A field system capable of making high-resolution measurements of rainfall is described. The system incorporates a disdrometer, an array of high-resolution raingauges, a general-purpose data acquisition system and ancillary equipment. In an evaluation of the disdrometer, a theory allowing calculation of the effects of windspeed on detection efficiency is presented which has wide applicability. The raingauges are an improved design allowing 10-s temporal resolution of rainfall intensity and 100 m spatial resolution of rain-patch size when used in the array. An extensive data base of measurements made using the field system is described. Duration of rainfall is shown to be approximately log-normal and is consistent with a log-normal distribution of precipitation region size. The fractional duration of rainfall above a threshold intensity varies considerably between rain periods, a result which may be important for electromagnetic attenuation models. Estimation of the shapes of rain patches using the raingauge array is demonstrated and shows considerable promise. Raindrop fallspeeds, measured using the disdrometer, generally show much less deviation from stagnant air terminal velocities than indicated by previously reported results. Much of the spread in the results is shown to be consistent with instrumentation errors although significant residual deviations are still apparent; the fallspeeds are generally slower than stagnant air values would suggest. Measurements of the arrival rate of raindrops at the disdrometer indicate clustering of drops rather than the often assumed Poisson distribution. The clustering is associated with small drops and has reasonable correlation with rainfall intensity. Examination of the cross-correlation of arrival rates of different sized drops show results in contradiction to previous results; small drops are found to lead other sized drops. Using a normalisation method, the shapes of raindrop size distributions measured are shown to be depressed in the mid-radius region.

Introduction: In this chapter the present status of our understanding of the interaction of five and fewer nucleons will be briefly summarized. Some of the gaps in our knowledge will be pointed out, making clear the motivation for the experimental work described in the remainder of the thesis. Since most of the experimental effort by the author has been directed towards scattering of neutrons by protons, the nucleon - nucleon problem will be considered in greater detail. In speaking of this, the simplest problem in nuclear physics, M.L. Goldberger made the following remarks at the 1960 Midwest Conference on Theoretical Physics: "There are few problems in modern theoretical physics which have attracted more attention than that of trying to determine the fundamental interaction between two nucleons. It is also true that scarcely ever has the world of physics owed so little to so many .. . . . . . In general, in surveying the field, one is oppressed by the unbelievable confusion and conflict that exists. It is hard to believe that many of the authors are talking about the same problem, or in fact that they know what the problem is."

A field system capable of making high-resolution measurements of rainfall is described. The system incorporates a disdrometer, an array of high-resolution raingauges, a general-purpose data acquisition system and ancillary equipment. In an evaluation of the disdrometer, a theory allowing calculation of the effects of windspeed on detection efficiency is presented which has wide applicability. The raingauges are an improved design allowing 10-s temporal resolution of rainfall intensity and 100 m spatial resolution of rain-patch size when used in the array. An extensive data base of measurements made using the field system is described. Duration of rainfall is shown to be approximately log-normal and is consistent with a log-normal distribution of precipitation region size. The fractional duration of rainfall above a threshold intensity varies considerably between rain periods, a result which may be important for electromagnetic attenuation models. Estimation of the shapes of rain patches using the raingauge array is demonstrated and shows considerable promise. Raindrop fallspeeds, measured using the disdrometer, generally show much less deviation from stagnant air terminal velocities than indicated by previously reported results. Much of the spread in the results is shown to be consistent with instrumentation errors although significant residual deviations are still apparent; the fallspeeds are generally slower than stagnant air values would suggest. Measurements of the arrival rate of raindrops at the disdrometer indicate clustering of drops rather than the often assumed Poisson distribution. The clustering is associated with small drops and has reasonable correlation with rainfall intensity. Examination of the cross-correlation of arrival rates of different sized drops show results in contradiction to previous results; small drops are found to lead other sized drops. Using a normalisation method, the shapes of raindrop size distributions measured are shown to be depressed in the mid-radius region.

