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Green's function for Helmholtz Equation

  • Thread starter Demon117
  • Start date
165
1
1. The problem statement, all variables and given/known data
Arfken & Weber 9.7.2 - Show that

[itex]\frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|}[/itex]

satisfies the two appropriate criteria and therefore is a Green's function for the Helmholtz Equation.


2. Relevant equations
The Helmholtz operator is given by

[itex]\nabla ^{2}A+k^{2}A[/itex]

Symmetricity of Green's functions.


3. The attempt at a solution
Right off the bat I am not sure what is mean't by "the two appropriate criteria" phrase. What exactly are the two appropriate criteria that they ask for in Arfken & Weber problem 9.7.2? Where can I find this criteria so that I know how to answer this question?
 
165
1
1. The problem statement, all variables and given/known data
Arfken & Weber 9.7.2 - Show that

[itex]\frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|}[/itex]

satisfies the two appropriate criteria and therefore is a Green's function for the Helmholtz Equation.


2. Relevant equations
The Helmholtz operator is given by

[itex]\nabla ^{2}A+k^{2}A[/itex]

Symmetricity of Green's functions.


3. The attempt at a solution
Right off the bat I am not sure what is mean't by "the two appropriate criteria" phrase. What exactly are the two appropriate criteria that they ask for in Arfken & Weber problem 9.7.2? Where can I find this criteria so that I know how to answer this question?
I answered my own question. I read the material 6 hours ago, looked at the assignment and did it but kept coming up with the solution that [itex](\nabla ^{2}+k^2)G=0[/itex], but I kept claiming that was incorrect. Well, looking back for the 100th time I realized this has to be true based on page 598 of Arfken & Weber, not only because it says so but also because the Helmholtz equation indicates a Green's function corresponding to an outgoing wave, which means that G(r1,r2) must satisfy a homogenous differential equation. Wow, that was a lot but I think it makes sense.
 
afkern and weber problem 9.7.3

Hey I am still struggling with the solution of the problem and trying to figure out your explanation. Can you explain it more elaborately.
 
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30
The solutions are given by hankel functions of first kind and second kind.A time dependence of exp(-iwt) is assumed.In physical cases only outgoing wave is present so only one is chosen and not the other.
 

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