The principle of superposition
When two pulses meet
Two friends hold the ends of a long rope. Each flicks a pulse towards the other. What happens when the pulses meet? Do they bounce off each other like two balls?
Watch closely. While the pulses overlap, the rope takes a new shape. Then each pulse comes out the other side, unchanged, as if the other had never been there.
During the overlap, the displacement at every point is simply the algebraic sum of the displacements the two pulses would give on their own.
This is the principle of superposition. If y one and y two are the separate displacements, the rope’s displacement is y equals y one plus y two. With more waves, we add them all.
Two equal crests meet. At the moment they overlap exactly, the rope rises to twice the height of either pulse.
Now a crest meets an equal trough. At the moment of exact overlap, the displacements cancel and the rope is perfectly flat.
Has the energy vanished? No. The rope is flat, but it is moving. At that instant all the energy is kinetic, and a moment later both pulses re-emerge.
Superposition is why you can pick out a friend’s voice in a noisy classroom. Many sound waves cross the same air at once, and each one passes through the others unharmed.
Interference of two waves
Now send two harmonic waves along the same string, in the same direction. They have the same amplitude a, the same ω and the same k, but they differ in phase by φ.
y one equals a sin of k x minus ω t. y two equals a sin of k x minus ω t plus φ.
Add them using sin A plus sin B equals 2 sin of half A plus B, times cos of half A minus B.
The result is y equals 2a cos of φ by 2, times sin of k x minus ω t plus φ by 2.
This is again a harmonic wave travelling in the same direction, with the same wavelength and frequency. Only its amplitude has changed: it is 2a cos of φ by 2.
If φ is zero, the waves are in phase. Crest falls on crest, and the amplitude is 2a. This is constructive interference.
If φ is π, the waves are exactly out of phase. Crest falls on trough, and the amplitude is zero. The string stays still. This is destructive interference.
In between, the amplitude takes in-between values. For φ equals 2π by 3, the amplitude is 2a cos 60 degrees, which is just a.
Noise-cancelling headphones use exactly this. A tiny microphone hears the noise, and the speaker plays the same wave shifted by π, so the two cancel near your ear.
Try it: two pulses
- Press Play, or drag the time slider to freeze the moment of overlap.
- Tick “Turn pulse 2 upside down”, then drag the time to 1.75 s: the string is flat for an instant.
- The dashed curves on the string are y₁ and y₂. The amber string is always their sum.
Try it: interference
- Slide φ from 0 to 360°. The amber amplitude A follows 2a cos(φ/2).
- At φ = 180° with equal amplitudes, the string stays flat: destructive interference.
- Make the amplitudes unequal. Now the best you can do at φ = 180° is |a₁ − a₂|.
- Try “Opposite ways” for a preview of the next lesson.
Where did the energy go?
A crest and an equal trough travel towards each other on a rope. At the instant they overlap exactly, the rope is perfectly straight. Which statement is correct?
- The energy of the pulses has been destroyed
- The energy has turned into heat, so the pulses will not reappear
- The rope is moving at that instant, so the energy is kinetic, and both pulses re-emerge
- The pulses reflect off each other and travel back
Answer
C. Superposition adds the displacements, so the shape is flat for an instant. But the rope elements are still moving: all the energy is kinetic at that moment. A moment later the pulses come out unchanged and carry on in their own directions.
Resultant amplitude
Two waves of the same frequency, each of amplitude 4 mm, travel in the same direction along a string with a phase difference of 120°. What is the amplitude of the resultant wave?
- 8 mm
- 6.9 mm
- 4 mm
- 2 mm
Answer
C. The resultant amplitude is A = 2a cos(φ/2) = 2 × 4 mm × cos 60° = 2 × 4 × 0.5 = 4 mm. Simply adding gives 8 mm, which is true only when φ = 0. Using sin instead of cos gives 6.9 mm; forgetting the factor 2 gives 2 mm.
Summary
- Principle of superposition:
y = y₁ + y₂ + … - Two equal waves, phase difference φ:
y = 2a cos(φ/2) sin(kx − ωt + φ/2) - Resultant amplitude:
A = 2a cos(φ/2)
- Where waves overlap, the resultant displacement is the algebraic sum of the separate displacements. Afterwards each wave carries on unchanged.
- Constructive interference: φ = 0, 2π, … gives A = 2a.
- Destructive interference: φ = π, 3π, … gives A = 0.
- The resultant of two waves of the same k and ω is a wave with the same k and ω; only the amplitude depends on φ.