Dispersion Relations

📐 Definition

For a linear PDE with Fourier symbol $\lambda(k)$ , the dispersion relation specifies frequency $\omega$ as a function of wavenumber $k$ , often via $\lambda(k) = -i\,\Omega(k)$ so that modes satisfy

u(t,k) = u(0,k)\,e^{-i\,\Omega(k) t}

Domain and Codomain

Defined for wavenumbers $k$ in $\mathbb{R}^d$ (or discrete grids). Output $\Omega(k)$ is real for nondissipative waves and complex when damping or growth is present.

⚙️ Key Properties

Phase and Group Velocity

v_p(k) = \frac{\Omega(k)}{|k|} \quad (k \ne 0), \qquad v_g(k) = \nabla_k \Omega(k)

Dispersion and Stability

Nonlinear $\Omega(k)$ introduces dispersion (different $k$ travel at different speeds). If $\Omega(k)=\Omega_r(k)+i\Omega_i(k)$ , then $e^{-i\Omega t}=e^{-i\Omega_r t}e^{\Omega_i t}$ , so $\Omega_i(k)<0$ corresponds to dissipation (decay) and $\Omega_i(k)>0$ to instability (growth).

🎯 Special Cases and Limits

Wave equation: $\Omega(k)=c|k|$ is nondispersive.
Schrödinger equation: $\Omega(k)=\tfrac{|k|^2}{2}$ yields dispersive spreading.
Small- $k$ / large- $k$ asymptotics give low- and high-frequency limits in composite or discrete media.

Spectral phase modulation applies phases $\phi(k) = \Omega(k)\Delta t$ during time stepping; Fourier transforms diagonalize linear PDEs enabling direct use of $\Omega(k)$ .

Usage in Oakfield

Oakfield encodes dispersion relations most directly in its spectral operators:

dispersion operator applies per-frequency phase advance in Fourier space (FFT → multiply by cexp(I * phase) → IFFT), using a configurable power law in $|k-k_0|$ .
Stability tuning: dispersion parameters interact with timestep choice; Oakfield’s integrators and step-metrics tracking help diagnose instability and step-size sensitivity.
Visualization/diagnostics: spectral stats (entropy/bandwidth) can highlight dispersion-induced spectral broadening.

Historical Foundations

📜 Waves and Fourier Modes

Dispersion relations arise from representing linear PDEs in Fourier space, where each mode evolves independently. The frequency–wavenumber relationship encodes how oscillations propagate and spread.

🌍 Modern Perspective

They are central diagnostics in spectral and pseudo-spectral methods, linking physical wave behavior to numerical stability and timestep constraints.

📚 References

Whitham, Linear and Nonlinear Waves
Strauss, Partial Differential Equations