Helmholtz Equation

The Helmholtz equation, named after Hermann von Helmholtz, is the linear partial differential equation. where is the Laplacian, is the amplitude, and is the wavenumber. The Helmholtz equation is also an eigenvalue equation. The Helmholtz differential equation can be solved by separation of variables in only 11 coordinate systems.

What is the Helmholtz Equation?

Helmholtz equation named after Hermann von Helmholtz, which is used in Physics and Mathematics. It is a partial differential equation and its mathematical formula is:

\(\bigtriangledown ^{2} A+k^{2}A=0\)

Where,

  • \(\bigtriangledown ^{2}\): Laplacian
  • k: wavenumber
  • A: amplitude

Helmholtz equation finds application in Physics problem-solving concepts like seismology, acoustics and electromagnetic radiation.

Helmholtz Equation Derivation

The derivation of Helmholtz equation is as follows:

\((\bigtriangledown ^{2}-\frac{1}{c^{2}}\frac{\partial^2 }{\partial x^2})u(r,t)=0\) (wave equation)

\(u(r,t)=A(r)T(t)\) (separation of variables)

\(\frac{\bigtriangledown ^{2}A}{A}=\frac{1}{c^{2}T}\frac{\mathrm{d^{2}T} }{\mathrm{d} t^{2}}\) (substitution into wave equation)

\(\frac{\bigtriangledown ^{2}A}{A}=-k^{2}\)

and

\(\frac{1}{c^{2}T}\frac{\mathrm{d^{2}T} }{\mathrm{d} t^{2}}=-k^{2}\) (above two are obtained equations)

\(\bigtriangledown ^{2}A+k^{2}A=(\bigtriangleup ^{2}+k^{2})A=0\) (Helmholtz equation after rearranging)

This was the helmholtz equation solution.

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Applications of Helmholtz Equation

Seismology: Scientific study of earthquake and its propagating elastic waves is known as seismology. Other study areas are tsunamis (due to environmental effects), volcanic eruptions (due to seismic source).

There are three types of seismic waves: body waves which have P-waves (primary waves) and S-waves (secondary or shear waves), surface waves and normal modes.

Helmholtz equation in seismology:

u=⛛φ+⛛*ψ, ⛛*ψ=0 (S-wave equation)

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