Acoustic and quantum wave propagators

Finding numerical solutions to complex problems is a vital goal in almost all areas of research, where analytic solutions are unavailable and only computer-generated approximations are possible. These computational solutions and simulations are themselves critically useful for studying complex problems, observing behaviours as systems evolve through time, and enabling the design of various devices and structures. This work examines the modified J-Chebyshev polynomial expansion scheme and demonstrates how it can be an effective, accurate and efficient means of providing numerical solutions to complex physical problems. It applies the expansion to the solution of two very different yet similarly structured wave propagation problems. The first is the acoustic wave equation, which governs the propagation of sound, and the second is the Schrödinger equation, which describes the quantum behaviour of electrons. Both of these exhibit wave behaviour and can be reduced to the action of an exponential operator on an initial state vector. This work develops algorithms to solve both of these, finding either good agreement with analytic solutions or potential flaws in the models. It uses these algorithms to model two respective important applications. The first is the design of industrial noise barriers, with a focus on an innovative new wave-trapping profile that reduces internal reflections. By comparing sound pressure levels and performing wave analysis, the wave-trapping barrier was shown to be an effective means of reducing internally reflected industrial noise. Further work is carried out to optimize its design for maximum effect in both short and long ranges, with gains of up to 1.33 dB over a flat barrier possible through geometric design alone. The second application is a simulation of electron transport in a surface acoustic wave through a semiconductor quasi-one-dimensional channel and over a potential barrier, for the purposes of a flying qubit quantum computation scheme. This is successfully simulated, but a severe limitation is discovered in the problem of false wave leakage, due to the moving potential introducing errors, showing that further development is necessary. It also examines the use of an efficient split region technique and its associated benefits and problems.