Physics 9630 Study Notes: Principle of Superposition and Stationary Waves
Welcome to one of the most exciting topics in the waves section! This chapter explains what happens when two or more waves meet. Does chaos ensue? No! Instead, they combine momentarily, leading to amazing phenomena like interference and the creation of musical notes in instruments. Understanding the Principle of Superposition is crucial, as it’s the foundation for everything from how noise-cancelling headphones work to how standing waves form on a guitar string.
Don't worry if this seems tricky at first—we’ll break down the concepts using simple, visual analogies.
1. The Principle of Superposition
The principle of superposition is the bedrock rule for all wave interactions. It dictates how waves behave when they pass through the same point in space at the same time.
What the Principle States
When two or more progressive waves meet at a point, the resultant (overall) displacement at that point is the vector sum of the displacements of the individual waves.
Displacement: Remember, this is the distance and direction (vector) of a point from its equilibrium position.
Vector Sum: You simply add the displacements. If the waves cause displacement in the same direction, you add them normally. If they cause displacement in opposite directions, you subtract them.
1.1 Types of Superposition (Interference)
The resulting wave pattern is called interference. There are two main types:
a) Constructive Interference:
This happens when two waves arrive at a point in phase (crest meets crest, or trough meets trough).
The displacements are in the same direction.
The resultant amplitude is maximum (equal to the sum of the individual amplitudes).
Analogy: Imagine two perfectly timed jumps on a trampoline. If you both hit the mat at the same time going down, the mat dips lower, creating a momentarily larger effect.
b) Destructive Interference:
This happens when two waves arrive at a point out of phase (crest meets trough).
The displacements are in opposite directions.
The resultant amplitude is minimum (equal to the difference between the individual amplitudes). If the waves have the same amplitude, the resultant displacement is zero.
Did you know? Noise-cancelling headphones work by generating a second sound wave (noise) that is precisely 180° out of phase with the unwanted external noise, causing destructive interference.
Key Takeaway 1
Superposition is just algebraic addition of displacements. The waves pass right through each other and continue unaffected after the meeting point.
2. Formation of Stationary Waves
Stationary waves (also known as standing waves) are special interference patterns formed under very specific conditions.
2.1 Conditions for Stationary Wave Formation
A stationary wave is formed when:
Two progressive waves travel in opposite directions.
The two waves have the same frequency (and therefore the same wavelength and speed).
The two waves have approximately the same amplitude.
In practice, this often happens when a wave reflects off a boundary, and the original wave interferes with its own reflection.
2.2 Stationary vs. Progressive Waves
The name "stationary" is key—these waves look like they are standing still, oscillating in place.
| Feature | Progressive Wave (Travelling Wave) | Stationary Wave (Standing Wave) |
|---|---|---|
| Energy Transfer | Transfers energy from source to absorber. | Does not transfer net energy; energy is stored (Potential and Kinetic). |
| Amplitude | All particles oscillate with the same amplitude. | Amplitude varies from zero (at nodes) to maximum (at antinodes). |
| Phase | Phase is gradually different for adjacent particles. | All particles between two adjacent nodes oscillate in phase. |
Key Takeaway 2
Stationary waves are the result of two identical waves overlapping while moving towards each other. They store energy but don't transport it, and crucially, not all points have the same amplitude.
3. Nodes and Antinodes
The fixed positions of zero and maximum amplitude are the defining feature of a stationary wave.
3.1 Defining Nodes and Antinodes
1. Nodes (N):
Points along the stationary wave where the resultant displacement is always zero.
These are locations of continuous destructive interference.
Particles at nodes are permanently stationary.
2. Antinodes (A):
Points along the stationary wave where the displacement reaches its maximum amplitude (the peak oscillation).
These are locations of continuous constructive interference.
Particles at antinodes oscillate with the greatest energy.
3.2 Relationship between Nodes, Antinodes, and Wavelength (\(\lambda\))
When studying stationary waves, these distance relationships are crucial:
Distance between two adjacent nodes (N to N) = \(\frac{\lambda}{2}\)
Distance between two adjacent antinodes (A to A) = \(\frac{\lambda}{2}\)
Distance between an adjacent node and antinode (N to A) = \(\frac{\lambda}{4}\)
Memory Trick: The pattern repeats every half-wavelength. Think of a sine wave cycle—the wave hits zero (Node) and then reaches its peak (Antinode) after a quarter cycle.
Key Takeaway 3
Nodes are fixed points of zero movement (destructive interference). Antinodes are fixed points of maximum movement (constructive interference).
