dingo_d
Jan26-10, 09:28 AM
1. The problem statement, all variables and given/known data
Plate in the shape of the circular halo (inner radius a, outer radius b>a), the inner edge is being kept at a constant temperature T_0, and the outer at the temperature given by the function f(\phi)=T_0\cos(2\phi). Find the equilibrium distribution of the heat everywhere inside the plate.
2. Relevant equations
Laplace equation:
\nabla^2 u(r,\phi)=0
Boundary conditions:
u(a,\phi)=T_0
u(b,\phi)=T_0\cos(2\phi)
3. The attempt at a solution
By separation of variables I get:
r^2\frac{R''}{R}+r\frac{R'}{R}=-\frac{\Phi''}{\Phi}=\lambda^2
For \lambda\neq 0 I have 2 DE, one is Euler's DE, and the other the equation of harmonic oscillator. Their solution is:
R(r)=Ar^\lambda+Br^{-\lambda}
\Phi(\phi)=C\cos(\lambda\phi)+D\sin(\lambda\phi)
With the condition of consistency of the azimuthal part:
\Phi(\phi)=\Phi(\phi+2\pi)
I get that \lambda=m\in\mathbb{Z}, the \lambda=0 gives different solutions.
So my first solution is:
u_1(r,\phi)=\sum_{m=1}^\infty r^m[A_m\cos(m\phi)+B_m\sin(m\phi)]+\sum_{m=1}^\infty r^{-m}[C_m\cos(m\phi)+D_m\sin(m\phi)]
For \lambda=0 I have these solutions:
R(r)=C_0\ln(r)+D_0
\Phi(\phi)=A_0\phi+B_0
With the condition of consistency I get:
A_0\phi+B_0=A_0\phi+A_02\pi+B_0\Rightarrow A_0=0
So the second solution is: (I have put the constants together)
u_2(r\phi)=C\ln(r)+D
The general solution is the superposition of the two solutions:
u(r,\phi)=C\ln(r)+D+\sum_{m=1}^\infty r^m[A_m\cos(m\phi)+B_m\sin(m\phi)]+\sum_{m=1}^\infty r^{-m}[C_m\cos(m\phi)+D_m\sin(m\phi)]
So here is where my problem starts. I know that I'm supposed to use formulas for Fourier summation but in the case of a string that was the more straightforward, for example:
f(x,0)=\sum_{m=0}^\infty A_m\sin\left(\frac{m\pi}{L}x\right)\Big/ \cdot \sin\left(\frac{n\pi}{L}x\right),\ \int_0^L dx
And the integral on the right would give me Kronecker delta which would 'kill' the sum so I would get the A_n from that. But here I don't have this simple way. How should I get the A_m,\ B_m,\ C_m,\ D_m?
Plate in the shape of the circular halo (inner radius a, outer radius b>a), the inner edge is being kept at a constant temperature T_0, and the outer at the temperature given by the function f(\phi)=T_0\cos(2\phi). Find the equilibrium distribution of the heat everywhere inside the plate.
2. Relevant equations
Laplace equation:
\nabla^2 u(r,\phi)=0
Boundary conditions:
u(a,\phi)=T_0
u(b,\phi)=T_0\cos(2\phi)
3. The attempt at a solution
By separation of variables I get:
r^2\frac{R''}{R}+r\frac{R'}{R}=-\frac{\Phi''}{\Phi}=\lambda^2
For \lambda\neq 0 I have 2 DE, one is Euler's DE, and the other the equation of harmonic oscillator. Their solution is:
R(r)=Ar^\lambda+Br^{-\lambda}
\Phi(\phi)=C\cos(\lambda\phi)+D\sin(\lambda\phi)
With the condition of consistency of the azimuthal part:
\Phi(\phi)=\Phi(\phi+2\pi)
I get that \lambda=m\in\mathbb{Z}, the \lambda=0 gives different solutions.
So my first solution is:
u_1(r,\phi)=\sum_{m=1}^\infty r^m[A_m\cos(m\phi)+B_m\sin(m\phi)]+\sum_{m=1}^\infty r^{-m}[C_m\cos(m\phi)+D_m\sin(m\phi)]
For \lambda=0 I have these solutions:
R(r)=C_0\ln(r)+D_0
\Phi(\phi)=A_0\phi+B_0
With the condition of consistency I get:
A_0\phi+B_0=A_0\phi+A_02\pi+B_0\Rightarrow A_0=0
So the second solution is: (I have put the constants together)
u_2(r\phi)=C\ln(r)+D
The general solution is the superposition of the two solutions:
u(r,\phi)=C\ln(r)+D+\sum_{m=1}^\infty r^m[A_m\cos(m\phi)+B_m\sin(m\phi)]+\sum_{m=1}^\infty r^{-m}[C_m\cos(m\phi)+D_m\sin(m\phi)]
So here is where my problem starts. I know that I'm supposed to use formulas for Fourier summation but in the case of a string that was the more straightforward, for example:
f(x,0)=\sum_{m=0}^\infty A_m\sin\left(\frac{m\pi}{L}x\right)\Big/ \cdot \sin\left(\frac{n\pi}{L}x\right),\ \int_0^L dx
And the integral on the right would give me Kronecker delta which would 'kill' the sum so I would get the A_n from that. But here I don't have this simple way. How should I get the A_m,\ B_m,\ C_m,\ D_m?