Progressive Waves
Wave Properties
A progressive wave transfers energy from one place to another without transferring matter. Key properties include:
| Property | Symbol | Unit | Definition |
|---|---|---|---|
| Amplitude | A | m | Maximum displacement from equilibrium |
| Wavelength | λ | m | Distance between two adjacent points in phase |
| Frequency | f | Hz (s−1) | Number of complete oscillations per second |
| Period | T | s | Time for one complete oscillation |
| Wave speed | v | m s−1 | Speed at which the wave profile moves |
The Wave Equation
Worked Example
A sound wave in air has a frequency of 440 Hz and a wavelength of 0.773 m. Calculate the speed of sound.
v = fλ = 440 × 0.773 = 340 m s−1
Phase Difference
Phase difference describes how much one wave is ahead or behind another. It is measured in radians or degrees.
Where d is the path difference (difference in distance travelled).
- In phase: phase difference = 0, 2π, 4π, ... (path difference = 0, λ, 2λ, ...)
- In antiphase: phase difference = π, 3π, ... (path difference = λ/2, 3λ/2, ...)
Key Facts
- Progressive waves transfer energy but not matter
- The intensity of a wave is proportional to the amplitude squared: I ∝ A2
- Frequency is determined by the source and does not change when a wave enters a new medium
Longitudinal & Transverse Waves
Comparing Wave Types
Transverse waves: The oscillation is perpendicular to the direction of wave travel. Examples: light, water surface waves, waves on a string, all EM waves.
Longitudinal waves: The oscillation is parallel to the direction of wave travel. They consist of compressions and rarefactions. Examples: sound, ultrasound, P-waves (seismic).
Polarisation
Polarisation restricts the oscillations of a transverse wave to a single plane. Only transverse waves can be polarised; longitudinal waves cannot.
Evidence for polarisation proves that light is a transverse wave. Applications include Polaroid filters, LCD screens, and reducing glare.
Exam Tip
If asked to prove that a wave is transverse, show that it can be polarised. Sound cannot be polarised, confirming it is longitudinal.
Superposition & Standing Waves
Principle of Superposition
When two or more waves meet at a point, the resultant displacement equals the vector sum of the individual displacements.
- Constructive interference: Waves in phase → amplitudes add → larger resultant
- Destructive interference: Waves in antiphase → amplitudes cancel → smaller (or zero) resultant
Stationary (Standing) Waves
A stationary wave is formed when two progressive waves of the same frequency and amplitude travel in opposite directions and superpose.
Nodes: Points of zero displacement (always stationary). Antinodes: Points of maximum displacement.
The distance between adjacent nodes (or adjacent antinodes) is λ/2.
Comparing Progressive and Stationary Waves
| Feature | Progressive wave | Stationary wave |
|---|---|---|
| Energy transfer | Transfers energy | No net energy transfer |
| Amplitude | Same for all particles | Varies: zero at nodes, maximum at antinodes |
| Phase | Changes continuously along wave | All points between two nodes are in phase |
| Wavelength | Distance between adjacent in-phase points | 2 × distance between adjacent nodes |
Harmonics on a String
For a string fixed at both ends, standing waves can only form at specific frequencies (harmonics). The fundamental (1st harmonic) has one antinode.
nth harmonic: fn = nf1
Exam Tip
When drawing standing wave patterns, count the number of antinodes to determine the harmonic. 1st harmonic = 1 antinode, 2nd = 2 antinodes, etc.
Refraction & Diffraction
Refraction and Snell's Law
Refraction occurs when a wave crosses a boundary between two media with different wave speeds. The wave changes direction (unless it enters at the normal).
The refractive index of a material is:
Where c = speed of light in vacuum and v = speed of light in the material. A higher refractive index means slower speed.
Total Internal Reflection
When light travels from a more optically dense medium to a less dense medium (n1 > n2), it refracts away from the normal.
At the critical angle (θc), the refracted ray travels along the boundary. Beyond this angle, total internal reflection occurs.
