Accurate series solutions for gravity-driven Stokes waves

In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that higher order behaviour of series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Pade approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Pade approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Exponential asymptotics with multiple stokes lines

When asymptotic series methods are applied in order to solve problems that arise in applied mathematics in the limit that some parameter becomes small, they are unable to demonstrate behaviour that occurs on a scale that is exponentially small compared to the algebraic terms of the asymptotic series. There are many examples of physical systems where behaviour on this scale has important effects and, as such, a range of techniques known as exponential asymptotic techniques were developed that may be used to examinine behaviour on this exponentially small scale. Many problems in applied mathematics may be represented by behaviour within the complex plane, which may subsequently be examined using asymptotic methods. These problems frequently demonstrate behaviour known as Stokes phenomenon, which involves the rapid switches of behaviour on an exponentially small scale in the neighbourhood of some curve known as a Stokes line. Exponential asymptotic techniques have been applied in order to obtain an expression for this exponentially small switching behaviour in the solutions to orginary and partial differential equations. The problem of potential flow over a submerged obstacle has been previously considered in this manner by Chapman & Vanden-Broeck (2006). By representing the problem in the complex plane and applying an exponential asymptotic technique, they were able to detect the switching, and subsequent behaviour, of exponentially small waves on the free surface of the flow in the limit of small Froude number, specifically considering the case of flow over a step with one Stokes line present in the complex plane. We consider an extension of this work to flow configurations with multiple Stokes lines, such as flow over an inclined step, or flow over a bump or trench. The resultant expressions are analysed, and demonstrate interesting implications, such as the presence of exponentially sub-subdominant intermediate waves and the possibility of trapped surface waves for flow over a bump or trench. We then consider the effect of multiple Stokes lines in higher order equations, particu- larly investigating the behaviour of higher-order Stokes lines in the solutions to partial differential equations. These higher-order Stokes lines switch off the ordinary Stokes lines themselves, adding a layer of complexity to the overall Stokes structure of the solution. Specifically, we consider the different approaches taken by Howls et al. (2004) and Chap- man & Mortimer (2005) in applying exponential asymptotic techniques to determine the higher-order Stokes phenomenon behaviour in the solution to a particular partial differ- ential equation.

Peter Alwast talks to Grant Stevens

This interview was published in the catalogue for Peter Alwast's solo exhibition, "Future Perfect", at the Institute of Modern Art, Brisbane, in August 2011.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Review : "Quality of life: the assessment, analysis and interpretation of patient-reported outcomes (2nd Ed.)" by Peter M. Fayers & David Machin

An updated version, this excellent text is a timely addition to the library of any nurse researching in oncology or other settings where individuals’ quality of life must be understood. Health-related quality of life should be a central aspect of studies concerned with health and illness. Indeed, considerable evidence has recently emerged in oncology and other research settings that selfreported quality of life is of great prognostic significance and may be the most reliable predictor of subsequent morbidity and mortality. From a nursing perspective, it is also gratifying to note that novel therapy and other oncology studies increasingly recognize the importance of understanding patients’ subjective experiences of an intervention over time and to ascertain whether patients perceive that a new intervention makes a difference to their quality of life and treatment outcomes. Measurements of quality of life are now routine in clinical trials of chemotherapy drugs and are often considered the prime outcome of interest in the cost/benefit analyses of these treatments. The authors have extensive experience in qualityof- life assessment in cancer clinical trials, where most of the pioneering work into quality of life has been conducted. That said, many of the health-related qualityof- life issues discussed are common to many illnesses, and researchers outside of cancer should find the book equally helpful.

Accused headmaster turned me into pedophile, says convicted cleric Peter Gilbert

A CLERGYMAN who allegedly raped boys at a north Queensland boarding school in the 1960s has claimed he was ordered to take female hormones by his headmaster, who encouraged the "romantic love" of children among staff. Former Anglican brother Peter Gilbert - sentenced to seven years' jail in 2006 for the rape and indecent assault of children in the 1980s in South Australia - has blamed St Barnabas headmaster the late Robert Waddington for turning him into a pedophile. In a statement to one of his alleged victims, Gilbert said Waddington was molesting children himself and the Anglican priest would absolve the young teacher of his abuse of children in the confessional. The account has emerged as the top ranks of the Anglican church in Australia and Britain have been rocked by allegations of the covering-up of complaints made in 1999 and 2003 about Waddington's abuse while headmaster at the boarding school in Ravenshoe, north Queensland, and later, when he was dean of Manchester. A full investigation is being demanded into the handling of the complaints by the former Archbishop of York, now Lord (David) Hope of Thornes, and Australian church officials, including the former bishop of north Queensland, Clyde Wood.

Author in waiting : self-portrait of Peter Goldsworthy as a boy

Review of His Stupid Boyhood by Peter Goldsworthy (Hamish Hamilton, 2013).