- #1

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**E**(

**r,**t) = (

**A**/r)exp{i(

**k.r**-ωt)

What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero.

Thanks!

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- Thread starter RESolo
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- #1

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What happens when t=0 and the waves originates at the origin, i.e. r=0 ... which I assume can't be right as you of course cannot divide by zero.

Thanks!

- #2

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The wave equation contains a Laplacian operator, which is undefined in spherical polar coordinates at the origin, so if the domain of the wave equation doesn't include the origin, you shouldn't assume that a solution exists there.

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BruceW

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The wave equation in Cartesian coordinates is isotropic and homogeneous.

But somewhere between converting to spherical polars and solving for some exp(i(k.r-wt)) type wave, the mathematics have become inhomogeneous in that there is no solution at r=0.

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BruceW

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WannabeNewton

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I understand how delta functions work, but I'm not even discussing the solution, just the mathematics of restricting your domain when you do a coordinate transform. Plus the wave equation is homogeneous, so you don't have any localised source term embedded in the equations like you do with Poisson's equation. r=0 essentially becomes a place where boundary conditions can be specified.

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BruceW

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- #12

BruceW

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1) the divergence in spherical polar coordinates has 1/r terms, so we should not be allowed to take the divergence at r=0 using spherical polar coordinates.

2) with a non-zero r, as we take the limit of r tends to zero, the divergence of the Poynting vector will diverge to infinity.

now, 2) is only true when we have a point source. This is the case for the solution in the O.P. This is a wave created by a point source, so close to the point source, there is an infinite amount of energy transmitted per volume (in loose terms). But 2) is not true when we have some finite charge distribution that causes spherical waves. In this case, the divergence of the Poynting vector is always finite, even when we allow r to tend to zero.

As for 1) yeah, I guess that is true. But this is not a fundamental problem. We can just switch to cartesian coordinates for the case r=0. So if we have a spherical wave that is created by a finite charge distribution, this will work, and we can calculate the divergence of the Poynting vector at r=0. And if our spherical wave is created by a point source at the origin, then again, quantities will diverge to infinity when we approach r=0.

- #13

Claude Bile

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So the op's point is moot.

Claude.

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BruceW

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You can, but that choice is not unique

- #16

BruceW

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edit: I am not a mathematician, so I am not certain about this stuff. But I swear I read somewhere that when using spherical polar coordinates, we can just give ##\theta## and ##\phi## arbitrary values at the origin, so that our co-ordinate system does not have any 'missing points'.

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