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.