Stationary Waves

A stationary wave (also called a standing wave) is formed by the superposition of two progressive waves of the same frequency, wavelength, and amplitude travelling in opposite directions. Unlike a progressive wave, a stationary wave does not transfer energy along its length.

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Formation of Stationary Waves

Stationary waves are typically formed by the superposition of an incident wave and its reflection. For example:

• A wave on a string reflects at a fixed end and interferes with the incoming wave.
• A sound wave reflects from a wall and interferes with the forward-travelling wave.
• A microwave reflects from a metal plate and interferes with the original microwave.

At certain frequencies, the pattern of constructive and destructive interference creates a stable, fixed pattern with positions that always have zero displacement (nodes) and positions that oscillate with maximum displacement (antinodes).

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Nodes and Antinodes

Nodes

A node is a point on a stationary wave that has zero displacement at all times. Nodes are caused by persistent destructive interference at that point. Adjacent nodes are separated by λ/2.

Antinodes

An antinode is a point on a stationary wave that oscillates with maximum amplitude. Antinodes are caused by persistent constructive interference. They are located midway between adjacent nodes.

Key Relationships

• Distance between adjacent nodes = λ/2
• Distance between adjacent antinodes = λ/2
• Distance between a node and the nearest antinode = λ/4

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Comparison: Stationary vs Progressive Waves

Property	Progressive Wave	Stationary Wave
Energy transfer	Transfers energy along the wave	No net energy transfer
Amplitude	All points oscillate with the same amplitude	Amplitude varies: zero at nodes, maximum at antinodes
Phase	Phase varies continuously along the wave	All points between two adjacent nodes are in phase; points in adjacent segments are in antiphase
Wavelength	Distance for one full cycle	Distance between alternate nodes = λ
Frequency	All points oscillate at the same frequency	All points oscillate at the same frequency

Exam Tip: A key distinguishing feature of stationary waves is that the amplitude varies from point to point. In a progressive wave, all points have the same amplitude. This is a common exam question.

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Stationary Waves on Strings

When a string is fixed at both ends (e.g., a guitar string), standing waves can be set up. Both ends must be nodes because the string cannot move at a fixed end.

Harmonics

The different modes of vibration are called harmonics. The lowest frequency is the fundamental (1st harmonic).

1st Harmonic (Fundamental):

• 1 antinode, 2 nodes (at the ends)
• String length L = λ₁/2, so λ₁ = 2L
• Fundamental frequency: f₁ = v/(2L)

2nd Harmonic:

• 2 antinodes, 3 nodes
• L = λ₂, so λ₂ = L
• f₂ = v/L = 2f₁

3rd Harmonic:

• 3 antinodes, 4 nodes
• L = 3λ₃/2, so λ₃ = 2L/3
• f₃ = 3v/(2L) = 3f₁

General pattern for the nth harmonic:

• n antinodes, (n + 1) nodes
• λₙ = 2L/n
• fₙ = nv/(2L) = nf₁

Speed of a Wave on a String

The speed of a transverse wave on a string depends on the tension T and the mass per unit length μ:

v = √(T/μ)

where T is the tension (N) and μ = m/L is the mass per unit length (kg m⁻¹).

Worked Example 1 — A guitar string has a length of 0.65 m, mass per unit length 3.5 × 10⁻³ kg m⁻¹, and is under a tension of 80 N. Calculate the fundamental frequency.

v = √(T/μ) = √(80/(3.5 × 10⁻³)) = √(22857) = 151.2 m s⁻¹

f₁ = v/(2L) = 151.2/(2 × 0.65) = 151.2/1.30 = 116 Hz

Worked Example 2 — The same string vibrates in its 3rd harmonic. What is the frequency and wavelength?

f₃ = 3f₁ = 3 × 116 = 349 Hz