Helmholtz Eq Solution from Laplace Eq Solution?

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Prequel: I am trying to develop a solution for a scattering the problem with boundary conditions on a torus. Suitably enough there are toroidal coordinates. Unfortunatelly Helmholtz equation is not separable in these coordinates, but Laplace equation is. My question is therefore as follows.



Question: Given a family of solutions to Laplace equation ($phi_n$):



$nabla^2 phi_nleft(vecrright) =0$



Is there a general way to construct a solution to Helmholtz equation out of them? i.e. find $F=Fleft(vecrright)$ s.t.



$left(nabla^2+k^2right)F=0,quad k in mathbbR$



For example, if $phi=1/r$ it is well-known that $F=expleft(pm i k/phiright)phi=fracexpleft(pm i k rright)r$ is a solution to Helmholtz equation (for $rneq 0$).



I will appreciate any pointers on this matter!



PS: I understand, of course, that the solution to Helmholtz equation will still not be separable in toroidal coordinates.



Thank you




functional-analysis differential-equations mathematical-physics




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  • I highly doubt this will be possible; the Helmholtz equation plays much less nicely with symmetry.
    – Ian
    Aug 22 at 15:33














up vote
1
down vote

favorite












Prequel: I am trying to develop a solution for a scattering the problem with boundary conditions on a torus. Suitably enough there are toroidal coordinates. Unfortunatelly Helmholtz equation is not separable in these coordinates, but Laplace equation is. My question is therefore as follows.



Question: Given a family of solutions to Laplace equation ($phi_n$):



$nabla^2 phi_nleft(vecrright) =0$



Is there a general way to construct a solution to Helmholtz equation out of them? i.e. find $F=Fleft(vecrright)$ s.t.



$left(nabla^2+k^2right)F=0,quad k in mathbbR$



For example, if $phi=1/r$ it is well-known that $F=expleft(pm i k/phiright)phi=fracexpleft(pm i k rright)r$ is a solution to Helmholtz equation (for $rneq 0$).



I will appreciate any pointers on this matter!



PS: I understand, of course, that the solution to Helmholtz equation will still not be separable in toroidal coordinates.



Thank you




functional-analysis differential-equations mathematical-physics




share|cite|improve this question




















  • I highly doubt this will be possible; the Helmholtz equation plays much less nicely with symmetry.
    – Ian
    Aug 22 at 15:33












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Prequel: I am trying to develop a solution for a scattering the problem with boundary conditions on a torus. Suitably enough there are toroidal coordinates. Unfortunatelly Helmholtz equation is not separable in these coordinates, but Laplace equation is. My question is therefore as follows.



Question: Given a family of solutions to Laplace equation ($phi_n$):



$nabla^2 phi_nleft(vecrright) =0$



Is there a general way to construct a solution to Helmholtz equation out of them? i.e. find $F=Fleft(vecrright)$ s.t.



$left(nabla^2+k^2right)F=0,quad k in mathbbR$



For example, if $phi=1/r$ it is well-known that $F=expleft(pm i k/phiright)phi=fracexpleft(pm i k rright)r$ is a solution to Helmholtz equation (for $rneq 0$).



I will appreciate any pointers on this matter!



PS: I understand, of course, that the solution to Helmholtz equation will still not be separable in toroidal coordinates.



Thank you




functional-analysis differential-equations mathematical-physics




share|cite|improve this question












Prequel: I am trying to develop a solution for a scattering the problem with boundary conditions on a torus. Suitably enough there are toroidal coordinates. Unfortunatelly Helmholtz equation is not separable in these coordinates, but Laplace equation is. My question is therefore as follows.



Question: Given a family of solutions to Laplace equation ($phi_n$):



$nabla^2 phi_nleft(vecrright) =0$



Is there a general way to construct a solution to Helmholtz equation out of them? i.e. find $F=Fleft(vecrright)$ s.t.



$left(nabla^2+k^2right)F=0,quad k in mathbbR$



For example, if $phi=1/r$ it is well-known that $F=expleft(pm i k/phiright)phi=fracexpleft(pm i k rright)r$ is a solution to Helmholtz equation (for $rneq 0$).



I will appreciate any pointers on this matter!



PS: I understand, of course, that the solution to Helmholtz equation will still not be separable in toroidal coordinates.



Thank you





functional-analysis differential-equations mathematical-physics





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asked Aug 21 at 13:36









Cryo

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  • I highly doubt this will be possible; the Helmholtz equation plays much less nicely with symmetry.
    – Ian
    Aug 22 at 15:33
















  • I highly doubt this will be possible; the Helmholtz equation plays much less nicely with symmetry.
    – Ian
    Aug 22 at 15:33















I highly doubt this will be possible; the Helmholtz equation plays much less nicely with symmetry.
– Ian
Aug 22 at 15:33




I highly doubt this will be possible; the Helmholtz equation plays much less nicely with symmetry.
– Ian
Aug 22 at 15:33















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