10.1 The plane-wave ansatz and dispersion relations

A wave equation is solved, in the first instance, by a single building block: the plane wave. Substituting it into any linear wave equation collapses the calculus into algebra and produces the equation’s dispersion relation — the function $\omega(k)$ that is the medium’s entire wave signature. This chapter studies the properties waves carry regardless of what is waving, and it starts with that signature.

The plane wave

\psi(\mathbf{r}, t) \;=\; A\,e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}

is the elementary eigensolution of every linear, homogeneous wave equation.

where

$\psi$: the wave field (pressure, displacement, potential, …) varies
$A$: complex amplitude varies
$\mathbf{k}$: wavevector; $|\mathbf{k}|$ is radians of phase per metre rad/m
$\omega$: angular frequency, radians per second rad/s

Surfaces of constant phase $\mathbf{k}\cdot\mathbf{r} - \omega t = \text{const}$ are planes perpendicular to $\mathbf{k}$ , marching in the $\mathbf{k}$ direction. Because linear wave equations superpose their solutions, and because Fourier analysis expresses any field as a sum of plane waves, understanding this one building block is enough to understand any wave the equation admits.

From PDE to dispersion relation

The plane wave’s power is that it turns derivatives into multiplications. Acting on $e^{i(kx-\omega t)}$ ,

\frac{\partial}{\partial t} \to -i\omega, \qquad \frac{\partial}{\partial x} \to ik.

Every derivative in a linear wave equation becomes a factor, so the differential equation collapses to an algebraic relation between $\omega$ and $k$ — the dispersion relation $\omega(k)$ .

PDE:

1. PDE

∂²ψ/∂t² = c² ∇²ψ

2. ansatz

ψ = A e^{i(kx − ωt)}

3. substitute

(−iω)² A e^{i(kx−ωt)} = c² (ik)² A e^{i(kx−ωt)}

4. dispersion relation
ω² = c² k²   →   ω = ±c k  (linear)

The plane-wave ansatz turns derivatives into multiplications: ∂/∂t → −iω, ∂/∂x → ik. The PDE becomes an *algebraic* relation between ω and k — the dispersion relation. Real ω means propagating waves; imaginary ω means decay (diffusion). The shape ω(k) determines whether the medium is dispersive (waves spread) or non-dispersive (waves propagate rigidly).

Three equations show the range of outcomes:

The wave equation $\psi_{tt} = c^2\psi_{xx}$ gives $\omega = ck$ — linear, real, non-dispersive.
The heat equation $\psi_t = \alpha\psi_{xx}$ gives $\omega = -i\alpha k^2$ — purely imaginary, so $\psi$ decays rather than propagates; diffusion, not a wave.
The Schrödinger equation gives $\omega = (\hbar/2m)k^2$ — real but quadratic, a genuine wave whose modes travel at different speeds.

The shape of $\omega(k)$ — linear, quadratic, square-root, or complex — is what classifies the medium as non-dispersive, dispersive, or dissipative. A real $\omega(k)$ means propagation without loss; an imaginary part means attenuation; curvature in the real part means dispersion, the subject of the next lesson. Everything else about wave propagation follows from reading this one function.