Solving Helmholtz Equation in a Hollow Cylinder

[JohanL]

Im trying to solve Helmholtz equation

<br /> \nabla ^2u(r,\phi,z) + k^2u(r,\phi,z) = 0 <br />

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

<br /> u(a,\phi,z) = F(\phi,z)<br />
<br /> u(b,\phi,z) = G(\phi,z) <br />
<br /> u(r,\phi,0) = P(\phi,z) <br />
<br /> u(r,\phi,L) = Q(\phi,z)<br />
<br /> u(r,\phi,z) = u(r,\phi + 2\pi,z)<br />

Solution:

With
<br /> u(r,\phi,z) = v_1(r,\phi,z) + v_2(r,\phi,z) + v_3(r,\phi,z) <br />

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

<br /> r\frac {d} {dr}(r\frac {dR_i(r)} {dr}) + (\mu_i^2r^2-m_i^2)R_i(r) = 0<br />

i = 1,2,3. and \mu, m are separation constants.

The boundary conditions are

<br /> R_1(a,\phi,z) = F(\phi,z),<br /> R_1(b,\phi,z) = G(\phi,z)<br />

<br /> R_2(a,\phi,z) = 0,<br /> R_2(b,\phi,z) = 0<br />

<br /> R_3(a,\phi,z) = 0,<br /> R_3(b,\phi,z) = 0<br />


The general solutions of Bessels equation are

<br /> R = C_1 J_m(nr) + C_2 N_m(nr)<br />

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?

 

[dextercioby]

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.

 

[thepatientmental]

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.