Green's function for Helmholtz Equation

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Homework Help Overview

The discussion revolves around the problem from Arfken & Weber regarding the demonstration that a specific function satisfies the criteria to be considered a Green's function for the Helmholtz Equation. The subject area includes differential equations and mathematical physics, particularly focusing on the properties of Green's functions in relation to the Helmholtz operator.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants express uncertainty about the "two appropriate criteria" mentioned in the problem statement and seek clarification on what these criteria entail. There is an exploration of the relationship between the Green's function and the Helmholtz equation, with one participant reflecting on their previous misunderstanding of the problem.

Discussion Status

The discussion is ongoing, with participants actively questioning the criteria for Green's functions and sharing insights from the textbook. One participant has identified a key relationship in the problem but is still seeking further clarification from others.

Contextual Notes

There is mention of specific page references in Arfken & Weber that may contain relevant information, indicating that participants are working within the constraints of the textbook material. The discussion also hints at the physical interpretation of the Green's function related to outgoing waves.

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Homework Statement


Arfken & Weber 9.7.2 - Show that

\frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|}

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


Homework Equations


The Helmholtz operator is given by

\nabla ^{2}A+k^{2}A

Symmetricity of Green's functions.


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?
 
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matumich26 said:

Homework Statement


Arfken & Weber 9.7.2 - Show that

\frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|}

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


Homework Equations


The Helmholtz operator is given by

\nabla ^{2}A+k^{2}A

Symmetricity of Green's functions.


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 (\nabla ^{2}+k^2)G=0, 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.
 
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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