short dashes distinguish the components; solid blue is their point-by-point sum. Every panel shares the same axes.

t = 0

Standing-wave construction at t = 0The horizontal axis is position from zero to 0.80 metres; the vertical axis is signed displacement in millimetres, from negative four to positive four. Components each have amplitude two millimetres and travel oppositely. The two component profiles coincide and are labelled together. The summed amplitude at an antinode is twice the component amplitude. At half a period all displacements reverse from the initial profile. Node positions are fixed in every panel. No envelope is being substituted for the instantaneous sum.0.000.200.400.600.80-4-20+2+4Displacement / mmPosition x / m

Component 1 = component 2 in this snapshot, so their dashed profiles coincide. The blue resultant is twice either component.

t = T/4 = 0.050 s

Standing-wave construction at t = T/4 = 0.050 sThe horizontal axis is position from zero to 0.80 metres; the vertical axis is signed displacement in millimetres, from negative four to positive four. Components each have amplitude two millimetres and travel oppositely. At one quarter period they are equal and opposite, so the whole blue sum is instantaneously zero. Only the three marked positions zero, 0.40 and 0.80 metres are permanent nodes. The antinodes at 0.20 and 0.60 metres are moving downward and upward respectively, as the direction arrows show. Node positions are fixed in every panel. No envelope is being substituted for the instantaneous sum.0.000.200.400.600.80-4-20+2+4Displacement / mmPosition x / m

The small arrows show particle-motion directions at the antinodes. An instantaneous zero displacement everywhere does not make every point a node.

t = T/2 = 0.100 s

Standing-wave construction at t = T/2 = 0.100 sThe horizontal axis is position from zero to 0.80 metres; the vertical axis is signed displacement in millimetres, from negative four to positive four. Components each have amplitude two millimetres and travel oppositely. The two component profiles coincide and are labelled together. The summed amplitude at an antinode is twice the component amplitude. At half a period all displacements reverse from the initial profile. Node positions are fixed in every panel. No envelope is being substituted for the instantaneous sum.0.000.200.400.600.80-4-20+2+4Displacement / mmPosition x / m

Component 1 = component 2 in this snapshot, so their dashed profiles coincide. The blue resultant is twice either component.

The two progressive profiles and their sum share the same scales at three times. At the first and third times the component curves coincide; at T/4 they are equal and opposite. The zero resultant at T/4 is an instantaneous state, not a new set of nodes.
  • At t = 0, the resultant displacements at x = 0, 0.20, 0.40, 0.60 and 0.80 m are 0, +4.0, 0, -4.0 and 0 mm.
  • At t = T/4 = 0.050 s, the whole resultant profile is at equilibrium. The two component profiles are not both zero; they cancel point by point.
  • At t = T/2 = 0.100 s, the nonzero resultant displacements have reversed sign.

Identify amplitude, nodes and antinodes

A node has zero displacement at all times and therefore zero oscillation amplitude. The fixed nodes here are x = 0, 0.40 and 0.80 m. An antinode has the greatest oscillation amplitude: here 4.0 mm at x = 0.20 and 0.60 m.

Amplitude describes the full oscillation, not the displacement in one frame. At T/4, an antinode passes through equilibrium at its greatest speed; it has not become a node.

Why does the zero profile not make every point a node?

A node remains at equilibrium throughout the cycle. Most points in the T/4 frame have nonzero displacements before and after it. Their zero displacement is temporary; only the fixed node positions have zero oscillation amplitude.

Adjacent nodes: separation = λ/2
Adjacent antinodes: separation = λ/2
A node to its neighbouring antinode: λ/4

Points within one loop oscillate in phase. Points in neighbouring loops oscillate in antiphase: when one loop moves up, the next moves down. The nodes separating them do not oscillate.

The ideal standing pattern has no net time-averaged energy transport along it. Energy is still present and exchanges locally. Real driven demonstrations need an energy input to replace losses; this does not turn the fixed node pattern into travelling crests.

The graphical model explains formation. The string and microwave experiments and air-column experiment show how the pattern can be observed and measured.