## Verifying the Helmholtz Green’s function.

Posted by peeterjoot on December 9, 2011

# Motivation.

In class this week, looking at an instance of the Helmholtz equation

We were told that the Green’s function

that can be used to solve for a particular solution this differential equation via convolution

had the value

Let’s try to verify this.

# Guts

Application of the Helmholtz differential operator on the presumed solution gives

## When .

To proceed we’ll need to evaluate

Writing we start with the computation of

We see that we’ll have

Taking second derivatives with respect to we find

Our Laplacian is then

Now lets calculate the derivatives of . Working on again, we have

So we have

Taking second derivatives with respect to we find

So we find

or

Inserting this and into 2.8 we find

This shows us that provided we have

## In the neighborhood of .

Having shown that we end up with zero everywhere that we are left to consider a neighborhood of the volume surrounding the point in our integral. Following the Coulomb treatment in section 2.2 of [1] we use a spherical volume element centered around of radius , and then convert a divergence to a surface area to evaluate the integral away from the problematic point

We make the change of variables . We add an explicit suffix to our Laplacian at the same time to remind us that it is taking derivatives with respect to the coordinates of , and not the coordinates of our integration variable . Assuming sufficient continuity and “well behavedness” of we’ll be able to pull it out of the integral, giving

Recalling the dependencies on the derivatives of in our previous gradient evaluations, we note that we have

so with , we can rewrite our Laplacian as

This gives us

To complete these evaluations, we can now employ a spherical coordinate change of variables. Let’s do the volume integral first. We have

To evaluate the surface integral we note that we’ll require only the radial portion of the gradient, so have

Our area element is , so we are left with

Putting everything back together we have

But this is just

This completes the desired verification of the Green’s function for the Helmholtz operator. Observe the perfect cancellation here, so the limit of can be independent of how large is made. You have to complete the integrals for both the Laplacian and the portions of the integrals and add them, *before* taking any limits, or else you’ll get into trouble (as I did in my first attempt).

# References

[1] M. Schwartz. *Principles of Electrodynamics*. Dover Publications, 1987.

## loiosu said

here can be applied in more cases

## loiosu said

The case of perfect contact is probably what most applications require when two materials are joined. This case also has the advantage of being simpler than the general case where heat can be obstructed at the boundary and the presentation is not encumbered by tedious algebraic manipulations and considered by prof dr mircea orasanu. The common approach to the overall problem is to form a solution by adding regular solutions from the diffusion equations to this source solution and use the parameters to satisfy boundary and initial conditions

## prof dr mircea orasanu said

Green’s Function for the Helmholtz Equation

If we fourier transform the wave equation, or alternatively attempt to find solutions with a specified harmonic behavior in time , we convert it into the following spatial form: (for example, from the wave equation above, where , , and by assumption). This is called the inhomogeneous Helmholtz equation (IHE).The Green’s function therefore has to solve the PDE: Once again, the Green’s function satisfies the homogeneous Helmholtz equation (HHE). Furthermore, clearly the Poisson equation is the limit of the Helmholtz equation. It is straightforward to show that there are several functions that are good candidates for .