5.6 - Interference of Waves

Superposition and Coherent Sources

The principle of superposition states that when two waves overlap, their resultant displacement is the sum of their individual displacements:

y = y_{1} + y_{2}

Interference is the superposition of two or more waves from coherent sources.
Two sources are coherent when their waves have the same frequency and a constant phase difference.
After overlapping, the individual wave profiles continue to propagate.

Type	Superposition of equal-amplitude waves
Constructive interference	Crest with crest gives $A + A = 2 A$ ; trough with trough gives $(- A) + (- A) = - 2 A$ . The resultant amplitude is larger.
Destructive interference	Crest with trough gives $A + (- A) = 0$ . Equal and opposite displacements cancel.

Water waves: two spherical dippers driven by the same ripple generator produce coherent circular waves. Constructive interference forms regions of alternating bright and dark fringes; destructive interference forms uniformly grey regions.
Light waves: laser light diffracted through a double slit produces two coherent waves. Their superposition produces alternating bright fringes from constructive interference and dark fringes from destructive interference.
Young’s double-slit experiment demonstrates the wave nature of light.
Sound waves: two identical loudspeakers connected to the same audio generator produce coherent waves. Constructive interference produces loud sound; destructive interference produces soft sound. A microphone and cathode ray oscilloscope show the corresponding changes in signal amplitude.
In the illustrated sound arrangement, the speakers are initially $1.0 m$ apart and driven at $1000 Hz$ . The microphone moves along a line $0.6 m$ from them; changing the speaker separation to $0.5 m$ changes the interference pattern.

An antinode is a point of constructive interference; a node is a point of destructive interference.
Crest–crest and trough–trough intersections locate antinodes; crest–trough intersections locate nodes.
Joining successive antinodes gives an antinodal line; joining successive nodes gives a nodal line.
Interference patterns contain alternating nodal and antinodal lines.

Symbol	Meaning
$λ$	Wavelength.
$a$	Separation of two coherent sources or slits.
$x$	Separation of two adjacent antinodal lines or two adjacent nodal lines; for light, two adjacent bright fringes or two adjacent dark fringes.
$D$	Perpendicular distance from the sources to the position where $x$ is measured; for light, the slit-to-screen distance.

x = \frac{λ D}{a}, λ = \frac{a x}{D}

With the other quantities fixed, increasing $a$ decreases $x$ ; increasing $λ$ or $D$ increases $x$ .
Lines become further apart at greater distances from the sources.
For a double slit with $a = 0.4 mm$ , $x = 4.5 mm$ and $D = 3.0 m$ :

λ = \frac{( 0.4 \times 1 0 ^{- 3} ) ( 4.5 \times 1 0 ^{- 3} )}{3.0} = 6.0 \times 1 0^{- 7} m

Bulbous bow of a ship: creates water waves that interfere destructively with waves around the hull, making the surrounding water calmer and reducing water drag.
Anti-reflection coating on lenses: reflected light interferes destructively, improving vision and reducing reflected images on spectacle lenses.
Headphones used on aircraft: a microphone and transmitter system produces sound waves that interfere destructively with surrounding noise.