Two sources 1.5 m apart emit waves of λ = 0.5 m. At what angles do the first two constructive interference maxima occur?

The correct answer is: θ₁ = 19.5°, θ₂ = 41.8°. D sin θ = mλ. Solve for θ at m=0,1,2.

A standing wave on a 0.6 m string vibrates in its 3rd harmonic. What is the wavelength? How many nodes are there?

The correct answer is: λ = 0.4 m, 4 nodes. For nth harmonic: λₙ = 2L/n. Nodes = n+1.

If you double the wavelength of both sources, how does the spacing of maxima change?

The correct answer is: Spacing doubles. D sin θ = mλ — wider spacing when λ is larger.

Two sources emit waves in phase. A point P is 3.25λ from source 1 and 1.75λ from source 2. Is P a node or antinode?

The correct answer is: Node (destructive). Path difference = 3.25λ - 1.75λ = 1.5λ = (m + ½)λ. This is destructive.

A string fixed at both ends has v = 120 m/s and L = 0.8 m. Find all resonant frequencies up to 500 Hz.

The correct answer is: 75, 150, 225, 300, 375, 450 Hz. Fₙ = nv/(2L). Calculate for n = 1,2,3,... until fₙ > 500 Hz.

Pro 🔒~20 min

Wave Interference

Superposition, standing waves, and double-slit patterns

How it works

When two waves overlap, they superpose: the total displacement at any point is the sum of individual displacements. If two crests meet, they add (constructive interference). If a crest meets a trough, they cancel (destructive interference). Standing waves form when reflected waves superpose with incident waves — nodes (zero displacement) and antinodes (maximum displacement) form at fixed positions. The double-slit experiment demonstrates that light (and matter) behaves as waves, with bright fringes where path lengths differ by whole wavelengths.

Can two sounds perfectly cancel each other out to create silence?

Noise-cancelling headphones do exactly this — they generate an inverted copy of ambient sound so the two waves destructively interfere, leaving near-silence.

What you'll learn

The Superposition Principle. When two waves occupy the same space, the resulting displacement is simply the sum of the individual displacements at every point. This principle holds for all linear waves — sound, light, water, and quantum matter waves. It is the foundation of all interference phenomena.
Constructive Interference. When two waves arrive in phase (crest meets crest), their amplitudes add together. This happens when the path length difference between the two sources is a whole number of wavelengths. The result is a bright fringe or loud sound — amplified beyond either source alone.
Destructive Interference. When two waves arrive exactly out of phase (crest meets trough), they cancel. This occurs when the path difference is a half-integer multiple of the wavelength. Perfect cancellation requires equal amplitudes — which is why noise-cancelling headphones work best for steady, low-frequency sounds.
Standing Waves. When a wave reflects back on itself in a confined space, superposition creates a standing wave — fixed nodes (zero displacement) and antinodes (maximum displacement). Only specific frequencies resonate: the harmonics. A guitar string, organ pipe, and microwave oven all exploit standing waves.

Step-by-step

Two point sources emit circular waves.
Watch the interference pattern emerge — bright lines are constructive, dark lines are destructive.
Adjust wavelength and separation to move the pattern.
Use the phase difference slider (Pro) to shift between constructive and destructive at center.
Switch to standing wave mode to see harmonics.

Key formulas

$y(x,t) = A\sin(kx - \omega t)$ Traveling wave equation
$y_{total} = y_1 + y_2$ Superposition principle
$\Delta r = d\sin\theta$ Path length difference (double-slit)
$\Delta r = m\lambda \quad \text{(constructive)}$ Constructive interference condition
$\Delta r = \left(m + \frac{1}{2}\right)\lambda \quad \text{(destructive)}$ Destructive interference condition
$f_n = \frac{nv}{2L} \quad (n = 1, 2, 3, \ldots)$ Standing wave harmonics on a string
$v = f\lambda$ Wave speed relationship

Frequently asked questions

Two sources 1.5 m apart emit waves of λ = 0.5 m. At what angles do the first two constructive interference maxima occur?: The correct answer is: θ₁ = 19.5°, θ₂ = 41.8°. D sin θ = mλ. Solve for θ at m=0,1,2.
A standing wave on a 0.6 m string vibrates in its 3rd harmonic. What is the wavelength? How many nodes are there?: The correct answer is: λ = 0.4 m, 4 nodes. For nth harmonic: λₙ = 2L/n. Nodes = n+1.
If you double the wavelength of both sources, how does the spacing of maxima change?: The correct answer is: Spacing doubles. D sin θ = mλ — wider spacing when λ is larger.
Two sources emit waves in phase. A point P is 3.25λ from source 1 and 1.75λ from source 2. Is P a node or antinode?: The correct answer is: Node (destructive). Path difference = 3.25λ - 1.75λ = 1.5λ = (m + ½)λ. This is destructive.
A string fixed at both ends has v = 120 m/s and L = 0.8 m. Find all resonant frequencies up to 500 Hz.: The correct answer is: 75, 150, 225, 300, 375, 450 Hz. Fₙ = nv/(2L). Calculate for n = 1,2,3,... until fₙ > 500 Hz.