Welcome to the World of Stationary Waves!
Hello future physicist! This chapter, Superposition, can seem tricky because it deals with two waves interacting simultaneously. But don't worry, once you master the concept of stationary waves, you'll unlock the physics behind musical instruments, acoustic design, and even microwave ovens!
A stationary wave (or standing wave) isn't really moving. It looks like it's just vibrating in place, storing energy rather than transferring it. Let's dive in and see how these fascinating patterns are created.
1. The Foundation: Principle of Superposition
Before we can form a stationary wave, we need to understand the fundamental rule for combining waves.
What is Superposition? (Syllabus 8.1.1)
The Principle of Superposition states that when two or more waves of the same type overlap at a point, the resultant displacement at that point is the vector sum of the displacements of the individual waves.
Analogy: Imagine dropping two pebbles into a pond. Where the ripples meet, the water's height (displacement) is either increased (if two peaks meet) or cancelled out (if a peak meets a trough).
In mathematical terms, if Wave 1 has displacement \(y_1\) and Wave 2 has displacement \(y_2\), the resultant displacement \(Y\) is:
$$Y = y_1 + y_2$$
Quick Review: Types of Interference
When waves superpose, we get two main outcomes:
- Constructive Interference: Occurs when waves meet in phase (crest meets crest, or trough meets trough). The resulting amplitude is maximum (the displacements add up).
- Destructive Interference: Occurs when waves meet anti-phase (crest meets trough). The resulting amplitude is minimum, often zero (the displacements cancel out).
Key Takeaway: Superposition is just adding up the displacements of individual waves when they meet.
2. Formation and Characteristics of Stationary Waves (Syllabus 8.1.3)
How are Stationary Waves Formed?
A stationary wave is created when two identical progressive waves traveling in opposite directions meet and superpose.
Typically, this happens when a progressive wave hits a boundary and reflects back upon itself.
Example: When you pluck a guitar string, the wave travels to the bridge (fixed end), reflects, and travels back, interfering continuously with the new waves being generated.
Identifying Nodes and Antinodes
The most important feature of a stationary wave is that the positions of maximum and minimum oscillation are fixed.
1. Nodes (N)
- Definition: Points on the stationary wave where the amplitude is always zero.
- Physics: At nodes, the two progressive waves always interfere destructively (they are always in anti-phase).
- Visuals: Nodes do not move at all. They are points of minimum energy.
2. Antinodes (A)
- Definition: Points on the stationary wave where the amplitude is maximum.
- Physics: At antinodes, the two progressive waves always interfere constructively (they are always in phase).
- Visuals: Antinodes oscillate with the largest possible amplitude. They are points of maximum energy conversion (from kinetic to potential).
Did you know? While a progressive wave transfers energy continuously, a stationary wave effectively stores energy in the fixed segments between the nodes.
Graphical Explanation of Formation
Understanding the graphical formation is key (Syllabus 8.1.3). Let's follow two waves (A, moving right; B, moving left) over time:
Step 1: Time \(t=0\)
Wave A and Wave B are perfectly in phase (crest on crest, trough on trough). When they superpose, they interfere constructively, creating a maximum amplitude stationary wave pattern. The displacement is high everywhere, except at the fixed nodes (N).
Step 2: Time \(t=T/4\) (One quarter of a period later)
Wave A and Wave B have each shifted by a quarter wavelength. They are now everywhere in anti-phase. When they superpose, they interfere destructively, resulting in zero displacement everywhere. This is the moment the string/medium is perfectly straight, but it has maximum kinetic energy (it’s moving fastest through the equilibrium position).
Step 3: Time \(t=T/2\) (Half a period later)
The waves are back in phase, but inverted compared to \(t=0\). Constructive interference occurs again, creating the maximum amplitude pattern, but in the opposite direction.
Conclusion: The regions around the Nodes (N) never move, while the Antinodes (A) rapidly oscillate between maximum positive and maximum negative displacement.
Common Mistake Alert!
Don't confuse the amplitude of the stationary wave with the progressive waves that formed it. The amplitude at an antinode is double the amplitude of the individual progressive waves.
Key Takeaway: Stationary waves have fixed positions of zero displacement (Nodes) and maximum displacement (Antinodes) due to continuous constructive and destructive interference.
