# Solving the Helmholtz Equation for a Point Source

In summary, By integrating (2-55), over a small volume containing the origin, substituting ψ = Ce-jβr/r, and letting r approach zero, show that C = 1/4π, thus proving (2-58).In summary, by using the divergence theorem and considering a small volume containing the origin, it can be shown that the constant C in ψ = Ce-jβr/r is equal to 1/4π, thus proving (2-58). This involves evaluating the integral of the Laplacian of ψ over a spherical surface and using the fact that ψ diverges at the origin.

## Homework Statement

By integrating (2-55), over a small volume containing the origin, substituting ψ = Ce-jβr/r, and letting r approach zero, show that C = 1/4π, thus proving (2-58).

## Homework Equations

(2-55): ∇2ψ + β2ψ = -δ(x)δ(y)δ(z)
(2-58): ψ = e-jβr/(4πr)

## The Attempt at a Solution

I am SO lost on this.

I integrated the RHS first. I would assume I have to do this in spherical coordinates. I know from the internet that to convert the delta function between coordinate systems you need to incorporate the jacobian, but because it's at the origin (i.e., x,y,z = 0) I just assumed that:

∫∫∫(-δ(x)δ(y)δ(z))r2sin(θ)dr dθ d∅
= -∫∫∫r2sin(θ)dr dθ d∅
= -(4π/3)r03

Where I've integrated from r = 0 to r0, θ from 0 to pi, and ∅ from 0 to 2pi. Is this even correct?

For the next part, the LHS, I'm totally lost. I'm trying to follow the instructions chronologically, thus

∫∫∫(∇2 + β2)ψ r2sin(θ)dr dθ d∅

Where I'm using the same integration limits as before. I'm really, really unsure how to even do this, but I assumed that psi is regarded as scalar (or otherwise does not have dependency on r, theta or phi). Thus, starting with the r component:

∫∫∫(1/r2)(∂/∂r)[r2(∂/∂r)] (ψ r2sin(θ)dr dθ d∅)
= ∫∫∫(1/r2)(∂/∂r)[r2 (ψ 2r sin(θ)dr dθ d∅)]
= ∫∫∫(1/r2)(∂/∂r)[2r3 (ψ sin(θ)dr dθ d∅)]
= ∫∫∫(1/r2)[6r2 (ψ sin(θ)dr dθ d∅)]
= ∫∫∫[6 (ψ sin(θ)dr dθ d∅)]
= 4ψ∏r0

For the next part (θ component):
∫∫∫(1/[r2 sin(θ)])(∂/∂θ)[sin(θ)(∂/∂θ)] (ψ r2sin(θ)dr dθ d∅)
=∫∫∫(1/[r2 sin(θ)])(∂/∂θ)[sin(θ)cos(θ)] (ψ r2dr dθ d∅)
=∫∫∫(1/[r2 sin(θ)])(∂/∂θ)[(1/2)sin(2θ)] (ψ r2dr dθ d∅)
=∫∫∫(1/[r2 sin(θ)])[cos(2θ)] (ψ r2dr dθ d∅)
=∫∫∫(cos(2θ)/ sin(θ)]) (ψ dr dθ d∅)

But I can't integrate this because (cos(2θ)/ sin(θ)]) doesn't converge. I'm really lost on where to go from here and don't fully understand what I'm being asked to do.

ANY help is appreciated. Thank you. Please note that I'm an electrical eng so our notation may be different, and I may have some difficulty with more of the hardcore physics notation. I have posted this on the intro physics board but nothing doing, so I'm not sure if maybe this is more advanced. If it isn't, my apologies.

## Homework Statement

By integrating (2-55), over a small volume containing the origin, substituting ψ = Ce-jβr/r, and letting r approach zero, show that C = 1/4π, thus proving (2-58).

## Homework Equations

(2-55): ∇2ψ + β2ψ = -δ(x)δ(y)δ(z)
(2-58): ψ = e-jβr/(4πr)

## The Attempt at a Solution

I am SO lost on this.

