Resonant oscillations of gases and liquids in three dimensions
Although extensive work has been carried out on one-dimensional resonant oscillations of both liquids in a tank (where the free surface varies in only one spatial dimension) and gases in a resonator, little is known about two-dimensional solutions. This thesis aims to unite and extend the knowledge about one-dimensional solutions and also develop a theory for classifying two-dimensional motions and, as a consequence, understand the different types of responses that may be found in tanks and resonators of arbitrary geometry. To do this we focus on (i) the nonlinearity and (ii) the geometry (and, hence, the nature of the spectrum) and ignore dissipation to lowest order although it is, in general, important. However, we can easily include dissipative effects a posteriori and its initial absence makes it easier to analyse the new two-dimensional effects. For reasons which will become apparent, we will mainly consider cuboid-shaped geometries and perturb the sidewalls of such tanks and resonators, allowing for the gradual introduction of two-dimensional effects. The thesis is split into two parts, underlying the differences between the problems that arise when the spectrum of the relevant linear problem is commensurate or non-commensurate. After a general introduction in Chapter 1 and a discussion of the model and governing equations in Chapter 2, the first part, comprising Chapter 3, looks at oscillations in deep water where the response typically consists of a finite number of modes. The second part is more extensive, looking at shallow water sloshing and the analogies of this problem with acoustic oscillations, both of which have a spectrum containing an infinite set of commensurate frequencies and the solution is much more intricate. We develop the problem and its one-dimensional solutions in Chapter 4 and then extend these ideas to two-dimensions in Chapter 5. With all this in mind we then make some general remarks about oscillations in tanks and resonators of arbitrary geometry in Chapter 6.