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.
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