- #1

JohanL

- 158

- 0

[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?