$$ V(x, t) = L\Delta x\frac{\partial I}{\partial t}(x, t) + \frac{1}{C \Delta x}\int{I_c(x, t) \; dt} \tag{1} $$ $$ V(x + \Delta x, t) =\frac{1}{C \Delta x} \int {I_c(x, t) \; dt} \tag{2} $$ $$ I(x) = I_c + I(x + \Delta x) \tag{3} $$ This are the equations that describe the circuit once applied Kirchhoff Laws. Using equations 1 and 2: $$ 0 = L\frac{\partial I}{\partial t}(x, t) + \frac{V(x + \Delta x, t) - V(x, t)}{\Delta x} \\ $$ $$ \boxed{ L\frac{\partial I}{\partial t} + \frac{\partial V}{\partial x} = 0 }\tag 4 $$ Now using equations 3: $$ I(x) = I_c + I(x + \Delta x) \Rightarrow I(x + \Delta x) - I(x) = -I_c \\ I_c = C \Delta x \frac{\partial V_c}{dt} \\ $$ $$ \boxed { C \frac{\partial V_c}{\partial t} + \frac{\partial I}{\partial x} = 0 } \tag 5 $$ It would be nice if the $V_c = V$, because the equations would be "simetric". Let's try to proof that $V_c = V$: $$ V(x + \Delta x) = V(x) + \Delta x \frac{\partial V}{\partial x} = \frac{1}{C \Delta x}\int { I_c \; dt} \\ I_c = C \Delta x \frac{\partial V_c}{\partial t} \\ V(x) + \Delta x \frac{\partial V}{\partial x} = V_c \\ \lim_{\Delta x \to 0} \left [ V(x) + \Delta x \frac{\partial V}{\partial x} \right] = \lim_{\Delta x \to 0} V_c \\ $$ $$ \boxed{V(x, t) = V_c(x, t)} $$ Hence we have the simetry we wanted. Now, playing a little bit with equations 4 and 5 we get: $$ L \frac{\partial^2 I}{\partial t^2} + \frac{\partial }{\partial x}\left ( \frac{\partial V}{\partial t} \right) = 0 \\ \frac{\partial V}{\partial t} = -\frac{1}{C}\frac{\partial I}{\partial x} \\ L \frac{\partial^2 I}{\partial t^2} + \frac{\partial }{\partial x}\left ( -\frac{1}{C}\frac{\partial I}{\partial x} \right) = 0 \\ $$ We get the following wave equation (same happends for $V$): $$ \boxed{ LC \frac{\partial^2 I}{\partial t^2} = \frac{\partial^2 I}{\partial x^2} \\ \; \\ \; LC \frac{\partial^2 V}{\partial t^2} = \frac{\partial^2 V}{\partial x^2} } $$ With a propagation speed of: $$ v = \frac{1}{\sqrt{LC}} $$ Is this line of reasoning correct? Is there any step I skipped or something I've had not taken into account? Sorry if the derivation is a bit caotic, hope it's understandable.
