AS Level Physics (9702) Study Notes: Chapter 8 – Superposition
Hey physicists! Ready to dive into the chapter where waves show off their social skills? Superposition is all about what happens when waves meet. This concept is fundamental to understanding light, sound, and even radio signals. Don't worry if it seems tricky at first—we'll break it down into simple, visual steps!
8.1 The Principle of Superposition
The foundation of this entire chapter is one simple idea: what happens when two waves overlap?
Definition and Application
The Principle of Superposition states that when two or more waves meet at the same point, the resultant displacement at that point is the algebraic sum of the displacements of the individual waves.
- Example: Imagine a rope with a crest (positive displacement) traveling toward a trough (negative displacement). When they meet, their displacements add up.
- Waves pass right through each other and continue their journey unchanged after the overlap. They don't crash and disappear!
1. Constructive Interference
This occurs when waves meet in phase (crest meets crest, or trough meets trough). The displacements add together to create a wave with a larger amplitude.
Result: Maximum Displacement (Antinode)
2. Destructive Interference
This occurs when waves meet out of phase (crest meets trough). The displacements partially or completely cancel each other out.
Result: Minimum or Zero Displacement (Node)
Quick Review: The Algebraic Sum
If wave A has displacement \(y_A = +2\, \text{cm}\) and wave B has displacement \(y_B\).
- If they interfere constructively (\(y_B = +3\, \text{cm}\)): Resultant \(y = +2 + 3 = +5\, \text{cm}\).
- If they interfere destructively (\(y_B = -2\, \text{cm}\)): Resultant \(y = +2 + (-2) = 0\, \text{cm}\).
Key Takeaway: Superposition is the rule for adding wave displacements, leading to the phenomena of interference and stationary waves.
8.2 Interference and Coherence
Interference is the observable pattern (like light and dark fringes, or loud and quiet spots) that results from the superposition of waves.
Conditions for Observable Interference Fringes (8.3.3)
To see a stable, sustained interference pattern (like the beautiful fringes in a Young’s slit experiment), the two sources must be coherent.
Coherence means the waves must have:
- The same frequency (and therefore the same wavelength).
- A constant phase difference between them. (They don't have to be zero phase difference, but the difference must not change over time).
Memory Aid: Coherence = Constant phase difference.
Path Difference and Interference Type
Whether interference is constructive or destructive depends on the path difference—the difference in the distance traveled by the two waves from their sources to the point where they meet.
Constructive Interference (Bright Fringe/Loud Sound):
The path difference is an integer multiple of the wavelength (waves arrive in step).
\(\text{Path Difference} = n\lambda\), where \(n = 0, 1, 2, 3, \dots\)
Destructive Interference (Dark Fringe/Quiet Spot):
The path difference is an odd multiple of half a wavelength (waves arrive out of step).
\(\text{Path Difference} = (n + \frac{1}{2})\lambda\), where \(n = 0, 1, 2, 3, \dots\)
Did you know? Two standard light bulbs cannot produce visible interference fringes because they are incoherent. The light from each atom is emitted randomly, meaning the phase difference between the bulbs changes millions of times per second!
Young’s Double-Slit Experiment (8.3.4)
The double-slit setup allows a single source of light to pass through two slits, creating two coherent sources that then interfere.
The equation relating the geometry of the setup to the wavelength is:
$$ \lambda = \frac{ax}{D} $$Where:
- \(\lambda\) = Wavelength of the light (in m)
- \(a\) = Separation between the two slits (in m)
- \(x\) = Fringe separation (the distance between the centres of two adjacent bright fringes or two adjacent dark fringes) (in m)
- \(D\) = Distance from the double slits to the screen (in m)
Common Mistake to Avoid: Make sure that all units (a, x, D, \(\lambda\)) are in metres before calculation. If you are given slit separation \(a\) in mm, convert it to m!
Key Takeaway: Coherence (constant phase difference) is essential to create a steady interference pattern, which can be quantified using the double-slit formula \(\lambda = ax/D\).
8.3 Diffraction (8.2)
Definition of Diffraction (8.2.1)
Diffraction is the bending or spreading of waves as they pass through an aperture (a gap) or around an edge of an obstacle.
Qualitative Effect of Gap Width (8.2.2)
The extent of diffraction (how much the wave spreads out) depends dramatically on the relationship between the wavelength (\(\lambda\)) and the size of the aperture or obstacle (\(w\)).
- Maximum Diffraction: When the gap size \(w\) is approximately equal to the wavelength \(\lambda\) (\(w \approx \lambda\)). The wave spreads out almost spherically from the gap.
- Noticeable Diffraction: When \(w\) is a few times larger than \(\lambda\).
