Analysis of Telegraph Equation for Propagating Waves with Dispersion and Attenuation

The structural vibration of industrial droplet dispensers can be modeled by telegraph-like equations to a good approximation. We reinterpret the telegraph equation from the standpoint of an electric-circuit system consisting of an inductor and a resistor, which is in interaction with an environment, say, a substrate. This interaction takes place through a capacitor and a shunt resistor. Such interactions serve as leakage. We have performed analytical investigation of the frequency dispersion of telegraph equations over unbounded one-dimensional domain. By varying newly identified key parameters, we have not only recovered the well-known characteristics but also discovered crossover phenomena regarding phase and group velocities. We have examined frequency responses of the electric circuit underlying telegraph equations, thereby confirming the role as low-pass filters. By identifying a set of physically meaningful reduced cases, we have laid foundations on which we could further explore wave propagations over finite domain with appropriate side conditions.