A shaped charge is an explosive device used to penetrate thick targets using a high velocity jet. A typical shaped charge contains explosive material behind a conical hollow. The hollow is lined with a compliant material, such as copper. Extremely high stresses caused by the detonation of the explosive have a focusing effect on the liner, turning it into a long, slender, stretching jet with a tip speed of up to 12km/s. A mathematical model for the penetration of this jet into a solid target is developed with the goal of accurately predicting the resulting crater depth and diameter. The model initially couples fluid dynamics in the jet with elastic-plastic solid mechanics in the target. Far away from the tip, the high aspect ratio is exploited to reduce the dimensionality of the problem by using slender body theory. In doing so, a novel system of partial differential equations for the free-boundaries between fluid, plastic and elastic regions and for the velocity potential of the jet is obtained. In order to gain intuition, the paradigm expansion-contraction of a circular cavity under applied pressure is considered. This yields the interesting possibility of residual stresses and displacements. Using these ideas, a more realistic penetration model is developed. Plastic flow of the target near the tip of the jet is considered, using a squeeze-film analogy. Models for the flow of the jet in the tip are then proposed, based on simple geometric arguments in the slender region. One particular scaling in the tip leads to the consideration of a two-dimensional paradigm model of a ``filling-flow'' impacting on an obstacle, such as a membrane or beam. Finally, metallurgical analysis and hydrocode runs are presented. Unresolved issues are discussed and suggestions for further work are presented.

This thesis examines certain aspects of diffraction and scattering of high frequency waves, utilising and extending upon the Geometrical Theory of Diffraction (GTD). The first problem considered is that of scattering of electromagnetic plane waves by a perfectly conducting thin body, of aspect ratio O(k^1/2), where k is the dimensionless wavenumber. The edges of such a body have a radius of curvature which is comparable to the wavelength of the incident field, which lies inbetween the sharp and blunt cases traditionally treated by the GTD. The local problem of scattering by such an edge is that of a parabolic cylinder with the appropriate radius of curvature at the edge. The far field of the integral solution to this problem is examined using the method of steepest descents, extending the recent work of Tew [44]; in particular the behaviour of the field in the vicinity of the shadow boundaries is determined. These are fatter than those in the sharp or blunt cases, with a novel transition function. The second problem considered is that of scattering by thin shells of dielectric material. Under the assumption that the refractive index of the dielectric is large, approximate transition conditions for a layer of half a wavelength in thickness are formulated which account for the effects of curvature of the layer. Using these transition conditions the directivity of the fields scattered by a tightly curved tip region is determined, provided certain conditions are met by the tip curvature. In addition, creeping ray and whispering gallery modes outside such a curved layer are examined in the context of the GTD, and their initiation at a point of tangential incidence upon the layer is studied. The final problem considered concerns the scattering matrix of a closed convex body. A straightforward and explicit discussion of scattering theory is presented. Then the approximations of the GTD are used to find the first two terms in the asymptotic behaviour of the scattering phase, and the connection between the external scattering problem and the internal eigenvalue problem is discussed.

The aim of this thesis is to derive and solve mathematical models for the flow of liquid in a foam. A primary concern is to investigate how so-called `Marangoni stresses' (i.e. surface tension gradients), generated for example by the presence of a surfactant, act to stabilise a foam. We aim to provide the key microscopic components for future foam modelling. We begin by describing in detail the influence of surface tension gradients on a general liquid flow, and various physical mechanisms which can give rise to such gradients. We apply the models thus devised to an experimental configuration designed to investigate Marangoni effects. Next we turn our attention to the flow in the thin liquid films (`lamellae') which make up a foam. Our methodology is to simplify the field equations (e.g. the Navier-Stokes equations for the liquid) and free surface conditions using systematic asymptotic methods. The models so derived explain the `stiffening' effect of surfactants at free surfaces, which extends considerably the lifetime of a foam. Finally, we look at the macroscopic behaviour of foam using an ad-hoc averaging of the thin film models.

