1.16 - Superposition of waves
Phase difference and superposition
● (let θ be phase-shift)
○ θ = 0˚ - in-phase
○ θ = (±)180˚ - anti-phase (ie completely out of phase)
○ Other values of θ - out of phase
● Principle of superposition of waves - when ≥2 waves overlap, total
displacement at a point = sum of individual displacements
● ***Path difference VS phase difference - path difference is the difference in the path
travelled by the 2 waves, while phase difference is the difference in phase angle of
the 2 waves
○ Path difference ∝ phase difference
Radians (rad)
●
180˚ π
○ ***1 rad = degrees (≈ 57.3˚); 1˚ = rad
π 180
1
, Interference
● ***Constructive interference: when a crest/trough meets up with another
crest/trough;
∆ r (change∈displacement )=n(no . of wave occurrences)× λ( wavelength)
○ One wave travels exactly whole numbers of wavelengths further
● ***Destructive interference: when a crest/trough meets up with another
1
trough/crest; ∆ r =( n+ )λ
2
○ One wave travels exactly (odd) halves of a wavelength further
Interferometers
● Use the patterns created by the recombination of a laser beam which has been
split into 2 separate beams
● Small changes in the path difference detected by a shift in fringe pattern
○ Even if you tap the interferometer and not physically moving it/the light
source, pattern of rings changes (as wavelength of [visible] light is very
small — only a few hundreds of nms — even the slightest change in
[physical] distance can be a major change to displacement of light waves)
● Light ring: constructive interference, dark: destructive
Standing waves
● ***Definition: waves that remain in a constant position; arise as a result of
interference between 2 waves of the same frequency + amplitude travelling in
opposite directions (usually achieved when a progressive wave is reflected back on
itself)
○ The resulting standing wave pattern doesn’t travel & ∴ doesn’t
transfer energy across the medium
Strings
●
○ Left: standing wave, right: common transverse wave
● If end of rope attached to a fixed point wall, the incoming wave exerts an upward
force on the wall; by Newton’s 3rd law, the wall ∴ exerts a downward force on
the rope, pushing it downwards — the reflected pulse ∴ experiences a
change of phase (phase difference = 180˚: both waves are in antiphase)
2
Phase difference and superposition
● (let θ be phase-shift)
○ θ = 0˚ - in-phase
○ θ = (±)180˚ - anti-phase (ie completely out of phase)
○ Other values of θ - out of phase
● Principle of superposition of waves - when ≥2 waves overlap, total
displacement at a point = sum of individual displacements
● ***Path difference VS phase difference - path difference is the difference in the path
travelled by the 2 waves, while phase difference is the difference in phase angle of
the 2 waves
○ Path difference ∝ phase difference
Radians (rad)
●
180˚ π
○ ***1 rad = degrees (≈ 57.3˚); 1˚ = rad
π 180
1
, Interference
● ***Constructive interference: when a crest/trough meets up with another
crest/trough;
∆ r (change∈displacement )=n(no . of wave occurrences)× λ( wavelength)
○ One wave travels exactly whole numbers of wavelengths further
● ***Destructive interference: when a crest/trough meets up with another
1
trough/crest; ∆ r =( n+ )λ
2
○ One wave travels exactly (odd) halves of a wavelength further
Interferometers
● Use the patterns created by the recombination of a laser beam which has been
split into 2 separate beams
● Small changes in the path difference detected by a shift in fringe pattern
○ Even if you tap the interferometer and not physically moving it/the light
source, pattern of rings changes (as wavelength of [visible] light is very
small — only a few hundreds of nms — even the slightest change in
[physical] distance can be a major change to displacement of light waves)
● Light ring: constructive interference, dark: destructive
Standing waves
● ***Definition: waves that remain in a constant position; arise as a result of
interference between 2 waves of the same frequency + amplitude travelling in
opposite directions (usually achieved when a progressive wave is reflected back on
itself)
○ The resulting standing wave pattern doesn’t travel & ∴ doesn’t
transfer energy across the medium
Strings
●
○ Left: standing wave, right: common transverse wave
● If end of rope attached to a fixed point wall, the incoming wave exerts an upward
force on the wall; by Newton’s 3rd law, the wall ∴ exerts a downward force on
the rope, pushing it downwards — the reflected pulse ∴ experiences a
change of phase (phase difference = 180˚: both waves are in antiphase)
2