4. Stationary Waves on Strings and Harmonics
Stationary waves are most easily visualized on a taut string fixed at both ends, like a guitar or violin string. The ends of the string must be nodes.
4.1 Harmonics (Resonance Patterns)
When a string vibrates, it naturally oscillates at certain specific frequencies called harmonics. The syllabus requires you to describe stationary waves in terms of these harmonics.
Let \(l\) be the length of the string.
First Harmonic (Fundamental Pattern):
This is the simplest vibration pattern.
It consists of two nodes (at the ends) and one antinode (in the middle).
The length of the string \(l\) is equal to half a wavelength (\(\frac{\lambda_1}{2}\)).
Wavelength: \(\lambda_1 = 2l\)
Second Harmonic:
This pattern has three nodes and two antinodes.
The string length \(l\) is equal to one full wavelength (\(\lambda_2\)).
Wavelength: \(\lambda_2 = l\)
The frequency is double the first harmonic frequency (\(f_2 = 2f_1\)).
Third Harmonic:
This pattern has four nodes and three antinodes.
The string length \(l\) is equal to one and a half wavelengths (\(\frac{3\lambda_3}{2}\)).
Wavelength: \(\lambda_3 = \frac{2l}{3}\)
The frequency is triple the first harmonic frequency (\(f_3 = 3f_1\)).
Note on Terminology: The syllabus specifically instructs us to use the terms "first harmonic," "second harmonic," etc., and not the terms "fundamental" or "overtone."
4.2 The First Harmonic Frequency Equation
The frequency of vibration for the first harmonic on a stretched string depends on the physical properties of the string:
$$f = \frac{1}{2l} \sqrt{\frac{T}{\mu}}$$
Where:
\(f\) is the frequency (Hz).
\(l\) is the length of the vibrating string (m).
\(T\) is the tension in the string (N).
\(\mu\) is the mass per unit length (kg m\(^{-1}\)). This is sometimes called the linear density.
4.3 Experimental Investigation of String Waves
You need to know how experiments investigate the variation of frequency (\(f\)) with \(l\), \(T\), and \(\mu\). The equipment typically used is a sonometer or a standing wave generator.
How does \(f\) change?
1. Length (\(l\)): Frequency is inversely proportional to length (\(f \propto \frac{1}{l}\)). (Shorter strings produce higher pitches—why a guitar player presses down on frets).
2. Tension (\(T\)): Frequency is proportional to the square root of the tension (\(f \propto \sqrt{T}\)). (Tighter strings produce higher pitches—why you tune a guitar by tightening the pegs).
3. Mass per unit length (\(\mu\)): Frequency is inversely proportional to the square root of the linear density (\(f \propto \frac{1}{\sqrt{\mu}}\)). (Thicker, heavier strings produce lower pitches, as they have a higher \(\mu\)).
Key Takeaway 4
The first harmonic has a wavelength \(\lambda = 2l\). The frequencies of higher harmonics are integer multiples of the first harmonic frequency (e.g., \(f_2 = 2f_1\), \(f_3 = 3f_1\)). Frequency is higher for shorter, tighter, and lighter strings.
5. Other Stationary Wave Examples
Stationary waves are not just limited to strings. They can be formed in any medium where reflection occurs, provided the conditions for superposition are met.
5.1 Stationary Microwaves
Microwaves are electromagnetic waves, meaning they are transverse waves. They can form stationary waves in a lab setup:
A microwave transmitter produces the progressive wave.
A metal plate acts as a reflector, sending the wave back in the opposite direction.
A microwave probe/detector can be moved between the transmitter and the plate to locate the positions of nodes (zero signal) and antinodes (maximum signal).
Common mistake: Students sometimes think microwaves bounce off the wall and disappear. Remember, reflection is the key mechanism for creating the opposing wave necessary for standing wave formation.
5.2 Stationary Sound Waves
Sound waves are longitudinal waves. Stationary sound waves are produced in air columns (like those inside organ pipes or resonance tubes) when the incident sound wave reflects off the closed or open end.
At a closed end (or solid boundary), air molecules cannot move, so a Node is always formed.
At an open end, air molecules can oscillate freely, so an Antinode is always formed.
Understanding these boundary conditions (N or A at the ends) is essential for predicting the harmonics possible in different types of musical wind instruments.
Quick Review: Key Concepts
Superposition: Algebraic addition of displacements when waves overlap.
Stationary Waves: Formed by two identical waves traveling in opposite directions.
Nodes: Zero displacement (always).
Antinodes: Maximum displacement (always).
First Harmonic: \(\lambda = 2l\). Frequency depends on length, tension, and mass per unit length.