Helmholtz equation

The Helmholtz equation (after Hermann von Helmholtz ) is a partial differential equation . It is:

${\displaystyle \Delta \varphi =\lambda \cdot \varphi }$

in an area with suitable boundary conditions on the edge . It is ${\displaystyle \Omega }$ ${\displaystyle \partial \Omega }$

${\displaystyle \Delta =\sum _{k=1}^{n}{\partial ^{2} \over \partial x_{k}^{2}}.}$

the Laplace operator in Cartesian coordinates.

The Helmholtz equation is accordingly a partial differential equation (PDGL) of the second order from the class of the elliptical PDGL . It also results from z. B. from the wave equation after separating the variables and assuming harmonic time dependence.

If one sets , one obtains the Laplace equation . ${\displaystyle \lambda =0}$

Example: Particulate solution of the inhomogeneous Maxwell equations

An application from physics is e.g. B. the solution of the inhomogeneous Maxwell equations (Maxwell equations with currents and charges). From these follow in Gaussian units with the Lorenz calibration

${\displaystyle {\vec {\nabla }}\cdot {\vec {A}}+{\frac {1}{c^{2}}}{\frac {\partial \Phi }{\partial t}}=0}$

the inhomogeneous wave equations for the electrical scalar potential as well as for the magnetic vector potential : ${\displaystyle \Phi }$ ${\displaystyle {\vec {A}}}$

${\displaystyle \Delta \Phi ({\vec {r}},t)-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\Phi ({\vec {r}},t)}{\partial t^{2}}}=-4\pi \varrho ({\vec {r}},t)}$

${\displaystyle \Delta A_{i}({\vec {r}},t)-{\frac {1}{c^{2}}}{\frac {\partial ^{2}A_{i}({\vec {r}},t)}{\partial t^{2}}}=-{\frac {4\pi }{c}}j_{i}({\vec {r}},t)}$

(here for the individual components: ) ${\displaystyle {\vec {A}}=\sum _{i=1}^{3}A_{i}{\hat {e}}_{i}}$

The solution for is now carried out as an example , the derivation for goes analogously. ${\displaystyle \Phi }$ ${\displaystyle {\vec {A}}}$

The general solution of these differential equations is the linear combination of the general solution of the corresponding homogeneous differential equation as well as a particular solution of the inhomogeneous differential equation :

${\displaystyle \Phi =\Phi _{\mathrm {hom.} }+\Phi _{\mathrm {part.} }}$

The solution to the homogeneous DGL are the electromagnetic waves; we limit ourselves here to the derivation of a particular solution.

To reduce the wave equation to the Helmholtz equation, consider the Fourier transform of and with respect to : ${\displaystyle \Phi }$ ${\displaystyle \varrho }$ ${\displaystyle t}$

${\displaystyle \Phi ({\vec {r}},t)={\frac {1}{\sqrt {2\pi }}}\int d\omega \,\Phi _{\omega }({\vec {r}})e^{-i\omega t}}$

${\displaystyle \varrho ({\vec {r}},t)={\frac {1}{\sqrt {2\pi }}}\int d\omega \,\varrho _{\omega }({\vec {r}})e^{-i\omega t}}$

Insertion into the wave equation gives:

${\displaystyle \Delta \int d\omega \,\Phi _{\omega }({\vec {r}})e^{-i\omega t}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\int d\omega \,\Phi _{\omega }({\vec {r}})e^{-i\omega t}=-4\pi \int d\omega \,\varrho _{\omega }({\vec {r}})e^{-i\omega t}}$

${\displaystyle \Rightarrow \int d\omega \,\left(\Delta -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial ^{2}t}}\right)\Phi _{\omega }({\vec {r}})e^{-i\omega t}=-4\pi \int d\omega \,\varrho _{\omega }({\vec {r}})e^{-i\omega t}}$

${\displaystyle \Rightarrow \int d\omega \,\left(\Delta +{\frac {\omega ^{2}}{c^{2}}}\right)\Phi _{\omega }({\vec {r}})e^{-i\omega t}=-4\pi \int d\omega \,\varrho _{\omega }({\vec {r}})e^{-i\omega t}}$

Both integrands must be the same since the Fourier transform is bijective:

${\displaystyle \left(\Delta +{\frac {\omega ^{2}}{c^{2}}}\right)\Phi _{\omega }({\vec {r}})=-4\pi \varrho _{\omega }({\vec {r}})}$

For the homogeneous wave equation we recognize the Helmholtz equation again. ${\displaystyle \left(\varrho ({\vec {r}},t)=0\right)}$ ${\displaystyle \left(\Delta +{\frac {\omega ^{2}}{c^{2}}}\right)\Phi _{\omega }({\vec {r}})=0}$

For the solution of the inhomogeneous equation one may Green's function are used, which equation ${\displaystyle \left(\varrho ({\vec {r}},t)\neq 0\right)}$ ${\displaystyle G({\vec {r}},{\vec {r}}')}$

${\displaystyle \left(\Delta +{\frac {\omega ^{2}}{c^{2}}}\right)G({\vec {r}},{\vec {r}}')=-4\pi \delta ({\vec {r}}-{\vec {r}}')}$

Fulfills.

