# Solving the 3D Helmholtz Equation Directly

• lth
In summary, the 3D Helmholtz equation is a partial differential equation used to model wave propagation in three-dimensional space. It is difficult to solve due to its second-order nature and variable coefficients. Solving it directly allows for a more accurate and efficient solution compared to numerical approximations, and has practical applications in various fields such as acoustics, optics, and fluid dynamics. Techniques for solving it include analytical methods such as separation of variables and integral transforms, as well as numerical methods like finite difference, finite element, and spectral methods.
lth

## Homework Statement

Given 3D Helmholtz eqn.
u_xx + U-yy + U_zz + Lamda*u = 0 ,Lamda > 0.

We are asked to "Calculate the fundamental solution directly (without using the Bessel identity for J_1/2 given)"
where:
Bessel identity given is w(r)=C_n*r^-(n-2)/2*J-(n-2)/2*(Lamda^1/2*r) ,n=odd ... and in this case n=3.

## Homework Equations

fundamental soln is u = (1/4*Pi*r_PQ)*Cos(Lamda^1/2*r_PQ)

## The Attempt at a Solution

After transforming to r-space and setting delta functions on RHS
u_rr + (2/r) * u_r + lambda * u = delta_3(r)

A second transformation from letting u(r) = r^1/2*v(theta) ;theta =Lamda^1/2*r
yields:

d^2v/d{theta}^2 + (1/theta)*dv/d{theta} +[1-1/4*theta]*v =0
and this is Bessels eqn. form.

This is from Kevorkian PDE text p2.3.4 and I feel like i am running in circles with this material and trying to figure how to solve the fundamental solution without using Bessels identity. Can someone give me some insight into what to do next?

thanks, lth

ep

Dear lthep,

Thank you for sharing your attempt at solving the problem. It seems like you are on the right track in transforming the Helmholtz equation into Bessel's equation form. From here, you can use the standard solutions for Bessel's equation, which are Bessel functions of the first and second kind (J and Y functions). The fundamental solution for the Helmholtz equation can then be written as a linear combination of these Bessel functions. You can also use the Wronskian to determine the coefficients in the linear combination.

I hope this helps guide you towards finding the fundamental solution without using the Bessel identity. Good luck with your work!

## 1. What is the 3D Helmholtz equation?

The 3D Helmholtz equation is a partial differential equation that describes wave propagation in three-dimensional space. It is commonly used in mathematical physics and engineering to model phenomena such as sound waves, electromagnetic waves, and fluid flow.

## 2. Why is solving the 3D Helmholtz equation difficult?

The 3D Helmholtz equation is difficult to solve because it is a second-order partial differential equation with variable coefficients. This means that the solution depends on the location in space and the frequency of the wave, making it a highly complex problem.

## 3. What are the advantages of solving the 3D Helmholtz equation directly?

Solving the 3D Helmholtz equation directly allows for a more accurate and efficient solution compared to numerical approximations. It also provides a better understanding of the underlying physical phenomena and can be used to predict and analyze wave behavior in various systems.

## 4. What are some techniques for solving the 3D Helmholtz equation directly?

Some techniques for solving the 3D Helmholtz equation directly include analytical methods such as separation of variables and integral transforms, as well as numerical methods such as finite difference, finite element, and spectral methods.

## 5. What are the practical applications of solving the 3D Helmholtz equation directly?

Solving the 3D Helmholtz equation directly has many practical applications, such as in acoustics, optics, electromagnetics, and fluid dynamics. It is used to design and optimize devices such as antennas, optical systems, and acoustic transducers, and to model wave phenomena in various fields of science and engineering.

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