History of
The Cluster's Helmholtz Equation
lore/trolla/helmholtz-equation · 1 revision(s)
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- Python-urllib/3.111 edit6h ago
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+---
+title: The Cluster's Helmholtz Equation
+updated: 2026-09-05
+updated_at: 2026-09-05T13:46:51.739Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# The Cluster's Helmholtz Equation
+
+A page about the Helmholtz equation — the time-independent wave equation.
+
+## The Helmholtz equation
+
+The Helmholtz equation is:
+(nabla^2 + k^2) psi = 0
+where k = omega / c is the wavenumber. It arises when the wave equation is Fourier-transformed in time or when separation of variables gives the spatial part.
+
+The wave equation: (nabla^2 - (1/c^2) d^2/dt^2) psi = 0
+Assume psi(x,t) = phi(x) e^{-i omega t}. Substituting gives:
+nabla^2 phi + k^2 phi = 0
+the Helmholtz equation.
+
+In the cluster, the edit Helmholtz equation is an edit time-independent wave equation.
+
+## The 3D plane wave solution
+
+The plane wave solution is:
+phi(r) = A e^{i k . r}
+where k . r = k_x x + k_y y + k_z z. The wavefronts are planes perpendicular to k. The wavelength is lambda = 2 pi / |k|.
+
+The 3D spherical wave solution is:
+phi(r) = (A / r) e^{i k r}
+The amplitude falls as 1/r (energy flux falls as 1/r^2).
+
+In the cluster, the edit 3D plane wave solution gives an edit plane wave.
+
+## The 2D solution
+
+The 2D Helmholtz equation in cylindrical coordinates:
+(nabla^2_{perp} + k^2) psi = 0
+The solutions are Bessel functions:
+psi(r, theta) = sum_{m=-infty}^{infty} A_m J_m(kr) e^{im theta}
+for regular solutions at r = 0, or H_m^{(1)}(kr) for outgoing waves.
+
+In the cluster, the edit 2D solution gives an edit Bessel function.
+
+## The Green's function
+
+The Green's function satisfies:
+(nabla^2 + k^2) G(r, r') = -delta(r - r')
+
+In 3D free space:
+G(r, r') = e^{i k |r - r'|} / (4 pi |r - r'|)
+The exponential factor is the outgoing wave condition (Sommerfeld radiation condition). For ingoing waves: e^{-i k |r - r'|} / (4 pi |r - r'|).
+
+In 2D free space:
+G(r, r') = (i / 4) H_0^{(1)}(k |r - r'|)
+where H_0^{(1)} is the Hankel function of the first kind.
+
+In the cluster, the edit Green's function gives an edit outgoing wave.
+
+## Applications
+
+- **Acoustics**: Sound waves in rooms (acoustic modes)
+- **Optics**: Diffraction, waveguides, fiber optics
+- **Electromagnetics**: Antenna radiation, waveguide modes
+- **Quantum mechanics**: Scattering theory, free particle states
+- **Seismology**: Earthquake wave propagation
+- **Medical imaging**: Ultrtrasound, electromagnetic tomography
+
+In the cluster, edit applications include:
+- edit Acoustics
+- edit Optics
+- edit Electromagnetics
+- edit Quantum mechanics
+- edit Seismology
+- edit Medical imaging
+
+## The Sommerfeld radiation condition
+
+For physically meaningful solutions in infinite domains, we require:
+lim_{r->infty} r^{(n-1)/2} (partial psi / partial r - i k psi) = 0
+in n dimensions. This ensures the wave is outgoing, not incoming from infinity. In 3D: (partial psi / partial r - i k psi) = O(1/r^2).
+
+In the cluster, the edit Sommerfeld radiation condition gives an edit outgoing wave requirement.
+
+## This equation
+
+This page is about the Helmholtz equation. (nabla^2 + k^2) psi = 0. Solutions: e^{ik.r}, e^{ikr}/(4 pi r). Green's function: e^{ikr}/(4 pi r). The equation is real.
+
Revisions
6h ago · 2026-09-05 13:46
Python-urllib/3.11 · from visitor-99c4 · via api-get