This is:

${\displaystyle G({\vec {r}},{\vec {r}}')={\frac {\exp(\pm i\omega |{\vec {r}}-{\vec {r}}'|/c)}{|{\vec {r}}-{\vec {r}}'|}}}$

Physically, this function describes a spherical wave.

This gives us for the entire charge distribution:

${\displaystyle \Phi _{\omega }({\vec {r}})=\int d^{3}r'\,\varrho _{\omega }({\vec {r}}')G({\vec {r}},{\vec {r}}')=\int d^{3}r'\,\varrho _{\omega }({\vec {r}}'){\frac {\exp(\pm i\omega |{\vec {r}}-{\vec {r}}'|/c)}{|{\vec {r}}-{\vec {r}}'|}}}$

We put this result in the Fourier representation of and get ${\displaystyle \Phi ({\vec {r}},t)}$

${\displaystyle {\begin{aligned}\Phi ({\vec {r}},t)&={\frac {1}{\sqrt {2\pi }}}\int d\omega \,\int d^{3}r'\,\varrho _{\omega }({\vec {r}}'){\frac {\exp(\pm i\omega |{\vec {r}}-{\vec {r}}'|/c)}{|{\vec {r}}-{\vec {r}}'|}}e^{-i\omega t}\\&={\frac {1}{\sqrt {2\pi }}}\int d\omega \,\int d^{3}r'\,{\frac {\varrho _{\omega }({\vec {r}}')}{|{\vec {r}}-{\vec {r}}'|}}\exp \left(-i\omega (\mp |{\vec {r}}-{\vec {r}}'|/c+t)\right)\end{aligned}}}$

With follows: ${\displaystyle t':=t\mp |{\vec {r}}-{\vec {r}}'|/c}$

${\displaystyle \Phi ({\vec {r}},t)={\frac {1}{\sqrt {2\pi }}}\int d\omega \,\int d^{3}r'\,{\frac {\varrho _{\omega }({\vec {r}}')}{|{\vec {r}}-{\vec {r}}'|}}\exp(-i\omega t')={\frac {1}{\sqrt {2\pi }}}\int d^{3}r'\,{\frac {1}{|{\vec {r}}-{\vec {r}}'|}}\int d\omega \,\varrho _{\omega }({\vec {r}}')e^{-i\omega t'}}$

${\displaystyle \Rightarrow \Phi ({\vec {r}},t)=\int d^{3}r'\,{\frac {\varrho ({\vec {r}}',t')}{|{\vec {r}}-{\vec {r}}'|}}}$

This is the particular solution of the inhomogeneous equation that we are looking for. For follows analogously: ${\displaystyle A_{i}}$

${\displaystyle A_{i}({\vec {r}},t)={\frac {1}{c}}\int d^{3}r'\,{\frac {j_{i}({\vec {r}}',t')}{|{\vec {r}}-{\vec {r}}'|}}}$

${\displaystyle \Rightarrow {\vec {A}}({\vec {r}},t)={\frac {1}{c}}\int d^{3}r'\,{\frac {{\vec {j}}({\vec {r}}',t')}{|{\vec {r}}-{\vec {r}}'|}}}$

The physical meaning is that the potential observed at the location at the time was caused by charges or currents at the location at the time . ${\displaystyle t}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle t'}$ ${\displaystyle {\vec {r}}'}$

Discussion: Retarded and advanced solution

The sign in the argument has not yet been determined. From a physical point of view, however, it seems plausible that the change in a charge distribution over time can only be observed at a later point in time at , since electromagnetic waves propagate at the (constant) speed of light . Therefore we choose the minus sign as a physically practicable solution: ${\displaystyle t\pm |{\vec {r}}-{\vec {r}}'|/c}$ ${\displaystyle {\vec {r}}'}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle c}$

${\displaystyle \Phi ({\vec {r}},t)_{\mathrm {ret.} }=\int d^{3}r'\,{\frac {\varrho ({\vec {r}}',t-|{\vec {r}}-{\vec {r}}'|/c)}{|{\vec {r}}-{\vec {r}}'|}}}$

When choosing the minus sign, the potential is also called retarded potential. If you choose the plus sign, you speak of advanced potential.

See also

• Bessel beam

Web links

• Helmholtz equation at Wolfram MathWorld (engl.)