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

\(\begin{array}{l}\bigtriangledown ^{2} A+k^{2}A=0\end{array} \)


  • \(\begin{array}{l}\bigtriangledown ^{2}\end{array} \)
    : Laplacian
  • k: wavenumber
  • A: amplitude

Helmholtz’s 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:

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

\(\begin{array}{l}u(r,t)=A(r)T(t)\end{array} \)
(separation of variables)

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

\(\begin{array}{l}\frac{\bigtriangledown ^{2}A}{A}=-k^{2}\end{array} \)


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

\(\begin{array}{l}\bigtriangledown ^{2}A+k^{2}A=(\bigtriangleup ^{2}+k^{2})A=0\end{array} \)
(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 waves.

Frequently Asked Questions – FAQs

What is Helmholtz function?

Helmholtz function is defined as the thermodynamic function of a system which is equal to the difference between the internal energy and the product of system’s temperature and entropy.

Who solved Helmholtz equation?

The Helmholtz equation was solved by many and the equation was used for solving different shapes. Simeon Denis Poisson used the equation for solving rectangular membrane. Equilateral triangle was solved by Gabriel Lame and Alfred Clebsch used the equation for solving circular membrane.

What is the difference between Helmholtz free energy and Gibbs free energy?

The Helmholtz free energy is defined as the work done which is extracted from the system such that the temperature and volume are constant. 

Whereas Gibbs free energy is defined as the maximum reversible work which is extracted from the system such that the temperature and pressure are constant.

What is the application of Gibbs Helmholtz equation?

The applications of Gibbs Helmholtz equation are as follows:

  • In the calculation of the temperature change effect on the equilibrium constant.
  • In the calculation of enthalpy change for a reaction when the temperature is not 298K.

What is the formula of Helmholtz free energy?

The formula of Helmholtz free energy is F = U – TS.

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