Helmholtz equation

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

In an area with suitable boundary conditions on the boundary. It is

Is the Laplace operator in Cartesian coordinates.

The Helmholtz equation is therefore a partial differential equation (PDE ) of the second order from the class of elliptic PDE. You also arises for example from the wave equation by separation of variables and assumption of harmonic time dependence.

Substituting, we obtain the Laplace equation.

Example: Particulate solution of the inhomogeneous Maxwell equations

An application of physics, for example, the solution of the inhomogeneous Maxwell equations ( Maxwell equations with currents and charges ). From these follow in Gaussian units with the Lorenz gauge the inhomogeneous wave equations for the electric scalar potential and the magnetic vector potential:

(in this case for the individual components: )

As an example, the solution is now performed for the derivation is analogous for.

The general solution of this differential equation is the linear combination of the general solution of the associated homogeneous differential equation and a particular solution of the inhomogeneous differential equation:

The solution of the homogeneous differential equation, the electromagnetic waves; we restrict ourselves to the derivation of a particular solution.

Due to the wave equation for the Helmholtz equation, we consider the Fourier transform and with respect to t:

Insertion into the wave equation yields:

Both integrands must be equal, since the integration extends over the same areas:

For the homogeneous wave equation, we identify with the Helmholtz equation again.

To the solution to the inhomogeneous equation can be used, a Green's function, which equation

Met.

This is:

Physically, this function describes a spherical wave.

Thus we obtain for the total charge distribution:

This result we have in the Fourier representation of a and obtain

With this:

This is the required particular solution of the inhomogeneous equation. For analogous follows:

Is the physical meaning that the time t at the location observed potential of charges or currents at the time t ' caused locally.

Discussion: Retarded and advanced solution

Even the sign is not clear in the argument. Physically but seems plausible that the temporal change in the charge distribution at at a later date when it can be observed, since electromagnetic waves propagate with the (constant) speed of light c. Therefore, we choose the minus sign as a physically viable solution:

It is called the potential for choice of the minus sign also retarded potential. If we choose the plus sign, then one speaks of the advanced potential.

See also

• Bessel beam