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- Problem Statement
- This is a problem from the book Physics of Waves. I have successfully obtained the dispersion relations as required but I have trouble working out the rest of the problem. The problem is as follows in the images.

- Relevant Equations
- Kirchhoff's Voltage Law

We know that the charge on capacitors as a function of time takes the general form of:

##Q(x,t)=qe^{ijka}e^{-i\omega t}##

The voltage at each capacitor:

##V_j = \frac 1 C (Q_j-Q_{i+1})##

From KVL we have differential equation of t-derivatives:

##LQ'' + RQ' = V_{j-1} - V_{j}##

##LQ''+RQ'= \frac 1 C (Q_{j-1}-Q_{j}) - \frac 1 C (Q_j-Q_{j+1}) ##

##(-\omega^2 L - i \omega R)qe^{ijka}e^{-i\omega t} = \frac 1 C ((qe^{i(j-1)ka}e^{-i\omega t}-qe^{ijka}e^{-i\omega t}) - (qe^{i(j)ka}e^{-i\omega t}-qe^{i(j+1)ka}e^{-i\omega t})##

Rearrange and cancel out the equivalent terms:

##(\omega^2 L + i \omega R) = \frac 1 C ((1-qe^{ika}) - (1-qe^{ika})e^{-ika}))##

##(\omega^2 L + i \omega R) = \frac 1 C (1-qe^{ika}) (1-e^{-ika})##

Multiply the above terms:

##(\omega^2 L + i \omega R) = \frac 1 C (2-2cos(ka))##

##(\omega^2 + \frac R L i \omega ) = \frac 2 {LC} (1-cos(ka))##

The above is answer for part A, for part B, I only got one boundary condition, but I can't get the other one and I don't know how to proceed

##Q_0(x,t) = 0##

For the other one, ##V_6 = V_0 cos(\omega t)##

Can someone help me?