Undular bores, or dispersive shock waves, are nonstationary waves propagating as oscillatory transitions between two basic states, in which the oscillatory structure gradually expands and grows in amplitude with distance traveled. In this work we report an important mechanism of generation of nonlinear dispersive shock waves in solids. We demonstrate, using high-speed pointwise photoelasticity, the generation of undular bores in solid (polymethylmethacrylate) prestrained bars by natural and induced tensile fracture. For the distances relevant to our experiments, the viscoelastic extended Korteweg-de Vries equation is shown to provide very good agreement with the key observed experimental features for suitable choice of material parameters, while some local features at the front of the bore are also captured reasonably well by the linearization near the nonzero prestrain level. The experimental and theoretical approaches presented open avenues and analytical tools for the study and applications of dispersive shock waves in solids.

We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear current. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive the modal equations from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (group speed, wave action conservation law) and to constructing more general ‘hybrid solutions’ consisting of a part of a ring wave and two tangent plane waves. The modal equations constitute a new spectral problem, and are analysed for a number of examples of surface ring waves in a homogeneous fluid and internal ring waves in a stratified fluid. Detailed analysis is developed for the case of a two-layered fluid with a linear shear current where we study their wavefronts and two-dimensional modal structure. Comparisons are made between the modal functions (i.e. eigenfunctions of the relevant spectral problems) for the surface waves in homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation. We also analyse the wavefronts of surface and interfacial waves for a large family of power-law upper-layer currents, which can be used to model wind generated currents, river inflows and exchange flows in straits. Global and local measures of the deformation of wavefronts are introduced and evaluated.

We study the evolution of the longitudinal release wave that is generated by induced tensile fracture as it propagates through solid rectangular polymethylmethacrylate (PMMA) bars of different constant cross-section. High-speed multi-point photoelasticity is used to register the strain wave at three distances from the fracture site in each experiment. In all cases, oscillations develop at the bottom of the release wave that exhibit the qualitative features of an undular bore. The pre-strain, post-strain, strain rate of the release wave and the cross-section dimensions determine the evolution of the oscillations. From the wave speed and strain rate close to the fracture site, we estimate the strain rate of the release wave as well as the growth of the amplitude and duration of the leading oscillation away from the fracture site by using formulae derived from the simple analytical solution of the linearized Gardner equation (linearized near the pre-strain level at fracture). Our estimates are then compared to experimental data, where good agreements of these three parameters are found between the predictions of the model and the experimental observations. Thus, we developed an approach to estimating the key characteristics of the developing unsteady undular bore based on the measured initial speeds of the longitudinal and shear waves. This does not require a prior knowledge of the elastic moduli for the conditions of the experiments, which in PMMA are known to be strain rate dependent.

Traditionally online assessments tend to focus on topics that require students to input algebraic and numeric responses. As such there is a paucity of questions that test students' knowledge of statistics, and what questions there are in our experience focus on computing specific values (mean, standard deviation, and so on). Through making use of a technology called JSXGraph that is supported within the STACK environment for online assessment of mathematical knowledge, we have developed statistics questions that aim to test conceptual knowledge. For example, by requiring students to adjust the bars in a graph in order to produce a dataset that has a required mean, median, mode and range. With careful design this approach enables open-ended questions that have more than one correct answer. In this paper we describe the questions we have designed, and report responses from a sample of students.

We study the evolution of the longitudinal release wave that is generated by induced tensile fracture as it propagates through solid rectangular Polymethylmethacrylate (PMMA) bars of different constant cross section. High speed multipoint photoelasticity is used to register the strain wave. In all cases, oscillations develop at the bottom of the release wave that exhibit the qualitative features of an undular bore. The pre-strain, post-strain, strain rate of the release wave and the cross section dimensions determine the evolution of the oscillations. From the wave speed and strain rate close to the fracture site, we estimate the strain rate of the release wave as well as the growth of the amplitude and duration of the leading oscillation away from the fracture site on using formulae derived from the simple analytical solution of the linearised Gardner equation (linearised near the pre-strain level at fracture), developed in our earlier work 1 . Our estimates are then compared to experimental data, where qualitative and good semi-quantitative agreements are established.