Stationary Waves

Overview

A stationary wave (standing wave) is formed when two progressive waves of the same frequency, wavelength, amplitude, and speed travel in opposite directions and superpose.

Unlike a progressive wave, a stationary wave does not transfer energy from one end to the other overall.

This topic is highly examinable in strings, air columns, resonance, and harmonics.

Definition

A stationary wave is produced when two identical progressive waves move in opposite directions and superpose.

Why It Matters

Stationary waves explain why strings, pipes, and resonant cavities vibrate only in certain modes. They connect superposition, phase, resonance, sound, and later quantum wave ideas.

Key Representations

Required Conditions

The two waves should have:

• same frequency
• same wavelength
• same speed
• same amplitude (ideal case)
• opposite directions of travel

Superposition Principle

When the two waves overlap, resultant displacement is the algebraic sum of individual displacements.

Some positions always cancel, while others always reinforce.

This creates a fixed pattern of nodes and antinodes.

Nodes and Antinodes

Node

A point of permanent zero displacement.

• amplitude = 0

Antinode

A point of maximum oscillation amplitude.

• amplitude = maximum

Spacing Rules

Adjacent nodes are separated by:

$$\frac{\lambda }{2}$$

Adjacent antinodes are separated by:

$$\frac{\lambda }{2}$$

Distance from node to nearest antinode:

$$\frac{\lambda }{4}$$

Phase Relationships

Between Points in the Same Segment

All particles between two adjacent nodes oscillate in phase.

Across a Node

Particles in neighbouring segments are in antiphase.

Phase difference:

$$\pi$$

Energy Transfer

A stationary wave has no net energy transfer along the medium.

Energy is stored and exchanged locally between kinetic and potential forms.

This is a key difference from progressive waves.

Mathematical Form

A typical stationary wave may be written as:

$$y=2A\sin (kx)\cos (\omega t)$$

Interpretation:

• $\sin (kx)$ controls amplitude with position
• $\cos (\omega t)$ controls oscillation with time

So different positions have different amplitudes.

Strings Fixed at Both Ends

For a stretched string of length $L$ fixed at both ends:

• ends are nodes

Allowed modes satisfy:

$$L=n\frac{\lambda }{2}$$

where:

$$n=1,2,3,\dots$$

Hence:

$${\lambda }_n=\frac{2L}{n}$$

and frequency:

$$f_n=\frac{nv}{2L}$$

Harmonics on a String

First Harmonic (Fundamental)

$$n=1$$

• one loop
• lowest frequency

Second Harmonic

$$n=2$$

• two loops

Third Harmonic

$$n=3$$

• three loops

Higher harmonics have higher frequency.

Air Columns

Open Pipe (both ends open)

Both ends are antinodes.

Allowed modes:

$$L=n\frac{\lambda }{2}$$

Closed Pipe (one end closed)

Closed end = node Open end = antinode

Allowed modes:

$$L=\frac{\lambda }{4},\frac{3\lambda }{4},\frac{5\lambda }{4},\dots$$

Only odd harmonics occur.

Worked Examples

Example 1: String Fundamental Frequency

String length:

$$L=0.80\text{ m}$$

Wave speed:

$$v=160{\text{ m s}}^{-1}$$

Fundamental:

$$f=\frac{v}{2L}=\frac{160}{1.60}=100\text{ Hz}$$

Example 2: Wavelength from Node Spacing

Adjacent nodes are $0.30\text{ m}$ apart.

Since node spacing is:

$$\frac{\lambda }{2}$$

Then:

$$\lambda =0.60\text{ m}$$

Example 3: Closed Pipe First Resonance

Tube length:

$$L=0.25\text{ m}$$

For first resonance:

$$L=\frac{\lambda }{4}$$

So:

$$\lambda =1.00\text{ m}$$

If speed is $340{\text{ m s}}^{-1}$:

$$f=\frac{340}{1.00}=340\text{ Hz}$$

Comparison: Progressive vs Stationary Waves

Feature	Progressive Wave	Stationary Wave
Energy transfer	Yes	No net transfer
Pattern moves	Yes	No
Amplitude	Same (ideal)	Depends on position
Nodes present	No	Yes
Antinodes present	No	Yes

Common Exam Pitfalls

• saying nodes are points of zero velocity at all times
• forgetting node spacing is $\lambda /2$
• forgetting node to antinode is $\lambda /4$
• assuming neighbouring segments are in phase
• using all harmonics for closed pipe
• forgetting ends of fixed string are nodes

Quick Revision Checklist

Ask:

• Is this a string, open pipe, or closed pipe?
• What are the boundary conditions?
• Where are nodes and antinodes?
• What harmonic number is shown?
• Is spacing $\lambda /2$ or $\lambda /4$?
• Is net energy transfer zero?

Related Links

• Waves
• Interference and Diffraction
• Sound Measurement Practicals
• Superposition of Waves

Links

• Main topic: Waves
• Related topic: Superposition of Waves
• Related concept: Interference and Diffraction
• Related concept: Sound Measurement Practicals

Summary

A stationary wave is formed by superposition of two identical opposite-travelling waves, producing nodes and antinodes with no net energy transfer.

Most questions reduce to:

• identifying boundary conditions
• using spacing rules
• selecting the correct harmonic relation