Welcome to Standing Waves
Have you ever watched a guitar string vibrate when plucked, or seen a skipping rope create loops that seem to stay in one place? What you are looking at is a standing wave (also known as a stationary wave). Unlike ordinary sound waves or ocean ripples that travel from one place to another, standing waves appear to stand completely still while vibrating up and down.
In this chapter of A2 4: Sound and Light, you will learn how standing waves form, what makes them unique compared to progressive waves, how harmonics work on stretched strings, and how to calculate wave speed and frequencies using tension and mass per unit length.
---1. How Are Standing Waves Formed?
A standing wave does not just appear on its own; it requires very specific conditions to form.
The Formation Recipe
A standing wave is formed when two progressive waves of equal frequency (or wavelength) and equal amplitude, travelling in opposite directions through the same medium, superpose (interfere with each other).
The Principle of Superposition
To understand how this happens, we rely on the Principle of Superposition:
"When two or more waves overlap, the resultant displacement at any point is the vector sum of the displacements caused by each individual wave."
As the two identical waves travel past each other in opposite directions, their peaks and troughs continuously reinforce or cancel each other out, creating a fixed pattern of vibration.
Key Takeaway: Standing waves are created by two identical waves travelling in opposite directions that meet and superpose.
---2. Nodes and Antinodes
When the waves superpose, they create two distinct types of points along the medium:
1. Nodes:
Points along the standing wave where the displacement is always zero. At a node, the two waves are always in complete antiphase, resulting in continuous destructive interference.
2. Antinodes:
Points along the standing wave where the displacement and amplitude reach their maximum value. At an antinode, the two waves reinforce each other, resulting in continuous constructive interference.
Crucial Distance Relationships
Remember these geometric relationships—examiners test them regularly!
• Distance between two adjacent nodes = \(\frac{\lambda}{2}\)
• Distance between two adjacent antinodes = \(\frac{\lambda}{2}\)
• Distance between an adjacent node and antinode = \(\frac{\lambda}{4}\)
Memory Trick: A full wave cycle has a wavelength of \(\lambda\). One single "loop" on a vibrating string represents half a wavelength (\(\frac{\lambda}{2}\)), bounded by a node at each end.
Key Takeaway: Nodes have zero amplitude (destructive interference), while antinodes have maximum amplitude (constructive interference). The distance between any two neighbouring nodes is always half a wavelength (\(\frac{\lambda}{2}\)).
---3. Progressive vs. Standing Waves
One of the most frequent examination questions asks you to compare progressive waves (travelling waves) with standing waves. Let's look at the three main differences:
A. Energy Transfer
• Progressive Waves: Propagate and transfer energy from one location to another through the medium.
• Standing Waves: Do not propagate energy along the medium; energy is stored or trapped within the vibrating pattern between nodes.
B. Amplitude
• Progressive Waves: Every particle along the path vibrates with the exact same maximum amplitude.
• Standing Waves: Amplitude varies depending on position, from zero at the nodes to a maximum at the antinodes.
C. Phase Differences
• Progressive Waves: Adjacent particles all vibrate with different phases.
• Standing Waves: All particles located between two adjacent nodes vibrate in phase (they reach their highest and lowest points at the exact same moment). Particles separated by a node vibrate in antiphase (they have a phase difference of \(180^\circ\) or \(\pi\text{ rad}\)).
Key Takeaway: Progressive waves travel and transfer energy; standing waves stay in place, trap energy, have varying amplitudes, and change phase by \(180^\circ\) when crossing a node.
---4. Harmonics on a Stretched String
When a string of length \(L\) is fixed at both ends (like a guitar or violin string), reflecting waves bounce back and forth. Because the ends are clamped tightly, boundary conditions apply:
• Fixed ends: Reflection causes phase inversion, so a node must always form at a fixed boundary.
• Free / open ends: An antinode forms at a free boundary.
For a string fixed at both ends, only specific resonant frequencies—called harmonics—can form stable standing wave patterns.
First Harmonic (Fundamental Mode, \(n = 1\))
This is the simplest pattern: one single loop with a node at each fixed end and one antinode in the middle.
