What would the right approach to perform a 2D discrete convolution for this problem ?

Questions
1

I am trying to perform a 2D discrete convolution of two terms; one a Gaussian and a differentiated Green’s function containing Hankel function of first kind and zero order. The convolution is defined as p(x,y,t) = f(x,y) * \frac{d}{dt} G(x,y,t) where,

  • f(x,y) = \epsilon e^{ -\alpha (x^2 + y^2) }
  • \frac{d}{dt}G(x,y,t) = \frac{\omega}{4} \frac{1}{c_0^2 } H_0^{(1)}\left [k\sqrt{x^2+y^2} \right ] e^{-i \omega t}

The common parameters in these terms are

  • \epsilon=0.5,
  • \alpha = \frac{log(2)}{2}
  • \omega = \frac{2\pi}{30}
  • c = 340.2626486
  • k = \omega/c
  • t = 90

These terms are from a paper in acoustics. I have tried to perform a discrete convolution but it fails to produced any output except the warning TerminatedEvaluation["RecursionLimit"]. I have attached the workbook which contains the usage of the commandDiscreteConvolve and the convolving variables I have used are \xi and \eta.

What would be right way to perform a convolution for these terms? Are there any gotchas which I need to be aware of ?

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2

You are not talking about the discrete convolution.
In Mathematica the discrete convolution is performed by ListConvolve.

You might try to tabulate your functions over some discrete x / y region an then in deed perform a discrete convolution via ListConvolve.

3

It seems that you are attempting a continuous convolution, not a discrete one. For lack of an NConvolve, you may try something like this:

cnvlv[x_?NumericQ, y_?NumericQ] :=
  NIntegrate[f[\[Xi], \[Eta]] DiffG[x - \[Xi], y - \[Eta]],
   {\[Xi], -Infinity, Infinity},
   {\[Eta], -Infinity, Infinity}];
cnvlv[5, 1]