3. Wavelength Determination (Syllabus 8.1.4)
The pattern of Nodes (N) and Antinodes (A) is regular and directly related to the wave's wavelength \(\lambda\).
We can use the position of these points to calculate the wavelength:
- The distance between two consecutive Nodes (N to N) is exactly half a wavelength.
$$\text{Distance (N to N)} = \lambda / 2$$ - The distance between two consecutive Antinodes (A to A) is also half a wavelength.
$$\text{Distance (A to A)} = \lambda / 2$$ - The distance between a Node and the very next Antinode (N to A) is one quarter of a wavelength.
$$\text{Distance (N to A)} = \lambda / 4$$
Memory Aid: Remember the sequence: N-A-N-A-N... Since it takes two steps (N to A, then A to N) to cover half a cycle, each step is \(\lambda/4\).
Key Takeaway: Measure the distance between nodes and double it to find the wavelength \(\lambda\).
4. Experimental Demonstration of Stationary Waves (Syllabus 8.1.2)
The syllabus requires understanding experiments demonstrating stationary waves in three contexts: strings, air columns, and microwaves. We assume end corrections are negligible in all cases.
4.1 Stationary Waves on a Stretched String
This is the classic example (like a violin or guitar).
Setup: A string is fixed at both ends (usually one end is attached to a vibration generator, and the other runs over a pulley with a weight attached for tension).
Boundary Conditions: Since the string is fixed at the ends, these points must be Nodes (N).
Observation: By adjusting the frequency of the generator or the tension, the string will vibrate with large amplitude when a standing wave pattern is achieved.
- First Harmonic (Fundamental): The string vibrates in one loop. It has N-A-N. The length L of the string is \(L = \lambda/2\).
- Second Harmonic: The string vibrates in two loops. It has N-A-N-A-N. The length L is \(L = 2(\lambda/2) = \lambda\).
4.2 Stationary Waves in Air Columns (Sound)
Stationary sound waves form in tubes (like organ pipes or flutes) when the sound wave reflects off the ends.
Boundary Conditions:
- Closed End: Air particles cannot move. This must be a Node (N) of displacement.
- Open End: Air particles can move freely (maximum vibration). This must be an Antinode (A) of displacement.
The Experiment: A loudspeaker produces sound near the open end of a tube. By varying the length of the tube or the frequency of the sound, standing wave resonance is detected when the sound intensity is maximum (this happens when an antinode forms at the opening).
Example - Tube Closed at One End:
The shortest possible standing wave must have a Node (N) at the closed end and an Antinode (A) at the open end.
$$\text{Length } L = \lambda / 4$$
4.3 Stationary Waves using Microwaves
This experiment uses electromagnetic waves (microwaves) to demonstrate the same principle.
Setup: A microwave transmitter produces progressive waves. A large, flat metal sheet acts as a reflector, sending the waves back. A microwave detector probe is moved between the transmitter and the reflector.
Boundary Condition: The reflector (metal plate) acts as a fixed boundary, meaning a Node (N) of electric field strength (and displacement) is formed right at the surface.
Observation:
- When the detector reads minimum intensity (often zero), it is at a Node (N).
- When the detector reads maximum intensity, it is at an Antinode (A).
By measuring the distance between successive nodes, \(x\), we find the wavelength:
$$\lambda = 2x$$
Key Takeaway: Stationary waves require a boundary (or boundaries) to reflect the progressive wave, establishing fixed points (N or A).
5. Final Review and Summary
Quick Check List for Stationary Waves
We've covered the crucial elements of stationary waves. Here is a summary of key facts you must know for the exam:
1. Formation:
Formed by the superposition of two identical progressive waves travelling in opposite directions.
2. Energy:
Energy is stored, not transferred.
3. Components:
Nodes (N): Points of zero displacement (always destructive interference).
Antinodes (A): Points of maximum displacement (always constructive interference).
4. Wavelength Relationship:
The distance between N and N (or A and A) is \(\lambda/2\).
The distance between N and A is \(\lambda/4\).
Keep practising identifying nodes and antinodes in different scenarios—especially in fixed strings and open/closed tubes—and you will master this topic! Good luck!