Applications: optical fibres, endoscopes, prisms in binoculars.
Worked Example
Calculate the critical angle for glass (n = 1.50) to air (n = 1.00).
sin θc = n2 / n1 = 1.00 / 1.50 = 0.667
θc = sin−1(0.667) = 41.8°
Diffraction
Diffraction is the spreading of waves as they pass through a gap or around an obstacle.
- Maximum diffraction occurs when the gap width is approximately equal to the wavelength
- If the gap is much larger than the wavelength, very little diffraction occurs
- Single slit diffraction produces a central maximum that is twice the width of the subsidiary maxima
Key Facts
- When light enters a denser medium, it slows down and bends towards the normal
- Frequency does not change during refraction; wavelength and speed change
- Total internal reflection requires: denser to less dense medium AND angle of incidence > critical angle
Young's Double Slit Experiment
The Experiment
Monochromatic, coherent light passes through two narrow slits separated by distance a. An interference pattern of bright and dark fringes appears on a screen at distance D.
Coherence means the sources have a constant phase difference and the same frequency.
The Double Slit Equation
Where:
- λ = wavelength
- a = slit separation
- x = fringe spacing (distance between adjacent bright fringes)
- D = distance from slits to screen
Worked Example
Light of wavelength 600 nm passes through two slits 0.50 mm apart. The screen is 1.5 m away. Calculate the fringe spacing.
Rearrange: x = λD / a
x = (600 × 10−9 × 1.5) / (0.50 × 10−3)
x = 9.0 × 10−4 / 5.0 × 10−4
x = 1.8 × 10−3 m = 1.8 mm
Conditions for Maxima and Minima
At a point on the screen, whether there is a bright or dark fringe depends on the path difference between waves from the two slits.
| Fringe type | Path difference | Phase difference |
|---|---|---|
| Bright (constructive) | nλ (n = 0, 1, 2, ...) | 0, 2π, 4π, ... |
| Dark (destructive) | (n + ½)λ | π, 3π, 5π, ... |
Exam Tip
The double slit equation λ = ax/D is only valid when D is much larger than a (D >> a). Always convert units: nm to m, mm to m.
Diffraction Gratings
The Diffraction Grating Equation
A diffraction grating has many equally spaced slits. It produces sharp, bright maxima at specific angles.
Where:
- d = slit spacing (distance between centres of adjacent slits)
- θ = angle of diffraction from the central maximum
- n = order number (0, 1, 2, 3, ...)
- λ = wavelength
If the grating has N lines per metre, then d = 1/N.
Maximum Number of Orders
The maximum order visible occurs when sin θ = 1 (i.e., θ = 90°).
Worked Example
A diffraction grating has 300 lines per mm. Light of wavelength 550 nm is incident normally. (a) Calculate the angle of the 2nd order maximum. (b) Find the total number of orders visible.
(a) d = 1 / (300 × 103) = 3.33 × 10−6 m
d sin θ = nλ
sin θ = nλ / d = (2 × 550 × 10−9) / 3.33 × 10−6 = 0.330
θ = sin−1(0.330) = 19.3°
(b) nmax = d / λ = 3.33 × 10−6 / 550 × 10−9 = 6.06
Maximum order = 6. Total maxima visible = 2 × 6 + 1 = 13 (6 on each side plus central)
Gratings vs Double Slit
Compared to the double slit pattern, a diffraction grating produces:
- Sharper, brighter maxima (more slits contributing)
- Wider spacing between maxima
- More precise measurements of wavelength
Applications include spectroscopy, measuring wavelengths of light, and analysing stellar spectra.
Key Facts
- d sin θ = nλ applies to each order of diffraction
- White light through a grating produces spectra, with violet diffracted least and red most
- The zero-order maximum (n = 0) is always at θ = 0 (straight through)
- More lines per mm means greater angular separation between orders
Exam Tip
Be careful with units when calculating d. If the grating has N lines per mm, then d = 1/(N × 103) metres. A common error is forgetting the mm to m conversion.
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