# Modelling a short square-wave pulse with Telegrapher's Equation

 P: 1 I am interested in looking at the voltage component of a short (possibly nanosecond width) pulse in a coaxial cable along the cable at a given distance, Z, along the line. I am "cheating" by using the steady state Telegrapher's equation [Eq. 1] and formulating a periodic pulse approximation. It may not be desirable to use this steady state equation, but some form of the Telegrapher's Eq. is advised (suggestions for alternatives are appreciated). This is unlike my "physical" setup in which I will send a single pulse of information (microwave signal). I have determined Fourier approximation [Eq. 2] of my input pulse and the Fourier Coefficient [A]. My understanding is that the infinite series of circuits, as described by the Telegrapher's Equation, is dependent on frequency. I also recognize that I am going to use he Coefficent [A] as my "weighted" distribution of frequencies to calculate my inherent, frequency dependent components. Eq. 1 $$\frac{\partial}{\partial x} V(x,t) =-L \frac{\partial}{\partial t} I(x,t) - R I(x,t)$$ $$\frac{\partial}{\partial x} I(x,t) = -C \frac{\partial}{\partial t} V(x,t) - G V(x,t)$$ Eq. 2$$\frac{1}{2T} + \sum_{n=1}^{200} a_n \cos(\frac{2\pi n t}{T}) + b_n \sin(\frac{2\pi n t}{T})$$ In my summation, 200 is the current limit as to not consume too much time approximating in Mathematica. where an is $$\frac{2}{T}\int^{\frac{T}{2}}_{0}{\cos(\frac{2 \pi n t}{T})$$ and bn is $$\frac{2}{T}\int^{\frac{T}{2}}_{0}{\sin(\frac{2 \pi n t}{T})$$ NB: I am only using an integration for half of my period because the square-wave is 1 from 0 to $$\frac{T}{2}$$ and 0 from $$\frac{T}{2}$$ to $$T$$; Therefore, the second half of the period has an integration of zero. My Fourier Coefficient is represented by $$b_n$$, since $$a_n$$ is effectively zero. Therefore, my "weighted" frequency formula, that determines the frequency dependent terms of the Telegrapher's Equation is $$b_n$$. Now, I have to admit, I know I should solve [Eq.1] with Dirichlet approach, due to the boundary conditions, but that is the limit of my knowledge on the matter. I want to know what my pulse looks like at some t, or z. Can anyone offer some insight into solving this initial equation and determining the dispersed signal's wave equation? NB2: As I am using Mathematica, I can attach the existing notebook if desired.