Application of Spectral Techniques in Solving the Wave Equation

Abstract

In this last chapter we apply spectral techniques to the wave equation and transmission lines. Transmission lines in a way supersede diffusion ones (last chapter) and in fact if we set the inductance to zero we fall back on the diffusion RC line. The wave equation governs the evolution of voltage and current along the T-line, both in space and time. It is second order both in space and time. To solve it we need initial conditions and boundary ones. We start the chapter with the un-driven line with either Neumann (fixed) or Dirichlet (tangential) boundary conditions and solve for various initial conditions. Then we move to the more general case with forced boundary conditions. The connecting link here is stimulating the line with a complex exponential and finding the solution which comes out in the form of forward propagating wave and backwards propagating one. Once that is known we can swap it with a general excitation and then use Fourier/Laplace transforms and superposition to find corresponding solution. In the chapter we examine physical parameters such as line input impedance, propagation constant, termination impedance, wave propagation, wave speed, wave reflection, and transmission. We also put emphasis on the impact of line dissipation (resistance) on wave propagation. In most cases we plot in 3D the wave propagation along the line for various time points and explain results. We also compare the two propagation phenomena of waves propagation and diffusion. We wrap the chapter with some notes on T-line input impedance and explain the meaning of zero or infinite impedance.