Granulation is an industrial process where fine particles are bound together into larger granules. The process has numerous applications including the manufacture of pharmaceuticals and the production of cosmetics, chemicals, detergents and fertilisers. This thesis studies aspects of wet granulation which involves the application of a viscous binder, usually in the form of a spray, to an agitated bed of powder particles. Individual powder particles may adhere together, joined by small quantities of binder fluid called liquid bridges. By a process of collision and adherence additional particles may join the newly formed agglomerates. Agglomerates may also coalesce together which is a process that leads to granule formation. On the completion of this process, granules are typically dried.This thesis studies wet granulation on three different levels. First, micro-level investigations of liquid bridges between two and three particles are performed. For the two-particle case, the fluid profile of static (stationary) and dynamic (moving) liquid bridges is investigated. For the static case, a numerical solution to the Young-Laplace equation is obtained; this relates the volume of binder fluid to liquid bridge properties such as the inter-particle force. An analytic solution is also obtained, providing the liquid bridge profile in terms of known mathematical functions. For both solutions, the radii of the (spherical) primary particles may be different. The dynamic case is then studied using the Navier-Stokes equations with the low Reynolds number approximation. The motion of the approaching particles is shown to be damped by the viscosity of the liquid bridge. Static liquid bridges between three equally sized primary particles are then studied. Symmetry of the problem is used to obtain a numerical solution to the Young-Laplace equation. Liquid bridge properties are calculated in terms of the binder fluid volume. Experimental agreement is provided.Secondly, a model to estimate the stickiness (fractional wet surface area) of agglomerates is proposed. Primary particles are approximated as spheres and are added one at a time in a closely packed arrangement. The model includes parameters to control the inter-particle separation distance and the fluid saturation state. Computational geometry is used to obtain results which relate the number of particles and the volume of binder fluid to the stickiness of the agglomerates.Finally, a population balance model for wet granulation is developed by extending an earlier model to incorporate the effects of binder fluid. Functions for the inter-particle collision rate and drying rate are proposed, including functions which are derived from the geometric model, described above, for the case of maximum particle consolidation. The model is solved numerically for a range of coalescence kernels and results are presented which show the effect of binder volume and the drying rate.

Introduction: This thesis is an account of the design, construction and testing of a particle accelerator which represents a minor variation on the now-familiar theme of the tandem van de Graaff. The machine has been very briefly described elsewhere (Naylor 1968).

Introduction: This thesis is an account of the design, construction and testing of a particle accelerator which represents a minor variation on the now-familiar theme of the tandem van de Graaff. The machine has been very briefly described elsewhere (Naylor 1968).

Introduction: This thesis is an account of the design, construction and testing of a particle accelerator which represents a minor variation on the now-familiar theme of the tandem van de Graaff. The machine has been very briefly described elsewhere (Naylor 1968).

Introduction: This thesis is an account of the design, construction and testing of a particle accelerator which represents a minor variation on the now-familiar theme of the tandem van de Graaff. The machine has been very briefly described elsewhere (Naylor 1968).