I integrated the RHS first. I would assume I have to do this in spherical coordinates. I know from the internet that to convert the delta function between coordinate systems you need to incorporate the jacobian, but because it's at the origin (i.e., x,y,z = 0) I just assumed that:

∫∫∫(-δ(x)δ(y)δ(z))r2sin(θ)dr dθ d∅
= -∫∫∫r2sin(θ)dr dθ d∅
= -(4π/3)r03

Where I've integrated from r = 0 to r0, θ from 0 to pi, and ∅ from 0 to 2pi. Is this even correct?

No, it's not correct. You've dropped the delta function in the second line ($\delta(\vec{x})\neq 1$ everywhere). There's no particular reason to integrate over spherical variables here on the RHS, but if you do, then you will actually need to use the Jacobian.

For the next part, the LHS, I'm totally lost. I'm trying to follow the instructions chronologically, thus

∫∫∫(∇2 + β2)ψ r2sin(θ)dr dθ d∅

Where I'm using the same integration limits as before. I'm really, really unsure how to even do this, but I assumed that psi is regarded as scalar (or otherwise does not have dependency on r, theta or phi).

$\psi$ is a scalar function, but it definitely depends on $r$. You are told in the problem to substitute the expression for it in 2-58 into 2-55. Why don't you actually try doing that?

∫∫∫(-δ(x)δ(y)δ(z))r2sin(θ)dr dθ d∅
= -∫∫∫r2sin(θ)dr dθ d∅
= -(4π/3)r03

Where I've integrated from r = 0 to r0, θ from 0 to pi, and ∅ from 0 to 2pi. Is this even correct?

Do the right hand side in Cartesian coordinates (you don't need to worry about limits as long as origin is inside the volume).

∫∫∫(1/r2)(∂/∂r)[r2(∂/∂r)] (ψ r2sin(θ)dr dθ d∅)
= ∫∫∫(1/r2)(∂/∂r)[r2 (ψ 2r sin(θ)dr dθ d∅)]
= ∫∫∫(1/r2)(∂/∂r)[2r3 (ψ sin(θ)dr dθ d∅)]
= ∫∫∫(1/r2)[6r2 (ψ sin(θ)dr dθ d∅)]
= ∫∫∫[6 (ψ sin(θ)dr dθ d∅)]
= 4ψ∏r0

That's not correct...
The radial part of the Laplacian is $$\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial \psi}{\partial r}\right)$$ The derivatives hit $\psi$.

The key here is to notice that ψ diverges in the origin, so you can't just plug it into the formulas and differentiate. The most intuitively clear way to proceed is to use the divergence theorem on the first term. You should have something like $$\int dV \nabla^2 \psi = \int dS \nabla \psi \cdot \hat{n}.$$ Take the surface to be a sphere, assume that ψ does not depend on the angles and you can evaluate the integral easily.

## 1. What is the Helmholtz equation and why is it important in science?

The Helmholtz equation, also known as the wave equation, is a partial differential equation that describes the propagation of waves in a medium. This equation is important in various fields of science, such as physics, engineering, and acoustics, as it allows us to understand and predict the behavior of wave phenomena.

## 2. How is the Helmholtz equation solved for a point source?

The Helmholtz equation for a point source can be solved using Green's function, which is a mathematical tool that helps us find the solution to a differential equation with a given source function. By using Green's function, we can represent the wave field as a superposition of infinitesimal point sources, which allows us to find the solution for a single point source.

## 3. What are the applications of solving the Helmholtz equation for a point source?

Solving the Helmholtz equation for a point source has various applications, including predicting the behavior of sound waves in a room, designing acoustic devices, and understanding the propagation of electromagnetic waves. It is also essential in fields such as medical imaging, seismology, and oceanography.

## 4. What are the challenges in solving the Helmholtz equation for a point source?

One of the main challenges in solving the Helmholtz equation for a point source is the high computational cost. As the number of point sources increases, the computational time and memory requirements also increase, making it challenging to solve for complex systems. Additionally, the numerical methods used to solve this equation may introduce errors, which can affect the accuracy of the results.

## 5. Are there any alternative methods for solving the Helmholtz equation for a point source?

Yes, there are alternative methods for solving the Helmholtz equation for a point source, such as the finite difference method, finite element method, and boundary element method. These methods use different approaches to approximate the solution to the equation and have their own advantages and limitations. The choice of method depends on the specific problem and the desired accuracy of the solution.

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