when two or more waves meet at a point, the resultant displacement is the sum of the displacements of the individual waves.
Stationary wave
the wave formed when two progressive waves of the same frequency (and similar amplitude) travelling in opposite directions superpose; no energy is transferred.
Node
a point on a stationary wave where the amplitude is always zero.
Antinode
a point on a stationary wave where the amplitude is a maximum.
Key points
Usually formed by a wave and its reflection travelling back along the same path.
Adjacent nodes (or antinodes) are λ/2 apart; node to adjacent antinode is λ/4.
All points between two adjacent nodes are in phase; points in adjacent loops are 180° out of phase.
Amplitude varies with position (zero at nodes); in a progressive wave every point has the same amplitude.
A stationary wave stores energy; a progressive wave transfers energy.
Wavelength of the stationary wave = wavelength of the two progressive waves that make it.
Stationary wave between fixed ends: solid and dashed shapes are half a period apart. Nodes N never move; antinodes A swing fully.
Phase in a stationary wave: P and Q (same loop) move in phase; R (next loop) is 180° out of phase with them.
Watch out!
Adjacent nodes are λ/2 apart, not λ. Three nodes on a string span one whole wavelength.
Points in the same loop are in phase even though their amplitudes differ.
8.1bStationary waves on strings, in air columns and with microwaves
Definitions
Fundamental (first harmonic)
the stationary wave of lowest frequency that can form on the string or in the air column.
Harmonic
a stationary wave mode whose frequency is a whole-number multiple of the fundamental frequency.
End correction
the small distance beyond the open end of a tube at which the antinode actually forms.
Key points
String fixed at both ends: nodes at both ends. Fundamental L = λ/2; then L = λ, 3λ/2 ...
String apparatus: vibration generator, string over a pulley, hanging masses; change f until loops appear.
Tube closed at one end: node at the closed end, antinode at the open end. Fundamental L = λ/4.
Tube open at both ends: antinodes at both ends. Fundamental L = λ/2.
Resonance tube: two successive resonance lengths differ by λ/2; subtracting cancels the end correction.
Microwaves: transmitter faces a metal plate; move a probe along the line. Adjacent minima (nodes) are λ/2 apart.
Measure across many nodes and divide, to reduce the percentage uncertainty.
Stationary waves on a string: the vibration generator sets f; the hanging masses set the tension. Here three loops, so L = 3λ/2.
Air columns (displacement envelopes): closed end = node, open end = antinode.
Microwaves reflected from a metal plate form a stationary wave; the probe finds nodes λ/2 apart.
Watch out!
A closed tube has only odd harmonics (L = λ/4, 3λ/4 ...); an open tube has all of them.
The first resonance length alone gives λ/4 plus the end correction; use the difference of two lengths.
8.2Diffraction
Definitions
Diffraction
the spreading of a wave as it passes through a gap or around the edge of an obstacle.
Key points
Spreading is greatest when the gap width is similar to the wavelength.
Gap much wider than λ: waves pass almost straight through, with slight spreading at the edges.
Wavelength, frequency and speed are unchanged by diffraction; only the direction of travel spreads.
Shown in a ripple tank: plane waves through a barrier gap become curved.
Sound (λ ~ 1 m) diffracts round doorways; light (λ ~ 5 × 10-7 m) needs a very narrow slit.
Diffraction at each slit is what lets waves from two slits overlap and interfere.
Ripple tank: a wide gap barely spreads the waves; a gap about one wavelength wide spreads them into semicircles. Wavefront spacing (λ) is unchanged.
Watch out!
A narrower gap spreads the wave more, not less (down to a gap about λ wide).
Do not draw a change in wavefront spacing after the gap: λ stays the same.
8.3aInterference and coherence
Definitions
Interference
the superposition of waves from two (or more) coherent sources to give regions of maximum and minimum amplitude.
Coherent
having a constant phase difference (so the waves have the same frequency).
Path difference
the difference in the distances travelled by two waves from their sources to a point.
Constructive interference
waves arrive in phase and superpose to give maximum amplitude.
Destructive interference
waves arrive in antiphase (180° out of phase) and superpose to give minimum amplitude.
Key points
A steady, observable pattern needs coherent sources; similar amplitudes give clear minima.
For in-phase sources: maximum where path difference = nλ; minimum where it is (n + ½)λ.
Equivalent phase differences: 0, 360° ... for a maximum; 180°, 540° ... for a minimum.
Complete cancellation only if the two amplitudes at that point are equal.
Demonstrations: ripple tank with two dippers on one bar; two loudspeakers on one signal generator; microwaves through two slits.
Two separate lamps are not coherent (random phase changes); use one laser, or one source and a double slit.
Two coherent sources in a ripple tank: crests (arcs) cross along lines of maximum (red, n = 0, 1); minima lie between them.
In phase: amplitudes add (maximum). In antiphase with equal amplitude: they cancel (minimum).
Watch out!
Coherent means constant phase difference, not necessarily zero phase difference.
If the sources are in antiphase the conditions swap: nλ gives a minimum.
8.3bThe double-slit experiment
Definitions
Fringe separation (x)
the distance between the centres of adjacent bright (or adjacent dark) fringes.
Slit separation (a)
the distance between the centres of the two slits.
Key points
Monochromatic light (e.g. a laser) through two narrow, close slits gives equally spaced bright and dark fringes.
The slits act as coherent sources: both are lit by the same wavefront; diffraction makes their light overlap.