- Minimal Diffraction: When the gap size \(w\) is much larger than the wavelength \(\lambda\) (\(w >> \lambda\)). The wave travels almost straight through.
Analogy: Imagine shouting around a corner. The long wavelength of sound diffracts easily around the corner, so someone can hear you. Light, which has a very tiny wavelength, doesn't diffract significantly around large objects like walls, which is why you can't see around corners!
Experiments (8.2.2): Ripple tanks are often used to demonstrate diffraction, showing how water waves spread out after passing through a narrow gap.
Key Takeaway: Diffraction is the spreading of waves. It is most effective when the size of the opening is similar to the wavelength of the wave.
8.4 Stationary Waves (Standing Waves) (8.1)
Stationary waves, or standing waves, are special interference patterns formed under very specific conditions.
Formation (8.1.3)
A stationary wave is formed when two progressive waves of:
- The same frequency (and wavelength).
- The same amplitude.
- Travel in opposite directions.
These progressive waves superpose, leading to fixed positions of zero amplitude and fixed positions of maximum amplitude.
Key Features: Nodes and Antinodes
A stationary wave does not transfer energy, and different points vibrate with different amplitudes.
Nodes (N):
- Points where destructive interference always occurs.
- Displacement is always zero.
- These points remain stationary.
Antinodes (A):
- Points where constructive interference always occurs.
- Displacement is maximum (vibrating with the largest amplitude).
Crucial relationship (8.1.4):
- The distance between two adjacent Nodes (N to N) is \(\frac{\lambda}{2}\).
- The distance between two adjacent Antinodes (A to A) is \(\frac{\lambda}{2}\).
- The distance between an adjacent Node and Antinode (N to A) is \(\frac{\lambda}{4}\).
Demonstrating Stationary Waves (8.1.2)
Stationary waves can be set up in different media:
- Stretched Strings: Using a vibration generator (fundamental mode and harmonics).
- Air Columns: Using resonance tubes and tuning forks (the boundary conditions define where nodes and antinodes must be placed, e.g., the closed end of a pipe must be a node).
- Microwaves: Using a microwave transmitter and a metal plate (reflector) to generate a reflected wave travelling in the opposite direction.
Important Note: For 9702 AS Level, we assume that end corrections for tubes and strings are negligible.
Key Takeaway: Stationary waves result from the superposition of identical waves travelling in opposite directions, creating fixed points of zero (Nodes) and maximum (Antinodes) amplitude.
8.5 The Diffraction Grating (8.4)
A diffraction grating is a piece of material (often glass or plastic) with many, many parallel, closely spaced slits (or lines) ruled onto it. It is far more effective than a double slit for separating light by wavelength.
Grating Spacing, d
The slits are separated by a constant distance \(d\), called the grating spacing.
If the grating has \(N\) lines per metre, then the spacing \(d\) is:
$$ d = \frac{1}{\text{number of lines per unit length}} $$For example, if a grating has 500 lines per mm, you must convert this:
$$ d = \frac{1}{500 \times 1000} = 2.0 \times 10^{-6}\, \text{m} $$The Grating Equation (8.4.1)
The formula governing the positions of the bright fringes (or maxima) produced by a diffraction grating is:
$$ d \sin \theta = n\lambda $$Where:
- \(d\) = Grating spacing (m)
- \(\theta\) = Angle of diffraction from the central maximum (in degrees or radians)
- \(n\) = Order number of the maximum (\(n=0\) is the central bright spot, \(n=1\) is the first order, \(n=2\) is the second order, etc.)
- \(\lambda\) = Wavelength of the light (m)
Central Maximum (n=0):
When \(n=0\), \(d \sin \theta = 0\). This means \(\theta=0\). All wavelengths of light meet at the central maximum, which is always the brightest spot.
Using the Grating to Determine Wavelength (8.4.2)
The diffraction grating is a powerful tool for finding the wavelength of light source (like a laser or a spectral lamp).
Step-by-Step Process:
- Measure the grating spacing (\(d\)) using the lines per unit length provided.
- Shine the light source (with unknown \(\lambda\)) onto the grating.
- Identify the central maximum (\(n=0\)).
- Measure the angle \(\theta\) to a higher order maximum (e.g., the first order, \(n=1\)). This measurement often involves finding the angle between the two first-order maxima and halving it.
- Calculate the wavelength using the rearranged formula: \(\lambda = \frac{d \sin \theta}{n}\).
Analogy: Think of white light (made of many wavelengths) hitting the grating. Since \(\lambda\) is different for each colour, the angle \(\theta\) must also be different (\(d\) and \(n\) are constant). This cleanly separates the colours, creating a spectrum.
Key Takeaway: The diffraction grating produces sharp interference maxima based on the equation \(d \sin \theta = n\lambda\). This allows precise measurement and separation of different wavelengths of light.