Principle of Superposition & Reflection of Waves
When two or more wave pulses overlap simultaneously in a medium, the resultant displacement of any element of the medium at any given instant is the vector sum of the displacements due to each individual wave:
This principle holds strictly for small-amplitude linear elastic waves (Hooke's law regime).
1. Interference of Two Harmonic Waves
Consider two harmonic waves of identical frequency ω and amplitude A1, A2 having a constant phase difference φ:
- y1 = A1 sin(kx − ωt)
- y2 = A2 sin(kx − ωt + φ)
By superposition, the resultant wave is: y = Ares sin(kx − ωt + θ), where:
Since intensity is proportional to the square of amplitude (I ∝ A2):
| Condition | Phase Difference φ | Path Difference Δx | Resultant Amplitude Ares | Resultant Intensity Ires |
|---|---|---|---|---|
| Constructive Interference (Maxima) | φ = 2nπ (0, 2π, 4π…) | Δx = n λ | Amax = A1 + A2 | Imax = (√I1 + √I2)2 (If A1=A2, Imax = 4I0) |
| Destructive Interference (Minima) | φ = (2n − 1)π (π, 3π…) | Δx = (2n − 1) λ / 2 | Amin = |A1 − A2| | Imin = (√I1 − √I2)2 (If A1=A2, Imin = 0) |
2. Reflection of Waves at Boundaries
When a travelling wave encounters an interface or boundary separating two media, part of the wave is reflected back into the first medium, and part is transmitted into the second medium.
1. Reflection at a Rigid (Fixed) Boundary
A boundary where displacement is strictly zero at all times (e.g. string tied to a rigid wall, closed end of an organ pipe):
- By Newton's third law, the wall exerts an equal and opposite reaction force on the string.
- The reflected wave suffers an abrupt phase change of π radians (180°).
- Crest reflects as a trough, and compression reflects as a compression (phase change in displacement, but pressure wave stays in phase).
Reflected: yr = −A sin(kx + ωt) = A sin(kx + ωt + π)
2. Reflection at an Open (Free) Boundary
A boundary where the end of the medium is completely free to move (e.g. ring sliding frictionlessly on a rod, open end of an organ pipe):
- The free end can move with twice the amplitude of the incident wave.
- The reflected wave undergoes NO phase change (Δφ = 0).
- Crest reflects as a crest, trough reflects as a trough.
Reflected: yr = A sin(kx + ωt)
3. Reflection and Transmission Amplitudes (Boundary Relations)
When a wave travels from medium 1 (wave speed v1) to medium 2 (wave speed v2):
- Rarer to Denser (v2 < v1): Ar is negative ⇒ π phase flip upon reflection.
- Denser to Rarer (v2 > v1): Ar is positive ⇒ zero phase change upon reflection.
- Transmitted wave At: Always in phase with the incident wave (no phase flip ever!).
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