Pertubations To Helmholtz Equation

In summary, the Helmholtz Equations with a perturbation p(r) involve a gradient operator and a forcing function in the left-hand side and a wave function in the right-hand side. Resources for solving and discussing this equation can be found, but most are focused on the case where p(r) = 0 and the RHS is a forcing function. Additional resources for the related wave equation with a perturbation are also mentioned, such as "Kumar's method (perturbation method)" which is often used in waveguide optics. One potential resource for this topic is a paper on "Perturbation theory for the Helmholtz equation" published in Optics Letters.
  • #1
gysush
26
0
Consider the Helmholtz Equations with a perturbation p(r)

[gradient^2 + p(r) + omega^2/c(r)^2 ]u(r,w) = 0

Does anyone know where I can find resources to the solutions/discussion of this equation? I can find many things such that p(r) = 0 , but the RHS = forcing function, but that is not what I want.

Likewise, any resources pertaining to the wave equation with a perturbation would also be nice

[gradient^2 + p(r) - 1/c(r)^2 * del^2/delT^2]u(r,t) = 0
 
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  • #2

1. What is the Helmholtz equation?

The Helmholtz equation is a partial differential equation that describes the relationship between the electric and magnetic fields in an electromagnetic wave. It is named after the German physicist Hermann von Helmholtz.

2. What are perturbations to the Helmholtz equation?

Perturbations to the Helmholtz equation refer to modifications or changes made to the original equation, often in the form of additional terms or coefficients. These perturbations can be used to model more complex physical systems or to account for external influences on the wave.

3. Why are perturbations important in the Helmholtz equation?

Perturbations are important in the Helmholtz equation because they allow for a more accurate description of real-world systems. The original Helmholtz equation assumes a perfect, idealized scenario, but perturbations can account for imperfections or external factors that may affect the behavior of the wave.

4. What are some common perturbations used in the Helmholtz equation?

Some common perturbations used in the Helmholtz equation include changes in the refractive index of the medium, non-uniformities in the medium, and the presence of external forces or sources. These perturbations can help model a variety of physical systems, such as waveguides, lenses, and scattering media.

5. How are perturbations to the Helmholtz equation solved?

The solution to perturbations in the Helmholtz equation can vary depending on the specific perturbation being considered. In some cases, analytical solutions may be possible, but in most cases, numerical methods must be used to find an approximate solution. These methods include finite element analysis, finite difference methods, and others.

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