• String length: \(L = \frac{1}{2}\lambda_1 \implies \lambda_1 = 2L\)
• Fundamental frequency: \(f_1 = \frac{v}{\lambda_1} = \frac{v}{2L}\)
Second Harmonic (First Overtone, \(n = 2\))
This pattern contains two loops: three nodes (two at the ends, one in the centre) and two antinodes.
• String length: \(L = \lambda_2 \implies \lambda_2 = L\)
• Frequency: \(f_2 = 2f_1 = \frac{v}{L}\)
Third Harmonic (Second Overtone, \(n = 3\))
This pattern contains three loops: four nodes and three antinodes.
• String length: \(L = \frac{3}{2}\lambda_3 \implies \lambda_3 = \frac{2L}{3}\)
• Frequency: \(f_3 = 3f_1 = \frac{3v}{2L}\)
General Formula for a String Fixed at Both Ends
For any harmonic number \(n\) (\(n = 1, 2, 3, \dots\)):
\(L = \frac{n\lambda_n}{2}\)
\(f_n = n f_1 = \frac{n v}{2L}\)
Key Takeaway: The \(n\)-th harmonic has \(n\) loops, a wavelength of \(\lambda_n = \frac{2L}{n}\), and a frequency that is an integer multiple of the fundamental frequency (\(f_n = n f_1\)).
---5. Wave Speed and String Tension Formula
What determines how fast a wave travels along a stretched wire or string? The wave velocity \(v\) depends on two factors: the tension in the string and its mass per unit length.
The Wave Speed Formula
\(v = \sqrt{\frac{T}{\mu}}\)
Where:
• \(v\) = wave speed in metres per second (\(\text{m}\cdot\text{s}^{-1}\))
• \(T\) = tension in the string in newtons (\(\text{N}\))
• \(\mu\) = mass per unit length (linear mass density) in kilograms per metre (\(\text{kg}\cdot\text{m}^{-1}\))
Calculating Linear Density (\(\mu\))
Linear density is simply the total mass of the string \(m\) divided by its total length \(L\):
\(\mu = \frac{m}{L}\)
Combining with Fundamental Frequency
By substituting \(v = \sqrt{\frac{T}{\mu}}\) into the fundamental frequency formula \(f_1 = \frac{v}{2L}\), we get:
\(f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}\)
What does this tell us?
• Tighter string (higher \(T\)): Higher pitch / higher frequency.
• Heavier string (higher \(\mu\)): Lower pitch / lower frequency.
• Shorter string (smaller \(L\)): Higher pitch / higher frequency.
Key Takeaway: Increasing tension increases wave speed and frequency, while using a heavier (thicker) wire decreases wave speed and frequency.
---6. Common Examiner Pitfalls & Mistakes to Avoid
• Pitfall 1: Wavelength Confusion: Do not confuse the distance between two nodes (\(\frac{\lambda}{2}\)) with the full wavelength (\(\lambda\)). One complete wave cycle consists of two loops, not one!
• Pitfall 2: Node-to-Antinode Distance: Remember that from a node to the very next antinode is a quarter of a wavelength (\(\frac{\lambda}{4}\)), not half.
• Pitfall 3: Phase Across a Node: Particles on opposite sides of a single node vibrate in antiphase (\(180^\circ\) or \(\pi\text{ rad}\)). They do not move in the same direction at the same time.
• Pitfall 4: Energy Misconception: Never state that standing waves transfer energy down a string. Standing waves store energy between nodes.
• Pitfall 5: Unit Errors in \(\mu\): Always ensure linear density \(\mu\) is in \(\text{kg}\cdot\text{m}^{-1}\). If mass is given in grams (\(\text{g}\)) or length in centimetres (\(\text{cm}\)), convert them to \(\text{kg}\) and \(\text{m}\) before calculating \(\mu\).
7. Quick Review Summary
• Formation: Superposition of two identical progressive waves travelling in opposite directions.
• Nodes: Zero displacement, destructive interference.
• Antinodes: Maximum displacement, constructive interference.
• Separation Distances: Node-to-node = \(\frac{\lambda}{2}\), Node-to-antinode = \(\frac{\lambda}{4}\).
• Harmonics on a String: \(\lambda_n = \frac{2L}{n}\) and \(f_n = \frac{n v}{2L}\).
• Wave Speed on Strings: \(v = \sqrt{\frac{T}{\mu}}\) and \(f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}\).