In this thesis, we discuss a geometric construction analogous to the Ward correspondence for the KP equations. We propose a Dirac operator based on the inverse scattering transform for the KP-II equation and discuss the similarities and differences to the Ward correspondence. We also consider the KP-I equation, describing a geometric construction for a certain class of solutions. We also discuss the general inverse scattering of the equation, how this is related to the KP-II equation and the problems with describing a single geometric construction that incorporates both equations. We also consider the Davey-Stewartson equations, which have a similar behaviour. We demonstrate explicitly the problems of localising the theory with generic boundary conditions. We also present a reformulation of the Dirac operator and demonstrate a duality between the Dirac operator and the first Lax operator for the DS-II equations. We then proceed to generalise the Dirac operator construction to generate other integrable systems. These include the mKP and Ishimori equations, and an extension to the KP and mKP hierarchies.

The theory of nonlinear mass-spring chains has a history stretching back to the now famous numerical simulations of Fermi, Pasta and Ulam. The unexpected results of that experiment have led to many new fields of study. Despite this, the mathematics of the lattice equations have proved sufficiently rich to attract continued attention to the present day. This work is concerned with the motions of an infinite one dimensional lattice with nearest-neighbour interactions governed by a generic potential. The Hamiltonian of such a system may be written $H = \sum_{i=-\infty}^{\infty} \, \Bigl(\frac{1}{2}p_i^2 + V(q_{i+1}-q_i)\Bigr)$, in terms of the momenta $p_i$ and the displacements $q_i$ of the lattice sites. All sites are assumed to be of equal mass. Certain generic conditions are placed on the potential $V$. Of particular interest are the solitary wave solutions which are known to exist upon such lattices. The KdV equation has long been known to emerge in a formal manner from the lattice equations as a continuum limit. More recently, the lattice's localized nonlinear modes have been rigorously approximated by the KdV's well-studied soliton solution, in the lattice's long wavelength regime. To date, however, little is known about how, and to what extent, lattice solitary waves differ from KdV solitons. It is proved in this work that a solution (which we prove to be unique) to a particular linear ordinary differential equation provides a correction to the KdV approximation. This gives, in an explicit way, the lowest order effect of lattice discreteness upon lattice solitary waves. It is also shown how such discreteness effects are propagated along the lattice both in isolation (single soliton case), and in the presence of another soliton correction (the bisoliton case). In the latter case their interaction is studied and the impact of lattice discreteness upon lattice solitary wave interactions is observed. This is possible by virtue of the discovery of an evolution equation for discreteness effects on the lattice. This equation is proved to have appropriate unique solutions and is found to be strikingly similar to corresponding equations known in both the theories of shallow water waves and ion-acoustic waves.

The Schrödinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrödinger equation is the gravity due to the density of $|\psi|^2$, where $\psi$ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the $(n+1)$th state has $n$ zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for $n$ quadruples of complex eigenvalues for the $(n+1)$th state, where a quadruple consists of $\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}$. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrödinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.

The classical theory of channelized subglacial drainage,due orginally to Röthlisberger (1972) and Nye (1976), considers water flow in an ice channel overlying a rigid, impermeable bed. At steady flow, creep closure of the channel walls is counteracted by melt-back due to heat dissipation, and this leads to an equilibrium relation between channel water pressure and discharge. More generally, such a balance exhibits an instability that can be used to describe the mechanics of catastrophic flood events known as `jökulhlaups'. In this thesis, we substantiate these developments by exploring a detailed model where the channel is underlain by subglacial till and the flow supports a sediment load. Attention is given to the physics of bed processes and its effect on channel morphology. In particular, we propose a theory in which the channel need not be semi-circular, but has independently evolving depth and width determined by a local balance between melting and closure, and in which sediment erosion and deposition is taken into account. The corresponding equilibrium relation indicates a reverse dependence to that in the classical model, justifying the possibility of the subglacial canals envisaged by Walder and Fowler (1994). Theoretical predictions for sediment discharge are also derived. Regarding time-dependent flood drainage, we demonstrate how rapid channel widening caused by bank erosion can explain the abrupt recession observed in the flood hydrographs. This allows us to produce an improved simulation of the 1972 jökulhlaup from Grímsvötn, Iceland, and self-consistently, a plausible estimate for the total sediment yield. We also propose a mechanism for the observed flood initiation lake-level at Grímsvötn. These investigations expose the intimate interactions between drainage and sediment transport, which have profound implications on the hydrology, sedimentology and dynamics of ice masses, but which have received little attention.

