Trying to calculate normal modes of nearly infinite network LC circuits

[slantz]

Homework Statement

The first circuit has a capacitor with capacitance c and an inductor with inductance L. In series with this is another capacitor which is connected to the next loop in the circuit.

It look something like http://imgur.com/YJDaD.png"
Sorry for the crude drawing.

Homework Equations

The first part of the problem was to prove the equation could be written as dI²/dt²=w₀²(I_i-1-2I_i+I_i+1)

So that is a relevant equation and I have managed to do that just fine with Kirchoff's laws.

The Attempt at a Solution

The second part is to find the normal frequencies. Now I understand that this is very similar to a beaded string problem, or a discrete wave, however in the wave equation there is a Sin(k*x_i) and I cannot for the life of me figure out what the equivalent equation is for a circuit.
I know that the current will repeat both with the individual loops and each loop will repeat with time, but I cannot figure out how to represent the space part of the wave equation in a circuit setting. This will be necessary for me to solve the differential equation to get the normal modes...

Help?

 

[diazona]

Unless I'm missing something, the equation you have there can be written as
\frac{\mathrm{d}^2}{\mathrm{d}t^2}\begin{pmatrix}I_1 \\ I_2 \\ \vdots \\ I_{n-1} \\ I_{n}\end{pmatrix} = \omega_0^2\begin{pmatrix}-2 & 1 & 0 & 0 & \ddots \\ 1 & -2 & 1 & \ddots & 0 \\ 0 & 1 & \ddots & 1 & 0 \\ 0 & \ddots & 1 & -2 & 1 \\ \ddots & 0 & 0 & 1 & -2\end{pmatrix}\begin{pmatrix}I_1 \\ I_2 \\ \vdots \\ I_{n-1} \\ I_{n}\end{pmatrix}
Are you familiar with equations of this type, i.e. would you know how to solve it? It's the same mathematical procedure that is used in the discussion of coupled oscillators.

 

[slantz]

I feel like to find the resonant frequency in the matrix fashion I would just take the determinant of the matrix and set it to zero revealing the eigenvalues.

However to answer your question, no, I wouldn't know how to solve it, but am willing to do some reading if you send me in the correct direction.

I have taken linear algebra so it shouldn't be too difficult to learn, this material just has not been presented to me in that fashion just yet.