- #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?
[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?