Lesson summary
8. Superposition — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Apply the principle of superposition to stationary waves, diffraction, interference, Young's double slit and the diffraction grating, and use the related equations…
- Water waves
- Drawing the resultant
- Stationary (standing) wave
- Reflection
- Air column
- Fundamental (first harmonic)
Section 1 of 10Principle of superposition
- Principle of superposition = when two or more waves overlap at a point, the resultant displacement is the sum of the displacements of the individual waves.
- Displacements may be positive or negative, so they are added algebraically like vectors.
- Pulses
- two pulses meeting give a resultant equal to their sum while overlapping, then continue unchanged.
- Water waves
- overlapping waves show areas of zero displacement (flat) and areas of increased displacement.
- Drawing the resultant
- find where the maxima and minima occur, add the displacements point by point and join them smoothly.
- Two equal displacements of 0.7 give a resultant of 1.4.
Section 2 of 10Formation of stationary waves
- Stationary (standing) wave = a wave formed by superposition of two waves of the same frequency and amplitude travelling in opposite directions, usually a wave and its reflection.
- The peaks and troughs do not move along the medium.
- Nodes
- fixed points of zero amplitude.
- Antinodes
- points of maximum amplitude, oscillating only in the vertical direction.
- Phase
- points between adjacent nodes are in phase.
- Energy
- the pattern does not move, so there is no net energy transfer.
- Frequency
- changing the frequency changes the number of nodes and antinodes, and therefore the wavelength.
- Lengths
- a string's length contains whole or half wavelengths, such as 3 nodes and 2 antinodes.
Section 3 of 10Demonstrating stationary waves
| System | How it works |
|---|---|
| Stretched string | A vibrator drives a string held taut by a mass over a pulley; different frequencies give different numbers of nodes and antinodes, as in stringed instruments |
| Microwaves | A source faces a reflecting plate; a detector moved between them finds minima (nodes) and maxima (antinodes) |
| Air column | A loudspeaker at one end of a tube containing fine powder; powder heaps at nodes, as in wind instruments |
- Reflection
- the reflected wave has the same frequency as the incident wave and overlaps it to form the stationary wave.
- Air column
- there is a node at the closed end and an antinode at the loudspeaker.
Section 4 of 10Harmonics on a string fixed at both ends
- Fundamental (first harmonic)
- one loop of two nodes and one antinode, which is half a wavelength, so λ1 = 2L.
- Second harmonic
- three nodes and two antinodes, so λ2 = L.
- Third harmonic
- four nodes and three antinodes, so λ3 = 2L/3.
- nth harmonic
- n antinodes and n + 1 nodes, with λn = 2L/n and fn = nv/(2L).
- Resonant frequencies
- they depend on the length L and the wave speed v.
Section 5 of 10Harmonics in air columns
| Column | End conditions | Fundamental | Harmonics present |
|---|---|---|---|
| Closed at one end | Node at the closed end, antinode at the open end | L = λ/4 | Odd harmonics only, L = nλ/4 (n = 1, 3, 5…) |
| Open at both ends | Antinode at each end | L = λ/2 | All harmonics, L = nλ/2 |
- Fundamental
- it counts as the first harmonic, n = 1, with a quarter or half wavelength.
- Matching
- check the numbers of nodes and antinodes before choosing the length–wavelength relation.
Section 6 of 10Diffraction
- Diffraction = the spreading of waves when they pass through a gap or round an obstacle.
- It happens with all waves, including sound, light and water.
- Gap size
- diffraction is greatest when the gap is about the same size as the wavelength.
- A gap much smaller than the wavelength gives strong spreading.
- A gap much larger than the wavelength gives little diffraction.
- What changes
- only the amplitude changes, because energy spreads out; wavelength, frequency and speed stay the same.
- Examples
- radio waves round buildings, water waves entering a harbour, and X-rays (wavelength about 10−10 m) through the gaps between atoms in a crystal.
- Ripple tank
- varying the gap width or wavelength shows the effect for a gap and for a barrier edge.
Section 7 of 10Interference and coherence
- Interference = the overlap of waves, with the resultant displacement equal to the sum of the individual displacements.
- Constructive
- waves in phase (peak on peak) give a larger amplitude.
- Destructive
- waves in antiphase (peak on trough) give a smaller or zero amplitude.
- Coherent sources = sources with the same frequency and a constant phase difference, needed for a stable interference pattern.
- A laser is coherent and monochromatic, while a filament lamp is not.
- Path difference
- constructive when path difference = nλ and destructive when path difference = (n + ½)λ, for n = 0, 1, 2…
- Path difference means how much longer one path is than the other, not the distance between the sources.
Section 8 of 10Two-source interference
- Water
- in a ripple tank, constructive interference occurs where peaks meet peaks and destructive where a peak meets a trough.
- (or troughs meet troughs)
- Counting wavelengths from each source to a point shows a whole-number difference for constructive interference.
- in a ripple tank, constructive interference occurs where peaks meet peaks and destructive where a peak meets a trough.
- Sound
- two speakers on the same source give loud and quiet regions.
- Destructive interference between compressions and rarefactions is the basis of noise-cancelling headphones.
- Microwaves
- a movable detector finds maxima and minima.
- Light
- a laser through two slits gives bright (constructive) and dark (destructive) fringes, and needs coherent, monochromatic light.
- Orders
- n = 0 is the central maximum and n = 1, 2… are the maxima on either side.
Section 9 of 10Young's double-slit experiment
- Set-up
- a laser passes through two narrow slits and the diffracted beams overlap on a distant screen, giving bright and dark fringes.
- Equation
- λ = ax/D.What the symbols mean: λ = wavelength (m) · a or s = slit separation (m) · x or w = fringe spacing (m) · D = slit-to-screen distance (m)
- a is the slit separation, usually in mm and the smallest quantity.
- x is the fringe separation between adjacent bright fringes, usually in cm.
- D is the distance from the slits to the screen, usually several metres.
- λ = ax/D.
- Fringe spacing
- x increases if the wavelength increases, if D increases or if the slit separation a decreases.
Section 10 of 10The diffraction grating
- Diffraction grating = a plate with a very large number of very thin, equally spaced parallel slits of spacing d.
- The regions of constructive interference are the regions of maximum intensity.
- Equation
- d sin θ = nλ, where θ is the angle from the central axis to the nth-order maximum.
- Slit spacing
- d = 1/N, where N is lines per metre.
- (or per mm, giving d in mm)
- d = 1/N, where N is lines per metre.
- Angle between orders
- θ increases with n, and the angle between two maxima is the difference of their angles.
- Highest order
- sin θ cannot exceed 1, so nmax = d/λ rounded down to a whole number.
- Measuring wavelength
- a laser is shone through the grating onto a screen and the maxima are measured:
- Shine a laser through the grating onto a screen a distance D away.
- Measure the distance h from the centre to the nth-order maximum.
- Find the angle from tan θ h/D.
- Calculate λ d sin θ / n.
- Improvements
- measure across several fringes and divide by their number, repeat 3 to 8 times, use a vernier scale, increase D and darken the room.
- A grating with more lines per mm makes h larger and reduces percentage uncertainty.
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