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MathGroup Archive 2005

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Solving 2D scalar wave equation?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60294] Solving 2D scalar wave equation?
  • From: AES <siegman at stanford.edu>
  • Date: Sat, 10 Sep 2005 06:46:43 -0400 (EDT)
  • Organization: Stanford University
  • Sender: owner-wri-mathgroup at wolfram.com

Can any of the math or DE eqn gurus on the NG give me pointers to 
references on algorithms or recommended numerical procedures for solving 
the 2D scalar wave equation in space and time, given an initial space 
distribution at t = 0?  I don't particularly want a canned package -- 
more like references to guidance and education on how to set up and do 
the job myself.

To define the problem more precisely, I want to plop an arbitrarily 
shaped blob of radiation u0[x,y] down in the middle of a theoretically 
unbounded flat planar waveguide -- that is, this "waveguide" is bounded 
and single mode in the perpendicular (z) direction, but has no finite 
boundaries in the transverse (x and y) directions -- at t=0, and then 
watch as this blob u[x,y,t] travels in time in the x,y space; spreads 
out in that space; and likely splits into multiple blobs because the 
initial distribution contains multiple kx and ky propagation vector 
components.  

Assuming this waveguide supports a 2D TE wave I think I can reduce 
Maxwell's eqns to a 2D scalar wave eqn

  ( d^2/dx^2 + d^2/dy^2 -  mu eps d^2/dt^2 ) u[x,y,t] == 0

and I'm further willing to assume that all the components of the blob 
have the same carrier frequency w0 so that I can write

      u[x,y,t] = u1[x,y,t] X Exp[I w0 t]

with u1 being "slowly varying" in t.  The blob will in general have 
multiple initial transverse kx and ky vector components, however, so I 
can't make a similar approximation in the x and y coordinates

I know a fair amount about algorithms for collimated laser beam 
propagation, and also guided optical waveguide propagation, both of 
these being cases where there is a dominant k vector to the radiation so 
that one can make paraxial-wave or small-angle approximations.  I've 
never encountered the totally unguided "blob" problem, however, and am 
looking for somewhere to get started.

Thanks for any pointers.


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