Multisymplectic integration is a relatively new addition to the field of geometric integration, which is a modern approach to the numerical integration of systems of differential equations. Multisymplectic integration is carried out by numerical integrators known as multisymplectic integrators, which preserve a discrete analogue of a multisymplectic conservation law. In recent years, it has been shown that various discretisations of a multi-Hamiltonian PDE satisfy a discrete analogue of a multisymplectic conservation law. In particular, discretisation in time and space by the popular symplectic Runge–Kutta methods has been shown to be multisymplectic. However, a multisymplectic integrator not only needs to satisfy a discrete multisymplectic conservation law, but it must also form a well-defined numerical method. One of the main questions considered in this thesis is that of when a multi-Hamiltonian PDE discretised by Runge–Kutta or partitioned Runge–Kutta methods gives rise to a well-defined multisymplectic integrator. In particular, multisymplectic integrators that are explicit are sought, since an integrator that is explicit will, in general, be well defined. The first class of discretisation methods that I consider are the popular symplectic Runge–Kutta methods. These have previously been shown to satisfy a discrete analogue of the multisymplectic conservation law. However, these previous studies typically fail to consider whether or not the system of equations resulting from such a discretisation is well defined. By considering the semi-discretisation and the full discretisation of a multi-Hamiltonian PDE by such methods, I show the following: • For Runge–Kutta (and for partitioned Runge–Kutta methods), the active variables in the spatial discretisation are the stage variables of the method, not the node variables (as is typical in the time integration of ODEs). • The equations resulting from a semi-discretisation with periodic boundary conditions are only well defined when both the number of stages in the Runge–Kutta method and the number of cells in the spatial discretisation are odd. For other types of boundary conditions, these equations are not well defined in general. • For a full discretisation, the numerical method appears to be well defined at first, but for some boundary conditions, the numerical method fails to accurately represent the PDE, while for other boundary conditions, the numerical method is highly implicit, ill-conditioned and impractical for all but the simplest of applications. An exception to this is the Preissman box scheme, whose simplicity avoids the difficulties of higher order methods. • For a multisymplectic integrator, boundary conditions are treated differently in time and in space. This breaks the symmetry between time and space that is inherent in multisymplectic geometry. The second class of discretisation methods that I consider are partitioned Runge– Kutta methods. Discretisation of a multi-Hamiltonian PDE by such methods has lead to the following two major results: 1. There is a simple set of conditions on the coefficients of a general partitioned Runge– Kutta method (which includes Runge–Kutta methods) such that a general multi- Hamiltonian PDE, discretised (either fully or partially) by such methods, satisfies a natural discrete analogue of the multisymplectic conservation law associated with that multi-Hamiltonian PDE. 2. I have defined a class of multi-Hamiltonian PDEs that, when discretised in space by a member of the Lobatto IIIA–IIIB class of partitioned Runge–Kutta methods, give rise to a system of explicit ODEs in time by means of a construction algorithm. These ODEs are well defined (since they are explicit), local, high order, multisymplectic and handle boundary conditions in a simple manner without the need for any extra requirements. Furthermore, by analysing the dispersion relation for these explicit ODEs, it is found that such spatial discretisations are stable. From these explicit ODEs in time, well-defined multisymplectic integrators can be constructed by applying an explicit discretisation in time that satisfies a fully discrete analogue of the semi-discrete multisymplectic conservation law satisfied by the ODEs. Three examples of explicit multisymplectic integrators are given for the nonlinear Schr¨odinger equation, whereby the explicit ODEs in time are discretised by the 2-stage Lobatto IIIA– IIIB, linear–nonlinear splitting and real–imaginary–nonlinear splitting methods. These are all shown to satisfy discrete analogues of the multisymplectic conservation law, however, only the discrete multisymplectic conservation laws satisfied by the first and third multisymplectic integrators are local. Since it is the stage variables that are active in a Runge–Kutta or partitioned Runge– Kutta discretisation in space of a multi-Hamiltonian PDE, the order of such a spatial discretisation is limited by the order of the stage variables. Moreover, the spatial discretisation contains an approximation of the spatial derivatives, and thus, the order of the spatial discretisation may be further limited by the order of this approximation. For the explicit ODEs resulting from an r-stage Lobatto IIIA–IIIB discretisation in space of an appropriate multi-Hamiltonian PDE, the order of this spatial discretisation is r - 1 for r = 10; this is conjectured to hold for higher values of r. For r = 3, I show that a modification to the initial conditions improves the order of this spatial discretisation. It is expected that a similar modification to the initial conditions will improve the order of such spatial discretisations for higher values of r.

The theory of radiative transfer is applied to the problem of multiple wave scattering in a one-dimensional multilayer. A new mathematical model of a multilayer is presented in which both the refractive index and width of each layer are randomized. The layer widths are generated by a new probability distribution which allows for strong layer width disorder. An expression for the transport mean free path of the multilayer is derived based on its single-scattering properties. It will be shown that interference between the field reflected from adjacent layer interfaces remains significant even in the presence of strong layer width disorder. It will be proven that even when the scattering is weak, the field in a random multilayer localizes at certain frequencies. The effect of increasing layer width randomization on this form of localization is quantified. The radiative transfer model of time-harmonic scattering in multilayers is extended to narrow-band pulse propagation in weakly scattering media. The tendency of pulses to broaden in this medium is discussed. A radiative transport model of the system is developed and compared to numerical solutions of the wave equation. It is observed that pulse broadening is not described by simple transfer theory. The radiative transfer model is extended by the addition of a Laplacian term in an attempt to model the effect of ensemble average pulse broadening. Numerical simulation results in support of this proposal are given, and applications for the theory suggested. Finally, the problem of acoustic wave scattering by planar screens is considered. The study was motivated by the idea that multiple scattering experiments may prove possible in a medium composed of such scatterers. Successful multiple scattering in a medium of planar scatterers will depend on the scattering cross-section at angles away from normal incidence. The scattering cross-section is calculated for a circular disc using a new technique for solving the acoustic wave equation on planar surfaces. The method is validated by comparison with available analytic solutions and the geometric theory of diffraction.