A note on the Stokes phenomenon in flow under an elastic sheet: Stokes Phenomenon in flow under a sheet

The Stokes phenomenon is a class of asymptotic behaviour that was first discovered by Stokes in his study of the Airy function. It has since been shown that the Stokes phenomenon plays a significant role in the behaviour of surface waves on flows past submerged obstacles. A detailed review of recent research in this area is presented, which outlines the role that the Stokes phenomenon plays in a wide range of free surface flow geometries. The problem of inviscid, irrotational, incompressible flow past a submerged step under a thin elastic sheet is then considered. It is shown that the method for computing this wave behaviour is extremely similar to previous work on computing the behaviour of capillary waves. Exponential asymptotics are used to show that free-surface waves appear on the surface of the flow, caused by singular fluid behaviour in the neighbourhood of the base and top of the step. The amplitude of these waves is computed and compared to numerical simulations, showing excellent agreements between the asymptotic theory and computational solutions. This article is part of the theme issue 'Stokes at 200 (part 2)'

Three-dimensional capillary waves due to a submerged source with small surface tension

Steady and unsteady linearised flow past a submerged source are studied in the small-surface-tension limit, in the absence of gravitational effects. The free-surface capillary waves generated are exponentially small in the surface tension, and are determined using the theory of exponential asymptotics. In the steady problem, capillary waves are found to extend upstream from the source, switching on across curves on the free surface known as Stokes lines. Asymptotic predictions and compared with computational solutions for the position of the free surface. In the unsteady problem, transient effects cause the solution to display more complicated asymptotic behaviour, such as higher-order Stokes lines. The theory of exponential asymptotics is applied to show how the capillary waves evolve over time, and eventually tend to the steady solution.Comment: 36 pages, 10 figure

Locating complex singularities of Burgers' equation using exponential asymptotics and transseries

Burgers' equation is an important mathematical model used to study gas dynamics and traffic flow, among many other applications. Previous analysis of solutions to Burgers' equation shows an infinite stream of simple poles born at t = 0^+, emerging rapidly from the singularities of the initial condition, that drive the evolution of the solution for t > 0. We build on this work by applying exponential asymptotics and transseries methodology to an ordinary differential equation that governs the small-time behaviour in order to derive asymptotic descriptions of these poles and associated zeros. Our analysis reveals that subdominant exponentials appear in the solution across Stokes curves; these exponentials become the same size as the leading order terms in the asymptotic expansion along anti-Stokes curves, which is where the poles and zeros are located. In this region of the complex plane, we write a transseries approximation consisting of nested series expansions. By reversing the summation order in a process known as transasymptotic summation, we study the solution as the exponentials grow, and approximate the pole and zero location to any required asymptotic accuracy. We present the asymptotic methods in a systematic fashion that should be applicable to other nonlinear differential equations.Comment: 30 pages, 6 figure

Kelvin wake pattern at small Froude numbers

The surface gravity wave pattern that forms behind a steadily moving disturbance is well known to comprise divergent waves and transverse waves, contained within a distinctive V-shaped wake. In this paper, we are concerned with a theoretical study of the limit of a slow-moving disturbance (small Froude numbers), for which the wake is dominated by transverse waves. We consider three configurations: flow past a submerged source singularity, a submerged doublet, and a pressure distribution applied to the surface. We treat the linearised version of these problems and use the method of stationary phase and exponential asymptotics to demonstrate that the apparent wake angle is less than the classical Kelvin angle and to quantify the decrease in apparent wake angle as the Froude number decreases. These results complement a number of recent studies for sufficiently fast-moving disturbances (large Froude numbers) where the apparent wake angle has been also less than the classical Kelvin angle. As well as shedding light on the wake angle, we also study the fully nonlinear problems for our three configurations under various limits to demonstrate the unique and interesting features of Kelvin wake patterns at small Froude numbers

Exponential asymptotics of woodpile chain nanoptera using numerical analytic continuation

Travelling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading-order behaviour. For many nonlinear systems, such as granular woodpile chains, it is not possible to calculate the leading-order solution explicitly. We show that accurate asymptotic approximations can be obtained using numerical approximation in place of the exact leading-order behaviour. We calculate the oscillation behaviour for Toda woodpile chains, and compare the results to exponential asymptotics based on tanh-fitting, Pad\'{e} approximants, and the adaptive Antoulas-Anderson (AAA) method. The AAA method is shown to produce the most accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish. This method is then applied to study granular woodpile chains, including chains with Hertzian interactions -- this method is able to calculate behaviour that could not be accurately approximated in previous studies.Comment: 24 pages, 17 figures, 1 tabl