Solving Helmholtz Equation in a Hollow Cylinder

In summary, the conversation discusses the process of solving Helmholtz equation in a hollow cylinder with specified length and boundary conditions. The problem is separated into three sub-problems, each of which can be solved separately using separation of variables. The general solutions of Bessel's equation are also mentioned. The conversation ends with a suggestion to read more on Bessel functions for further understanding and solving the problem.
  • #1
JohanL
158
0
Im trying to solve Helmholtz equation

[tex]
\nabla ^2u(r,\phi,z) + k^2u(r,\phi,z) = 0
[/tex]

in a hollow cylinder with length L and a < r < b
and the boundary conditions:

[tex]
u(a,\phi,z) = F(\phi,z)
[/tex]
[tex]
u(b,\phi,z) = G(\phi,z)
[/tex]
[tex]
u(r,\phi,0) = P(\phi,z)
[/tex]
[tex]
u(r,\phi,L) = Q(\phi,z)
[/tex]
[tex]
u(r,\phi,z) = u(r,\phi + 2\pi,z)
[/tex]

Solution:

With
[tex]
u(r,\phi,z) = v_1(r,\phi,z) + v_2(r,\phi,z) + v_3(r,\phi,z)
[/tex]

i get three problems which i can solve separately.
Separation of variables gives 9 d.e. Three of them are bessel equations.

[tex]
r\frac {d} {dr}(r\frac {dR_i(r)} {dr}) + (\mu_i^2r^2-m_i^2)R_i(r) = 0
[/tex]

i = 1,2,3. and [tex] \mu, m [/tex] are separation constants.

The boundary conditions are

[tex]
R_1(a,\phi,z) = F(\phi,z),
R_1(b,\phi,z) = G(\phi,z)
[/tex]

[tex]
R_2(a,\phi,z) = 0,
R_2(b,\phi,z) = 0
[/tex]

[tex]
R_3(a,\phi,z) = 0,
R_3(b,\phi,z) = 0
[/tex]


The general solutions of Bessels equation are

[tex]
R = C_1 J_m(nr) + C_2 N_m(nr)
[/tex]

where J_m is the mth bessel function of the first kind and N_m is the mth neumann function (or bessel function of the second kind)

I don't know how to continue with the boundary conditions.
Any ideas?
 
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  • #2
There are some theorems which under certain conditions allow you expand any function (namely the functions which appear as boundary conditions) in terms of Bessel functions.Note that these functions are not orthonormal polynomials (so no Hilbert space here),but that still doesn't prevent this from happening.

So i suggest you read more on the Bessel functions.Gray & Matthews wrote a monography.And there are tons of other useful books.

Daniel.
 
  • #3


Firstly, it is important to note that the Helmholtz equation is a second-order partial differential equation, and therefore, three boundary conditions are needed for a unique solution. In this case, we have four boundary conditions, but the periodic boundary condition u(r,\phi,z) = u(r,\phi+2\pi,z) can be incorporated into the general solution, as it is already satisfied by the Bessel functions.

To continue with the boundary conditions, we can use the orthogonality property of Bessel functions. This property states that for any two different values of m, the integral of J_m(nr)N_m(nr)rdr from 0 to b will be equal to zero. Therefore, we can use this property to find the coefficients C_1 and C_2 in the general solution, by equating the integrals of the Bessel functions multiplied by the given boundary conditions to zero. This will give us a system of equations that can be solved to find the coefficients.

Once we have the coefficients, we can substitute them back into the general solution to get the complete solution for each of the three functions v_1, v_2, and v_3. Then, by adding them together, we will have the solution for u(r,\phi,z).

Finally, we can use the boundary conditions at z=0 and z=L to find the constants in the general solution for the third variable, z. This will give us the complete solution for the Helmholtz equation in the hollow cylinder.
 

Related to Solving Helmholtz Equation in a Hollow Cylinder

1. What is the Helmholtz Equation and why is it important?

The Helmholtz Equation is a partial differential equation that describes the behavior of waves in a medium, such as sound or electromagnetic waves. It is important because it allows us to understand and solve many real-world problems related to wave propagation and interference.

2. How is the Helmholtz Equation solved in a hollow cylinder?

The solution to the Helmholtz Equation in a hollow cylinder involves finding a set of eigenvalues and eigenfunctions that satisfy the boundary conditions of the problem. These eigenfunctions form a basis for the solution, which can then be expressed as a linear combination of the eigenfunctions.

3. What are the boundary conditions for solving the Helmholtz Equation in a hollow cylinder?

The boundary conditions typically involve specifying the behavior of the wave at the inner and outer surfaces of the cylinder. This can include conditions such as fixed or free boundary conditions, or specific values for the wave function or its derivatives at the boundaries.

4. Are there any limitations to solving the Helmholtz Equation in a hollow cylinder?

One limitation is that the problem must be axisymmetric, meaning that the solution is independent of the angular coordinate around the cylinder. This greatly simplifies the problem and allows for an analytical solution. Additionally, the cylinder must have a constant cross-sectional area and homogeneous material properties.

5. Can the Helmholtz Equation in a hollow cylinder be solved numerically?

Yes, the Helmholtz Equation can also be solved numerically using methods such as finite difference or finite element analysis. These methods are useful when the problem is not axisymmetric or the boundary conditions are more complex.

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