Stationary waves form when identical waves travelling in opposite directions interfere. Within these patterns, nodes and antinodes represent fixed and oscillating points of energy transfer and displacement.

Understanding Stationary Waves

A stationary wave (or standing wave) arises when two progressive waves of identical frequency, wavelength, and amplitude travel in opposite directions and superpose. The interference between these waves produces regions where displacement either cancels out completely or varies between maximum and minimum values over time, creating distinctive points known as nodes and antinodes.

Unlike progressive waves, stationary waves do not transfer energy along their length. Instead, the energy oscillates locally within confined regions. This makes stationary waves essential in explaining sound in musical instruments, resonance in air columns, and vibration modes in strings.

Formation of Nodes and Antinodes

The positions of nodes and antinodes result from superposition, the principle that the resultant displacement of two overlapping waves equals the vector sum of their individual displacements.

At certain points, the two waves always meet in antiphase (180° out of phase), producing destructive interference, where displacements cancel out completely. At these points, the medium remains stationary, forming nodes.

At other positions, the waves meet in phase (0° or 360° phase difference), producing constructive interference. Here, the resultant displacement reaches its maximum possible value, forming antinodes. These alternating stationary points give the wave its distinctive appearance.

Between any two adjacent nodes lies a point of maximum displacement — an antinode.

Between two successive nodes, particles oscillate in the same direction and reach their maximum amplitude at the midpoint, the antinode. The distance between two consecutive nodes (or antinodes) is equal to half the wavelength (λ/2) of the original waves.

A labelled standing-wave profile showing nodes (zero displacement) and antinodes (maximum displacement) with separations of λ/2 between like points and λ/4 between a node and adjacent antinode. The plot also includes the incident, reflected, and resultant waves to emphasise superposition. Extra detail beyond the syllabus: the separate incident and reflected curves are shown to illustrate how the stationary pattern forms. Source.

Spatial Relationships within the Wave Pattern

The spatial distribution of nodes and antinodes defines the wave mode or harmonic pattern. These features determine how energy is distributed in a vibrating system.

• Node spacing: adjacent nodes are separated by λ/2.

• Antinode spacing: identical to node spacing — also λ/2.

• Node–antinode separation: the distance between a node and its nearest antinode equals λ/4.

The overall pattern does not travel through space but oscillates in time. Each point on the medium has a fixed amplitude determined by its distance from a node. The displacement of particles follows a sinusoidal time variation, but their amplitude depends on position.

Displacement and Phase Characteristics

The phase of oscillation is consistent within each segment of the stationary wave between two adjacent nodes but reverses direction across a node.

• All points between two nodes vibrate in phase, meaning they reach their maximum displacement simultaneously.

A standing-wave snapshot with nodes marked by red dots at integer multiples of λ/2 and antinodes marked by blue dots at odd multiples of λ/4. The visual emphasises fixed points of zero displacement versus oscillating maxima, aligning with the in-phase segments between nodes. (No extra detail beyond the syllabus.) Source.

• Points in adjacent segments (on either side of a node) oscillate 180° out of phase.

This alternating phase relationship is a hallmark of stationary wave behaviour and contrasts with progressive waves, where every point along the medium has the same amplitude but a continuously varying phase.

Visualising Node and Antinode Motion

To visualise the pattern:

• At one instant, all antinodes on the upper half of the medium reach maximum positive displacement.

• Half a period later, these same antinodes reach maximum negative displacement.

• Nodes, in contrast, remain fixed at all times — the medium at these points experiences no movement.

This oscillating pattern produces a wave that appears to “stand still,” giving rise to the term standing wave.

Practical Examples of Nodes and Antinodes

1. Vibrating Strings
When a stretched string is plucked, reflections at its fixed ends cause incident and reflected waves to superpose.

• The fixed ends act as nodes, since displacement is zero.

• Antinodes form midway between nodes, where displacement is maximum.
  The number of nodes and antinodes depends on the harmonic (or mode) of vibration.

2. Air Columns
In wind instruments, standing waves form in air columns.

• Closed end: acts as a node (no air movement, pressure antinode).

• Open end: acts as an antinode (maximum air displacement, pressure node).

Standing-wave patterns in an open–closed air column across successive modes. Displacement antinodes occur at the open end and displacement nodes at the closed end, matching the boundary conditions described. Extra detail beyond the syllabus: multiple overtones are shown to illustrate how the pattern repeats. Source.

This difference in boundary conditions produces distinct harmonic patterns between open and closed tubes.

Energy Distribution within the Pattern

In a stationary wave, energy does not propagate along the medium. Instead, energy oscillates locally between kinetic and potential forms.

• At nodes, all particles are momentarily stationary, so kinetic energy is zero.

• At antinodes, particles move with the greatest speed, giving maximum kinetic energy.

• Potential energy, associated with the medium’s deformation, alternates inversely with kinetic energy through the cycle.

Although energy transfers between forms, there is no net transfer along the wave — one of the key distinctions from progressive waves.

Mathematical Description

The general form of a stationary wave can be represented as:

EQUATION
—-----------------------------------------------------------------
Stationary Wave Equation (y) = 2A sin(kx) cos(ωt)
y = Displacement of particle at position x and time t (m)
A = Amplitude of each component progressive wave (m)
k = Wave number = 2π/λ (rad m⁻¹)
ω = Angular frequency = 2πf (rad s⁻¹)
t = Time (s)
x = Position along the medium (m)
—-----------------------------------------------------------------

This equation demonstrates that the amplitude at any point depends on position (via sin kx), while the temporal oscillation depends on cos ωt. Nodes occur where sin kx = 0, and antinodes occur where sin kx = ±1.

Key Observations and Relationships

• Nodes: zero displacement, maximum potential energy, occur every λ/2.

• Antinodes: maximum displacement, maximum kinetic energy, also spaced λ/2 apart.

• Adjacent node–antinode distance: λ/4.

• All particles between two nodes: oscillate in phase.

• Energy remains confined: no net energy flow along the medium.

These relationships underpin the physics of resonance and vibration modes in both mechanical and electromagnetic stationary systems, fulfilling the OCR requirement to define nodes and antinodes in stationary wave patterns.