This thesis is motivated by the need to calculate the electromagnetic fields produced by sources radiating in the presence of conductors. We begin by reviewing existing theory concerning sources in the presence of flat structures. Various extensions to the canonical Sommerfeld problem are considered. In particular we investigate the asymptotic solution for a finite source that focusses its energy at a point. In chapter 5 we review and extend the asymptotic results concerning illumination of a convex perfect conductor by an incident plane wave and outline the procedure for decoupling the electromagnetic surface field into two scalar modes. In chapter 6 we place a source on a perfect conductor and obtain a complete asymptotic solution for the fields. Special attention is paid to the asymptotic structure that smoothly matches between the leading order lit and shadow regions. We also investigate the degenerate case where one of the curvatures of the perfect conductor is zero. The case where the source is just off the surface is also investigated. In chapter 8 we use the Euler-Maclaurin summation formula to cheaply calculate the fields due to complicated arrays of point dipoles. The final chapter combines many earlier results to consider more general sources on the surface of a perfect conductor. In particular we must introduce new asymptotic regions for open sources. This then enables us to consider the focussing of the surface field due to a finite source. The nature of the surface and geometrical optics fields depends on the size of the source in comparison to the curvatures of the surface on which they lie. We discuss this in detail and conclude with the practical example of a spiral antenna.

Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. The principal aim of this thesis is to develop a series of mathematical models for somite formation. We begin by reviewing the current models for somitogenesis in the light of new experimental evidence regarding the presence of a segmentation clock and graded expression of FGF8. We conduct a preliminary investigation into the wavefront of FGF8 along the antero-posterior axis and integrate this model into the framework of an existing model for a signalling process. We demonstrate that this new “Clock and Wavefront” model can produce coherent series’ of somites in a manner that is tightly regulated in both space and time, and that it can also mimic the effects seen when FGF8 expression is perturbed locally. We then use the model to make some experimentally testable predictions. The latter part of the thesis concentrates on building more biologically accurate model for the FGF8 gradient. We move to consider a model for the FGF8 gradient which involves a complex network of biochemical interactions with negative feedback between FGF8 and retinoic acid. The resulting system of seven coupled non-linear equations, including both ordinary and partial differential equations, is difficult to analyse. To facilitate our understanding of the non-linear interactions between FGF8 and retinoic acid, we finally consider a reduced model which can display travelling wavefronts of opposing FGF8 and retinoic acid concentrations moving down the antero-posterior axis. The model allowed us to calculate a minimum wave speed for the wavefronts as a function of key model parameters such as the rate of FGF8 and retinoic acid decay; strong dependence on the values of these parameters is a result that is hypothesised to occur in vivo.

The aim of this thesis is to investigate the mathematics and modelling of the industrial crimper, perhaps one of the least well understood processes that occurs in the manufacture of artificial fibre. We begin by modelling the process by which the fibre is deformed as it is forced into the industrial crimper. This we investigate by presuming the fibre to behave as an ideal elastica confined in a two dimensional channel. We consider how the arrangement of the fibre changes as more fibre is introduced, and the forces that are required to confine it. Later, we apply the same methods to a fibre confined to a three dimensional channel. After the fibre has under gone a preliminary deformation, a second process known as secondary crimp can occur. This involves the `zig-zagged' material folding over. We model this process in two ways. First as a series of rigid rods joined by elastic hinges, and then as an elastic with a highly oscillatory natural configuration compressed by thrusts at each end. We observe that both models can be expressed in a very similar manner, and both predict that a buckle can occur from a nearly straight initial condition to an arched formation. We also compare the results to experiments performed on the crimped fibre. Throughout much of the process, the configuration of the fibre does not alter. This part of the process we call the block, and model the material in this region in two ways: as a series of springs; and as an isotropic elastic material. We discuss the coupling between the different regions and the process that occurs in the block, and consider both the steady state and stability of the system.