Phase-Space Representations of Elastic Wave Problems #

Rory Collett

Poster session

Abstract #

In this project we utilise phase space representation techniques to analyse solutions to elastic wave problems in engineering. In particular, analysing vibrations on a Bernoulli beam. Linear wave problems, such as the one considered here, are widespread in physics and engineering, from the Schrodinger equation to the wave equation of a string, they describe a variety of scenarios important for understanding the world around us. However, linear wave problems at high frequencies are notoriously difficult to model numerically.

To resolve this difficulty, we utilise high-frequency asymptotics to numerically model high-frequency problems where the wave length is small relative to the length scale of the system. These asymptotics lead directly to Hamilton’s equations, a subsection of dynamical systems and ODEs. As a result, these systems lack the typical wave behaviour we are used to observing; diffraction, interference, and resonances. So how can these methods be useful, if they don’t display some of the most basic characteristics which we know waves to exhibit?

Despite these shortfalls, we can extract useful information from these asymptotic methods through the calculation of so-called Wigner and Husimi Functions. These are well-known in the physics community for understanding semi-classical behaviour, where quantum meets classical. As such, they haven’t been applied to many engineering problems.