What exactly is 'dispersion,' and does it happen in real life?

Dispersion occurs when waves of different frequencies travel at different speeds through a medium. This is a common real-world phenomenon. For example, a prism disperses white light into a rainbow because the glass has an index of refraction (which determines the wave speed) that depends on the light's frequency or color. Similarly, surface waves on deep water are dispersive, which is why ocean swells can travel vast distances while sorting themselves by wavelength.

Why does the wave packet spread when α is not zero?

When α ≠ 0, the dispersion relation ω(k) is not simply proportional to k. This means each sinusoidal component (each specific k) has a slightly different phase velocity. The constructive interference that creates the localized packet at time t=0 relies on a precise alignment of the peaks of all components. As time progresses, the faster-moving components pull ahead and the slower ones lag behind, destroying the precise alignment and causing the packet to widen while its overall amplitude decreases.

What does the 'Wave on String PDE' refer to, and why doesn't that wave spread?

It refers to the classic one-dimensional wave equation, ∂²Ψ/∂t² = c² ∂²Ψ/∂x². This partial differential equation (PDE) has a linear dispersion relation, ω = ck, where the phase velocity ω/k = c is constant for all k. Since all Fourier components travel at the identical speed c, the shape of any wave—whether a pulse or a packet—is preserved as it propagates, with no spreading. This is an idealization for a perfectly flexible, uniform string under tension.

What is the difference between phase velocity and group velocity?

Phase velocity (v_ph = ω/k) is the speed of a single sinusoidal wave's crests and troughs. Group velocity (v_gr = dω/dk) is the speed at which the overall envelope or 'group' of waves (the packet) propagates, and it's the velocity at which energy or information is carried. In a dispersive medium, these velocities are different. In this simulator, the peak of the Gaussian envelope moves at the group velocity.

Wave Packet & Dispersion

Superpose cos(kx−ωt); ω=ck+αk²; spreading vs Wave on String PDE.

Live graphs

About this model

A wave packet, or wave group, is a localized disturbance formed by the superposition of many sinusoidal waves with different wavenumbers, k. This simulator visualizes the construction and evolution of such a packet. The core mathematical operation is a Fourier synthesis: the wave function Ψ(x,t) is calculated by summing (integrating) over a range of component plane waves, typically of the form cos(kx − ωt), each with a specific amplitude determined by a Gaussian distribution in k-space. The critical physics lies in the dispersion relation, ω(k), which connects the angular frequency ω of each component to its wavenumber. Here, we explore a general quadratic relation: ω = ck + αk². The constant 'c' is the phase velocity for a non-dispersive wave (when α=0), as described by the standard wave equation for an ideal string. The term 'αk²' introduces dispersion, meaning the phase velocity ω/k depends on k. When α ≠ 0, the different sinusoidal components travel at different speeds, causing the initially compact wave packet to spread, or disperse, over time. By comparing the cases α=0 (no spreading, simulating a wave on a string) and α>0 (positive dispersion, spreading), students directly observe the profound consequence of the dispersion relation on wave dynamics. Key principles demonstrated include the superposition principle, the concept of group velocity (the velocity of the packet's envelope), and the distinction between phase and group velocity. The model simplifies reality by considering a one-dimensional, lossless medium and a specific, analytic dispersion relation, allowing clear isolation of the dispersion effect.

Who it's for: Undergraduate physics and engineering students studying wave mechanics, particularly in courses covering Fourier analysis, wave packets, and dispersion in mediums like optics, quantum mechanics, or acoustics.

Key terms

Wave Packet
Dispersion Relation
Superposition Principle
Phase Velocity
Group Velocity
Fourier Synthesis
Wave Number (k)
Gaussian Envelope

How it works

Wave on a String in this lab solves a damped wave PDE with a driven boundary — good for propagation and reflections. This page instead builds a packet as a sum of sinusoids with a dispersion relation ω(k). When ω is linear in k (α = 0), the envelope travels unchanged; with α ≠ 0, components walk out of phase and the packet disperses.

Key equations

η(x,t) = Σₙ Aₙ cos(kₙx − ω(kₙ)t), ω(k) = ck + αk²

v_p = ω/k, v_g = dω/dk = c + 2αk

Frequently asked questions

What exactly is 'dispersion,' and does it happen in real life?: Dispersion occurs when waves of different frequencies travel at different speeds through a medium. This is a common real-world phenomenon. For example, a prism disperses white light into a rainbow because the glass has an index of refraction (which determines the wave speed) that depends on the light's frequency or color. Similarly, surface waves on deep water are dispersive, which is why ocean swells can travel vast distances while sorting themselves by wavelength.
Why does the wave packet spread when α is not zero?: When α ≠ 0, the dispersion relation ω(k) is not simply proportional to k. This means each sinusoidal component (each specific k) has a slightly different phase velocity. The constructive interference that creates the localized packet at time t=0 relies on a precise alignment of the peaks of all components. As time progresses, the faster-moving components pull ahead and the slower ones lag behind, destroying the precise alignment and causing the packet to widen while its overall amplitude decreases.
What does the 'Wave on String PDE' refer to, and why doesn't that wave spread?: It refers to the classic one-dimensional wave equation, ∂²Ψ/∂t² = c² ∂²Ψ/∂x². This partial differential equation (PDE) has a linear dispersion relation, ω = ck, where the phase velocity ω/k = c is constant for all k. Since all Fourier components travel at the identical speed c, the shape of any wave—whether a pulse or a packet—is preserved as it propagates, with no spreading. This is an idealization for a perfectly flexible, uniform string under tension.
What is the difference between phase velocity and group velocity?: Phase velocity (v_ph = ω/k) is the speed of a single sinusoidal wave's crests and troughs. Group velocity (v_gr = dω/dk) is the speed at which the overall envelope or 'group' of waves (the packet) propagates, and it's the velocity at which energy or information is carried. In a dispersive medium, these velocities are different. In this simulator, the peak of the Gaussian envelope moves at the group velocity.

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