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Interference

Tags
phys/wave
cegep/3
Word count
299 words
Reading time
2 minutes

As a result of superposition, the displacement of the medium at a point is the sum of the displacements due to each wave.

Δϕ=2πλΔx±2πTΔt+Δϕ0

[!abstract] in / out of phase
In phase: Δϕ0=0
Out of phase: Δϕ0=π

[!abstract] node / antinode
Locations where amplitude is always zero / maximal

[!abstract]+ nodal / antinodal line
Line of destructive / constructive interference in 2D

  • Nodal lines: Δr=±12λ,±32λ,
  • Antinodal lines: Δr=0,±λ,±2λ,
  • Δr is the same for all points on an antinodal line.

Types

Constructive

Superposition resulting in increased amplitude
Intrinsic phase difference is even multiple of π

Δϕ=2mπ

Destructive

Superposition resulting in decreased amplitude
Intrinsic phase difference is odd multiple of π

Δϕ=(2m+1)π

Partial

Between constructive and destructive

Examples

Two out-of-phase loudspeakers emit 515 Hz sound waves with no time delay. The air conditions on this day are normal. Speaker 1 is at 𝑥 = 1.00 m and speaker 2 at 𝑥 = 3.50 m.

  1. At the position 𝑥 = 1.25 m, what kind of interference occurs between these two signals?
Δx=x2x1=(1.251)(3.501.25)=2mT=1f=1515sλ=23mΔϕ=2πλΔx+2πTΔt+Δϕ0=2π23(2)+2π15150+π=5π=(2(3)+1)π

Destructive interference occurs at that position.

  1. What is the smallest path-length difference (∆𝑥) that would result in constructive interference?
(2m+1)π=2π23Δx+π2mπ=3πΔxΔx=2m3Δxmin=213=23m

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