Hi,
I am struggeing with a numeric implementation of the free field 2D Green function of the wave equation in space and time domain, which, according to all references I could find, is proportional to
( t2-(r/c)2 )-1/2
and thus, has a singularity at r/c=t. If I want to implement this function as numeric array that can be convolved numerically with a source distribution to get a wave field, I do not know how to deal with this singularity.
An option would be polynominal extrapolation, but I would prefer a mathematically correct attempt.
I thought I might have to analytically convolve the Green function it with a sinc function first, to attain a function that can be accutely sampled according to the Nyquist criterion, but this still did not resolve the singularity.
In consequence, I am also wondering if the 2D Green function is even integrable. I believe it should be in terms of energy conservation, but I ended up with an infinite integral.
Thanks for your time!
A few (trivial) remarks: the Green function is studied here, http://web.physics.ucsb.edu/~fratus/Phys100A/Boris/605_wveq_%288%29.pdf , and more details are given in the book by Evans. In general the Green function isn't a function, but a weaker object (a distribution) which can be locally expressed as derivatives of measures.
In order to cope with initial data, it must be a Dirac mass at t=0 (the identity of the convolution operator). Usually, one discretizes a Dirac mass in 1D by putting a weight multiplied by 1/DX, being DX the space step. The rationale is to ensure that, taking the antiderivative (-> Heaviside), it has the correct jump for any value of DX>0. This is a way to secure weak consistency.
Clearly, it's even more delicate to do so in 2d with radial functions ... I know that Anna-Karin Tornberg published years ago some papers dealing with efficient Dirac masses approximation at the numerical level, here is one of these.
Thanks Laurent,
I understand that, for t=0, the shape must be dirac like, but I can still not figure out how to deal with the singularity of the 2D Green function for t>0, which, to my understanding, cannot be modeled as a delta distribution (which I believe is the same as what you refer to as diract mass).
Thanks,
Hans-Martin
Hans-Martin, can you add some details about your application, please? For example: geometry, materials, boundary and initial conditions are necessary to define mathematics and physics of your problem.
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