Helmholtz equation ↩

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created: 2021-10-25 21:07:32
modified: 2022-04-23 12:31:11

Where is the eigenvalue and is the eigenfunction (eigenvector).

In the case of waves, is known as the wave number.

Application: wave equation

Consider the following wave equation:

We can follow through by separating variables if we assume that is in fact separable:

By substitution and the simplification, we get the following equation:

Now we notice that the left side of the expression only depends on and the right side only on . Then in general, the equation can only be valid for all values, if both sides of the equation are equal to a constant value. We choose this constant to be . From this we obtain the two independent equations:

Which is the Helmholtz equation
and

That describes the time dependence of the wave where the is angular frequency of the wave.

The second-order ordinary differential equation for the time can be solved as a linear combination of sine and cosine functions, based on the initial conditions.

The spatial part, the Helmholtz equation's solution will depend on the boundary conditions.

Paraxial approximation

For the Helmholtz equation

In the paraxial approximation, the complex amplitude is expressed as:

Where modulates the sinusoidal plane wave represented by the exponential factor.
Now under the paraxial approximation we say that , so:

Now if we substitute this form of into the Helmholtz equation:

We see that the term can be neglected in comparison to the term so the paraxial Helmholtz equation takes this form:

Where is the transverse part of the Laplacian.

This approximation is very useful in optics, particularly for describing Gaussian beams (most lasers emit Gaussian beams).