*Apr 13 2021*

16:00 - 17:15

#### Organiser:

**Stewart Haslinger**, University of Liverpool

The seminars take place bi-weekly on Tuesdays at 16:00 – 17:15 (GMT).

The University of Liverpool Research Centre in Mathematics and Modelling (RCMM) seminar series will cover a wide range of topics in the theory of partial differential equations that describe the propagation of waves in non-uniform media (both periodic and non-periodic), asymptotic analysis and solid mechanics.

### Next Seminar

13 April 2021

**John Chapman,** Keele University -* Fractional power series and the method of dominant balances*

- In this talk I'll give a general treatment of the method of dominant balances for a single polynomial equation, in which an arbitrary number of parameters is to be scaled in such a way that the maximum possible number of terms in the equation is in balance at leading order. This leads in general to a fractional power series (a `Puiseux series'), in which, surprisingly, there can be large and irregular gaps (lacunae) in the fractional powers actually occurring. A complete theory is given to determine the gaps, requiring the notion of a Frobenius set from number theory, and its complement, a Sylvester set. The starting point is the Newton polytope in arbitrarily many dimensions, and key tools for obtaining precise results are Faà di Bruno's formula for the high derivatives of a composite function, and Bell polynomials. Full account is taken of repeated roots, of arbitrary multiplicity, in the dominant balance which launches a Puiseux series. The fractional powers in these series can have remarkably large denominators, even for a polynomial of modest degree, and the nature of Frobenius sets is such that it can take hundreds of terms for long-run regularity to emerge. The talk is applied in outlook, as the method of dominant balances is widely used in physics and engineering, where it gives results of extraordinary accuracy, far beyond the expected range. The work has been conducted in a collaboration begun at the Isaac Newton Institute, Cambridge, with H. P. Wynn (London School of Economics). We believe the results are new. Despite hundreds of years of use of Puiseux series (since 1676), we are not aware of any previous attempt to give a complete quantitative account of their gaps.

Past Seminars

10 November 2020

**Artur Gower**, University of Sheffield *- Ensemble average waves in random materials of any geometry*

Which is more useful: knowing the effective properties of a material, or its effective wavenumbers? In a low frequency regime these are essentially the same, but when the wavelength becomes comparable to the microstructure’s size, they are not. Using a broad range of frequencies is needed to characterize materials and to design for broad wave speed control and attenuation.

24 November 2020

**Alexander Movchan, **University of Liverpool - *Waves in Chiral Discrete Systems*

The talk will provide an overview of modelling for time harmonic vibrations of solids with systems of chiral resonators. As well as models for finite-sized structures, the presentation will include infinite periodic systems and Floquet-Bloch waves in both continuous and discrete systems.ppose we perform an experiment that measures the scattered field from a material with a random microstructure. We could then find what effective properties would lead to the same scattered field we measured. So far so good. But what if we took that same microstructure and molded it into a different geometry and then repeated the experiment? The bad news is that even when using the same frequency we would potentially find a different set of effective properties. In contrast, in this talk I show how effective wavenumbers are inherently related to the material microstructure and not its geometry. I will also show how to calculate the average scattered field, including the field scattered from a pipe geometry filled with particles and a spherical droplet filled with particles.

8 December 2020

**Anastasia Kisil,** The University of Manchester - *Edges and Point Scatterers: a simple model for a metamaterial with edges*

This talk will outline some of the challenges that are present for the understanding of acoustic scattering by obstacles that have periodic structure. This provides a simple model to understand some interesting effects such as Wood anomaly. Wood anomaly leads to counterintuitive effects which could be used in the design of metamaterials. Different configuration of point scatters will be considered. Some analytic methods, their generalisation and limitations will be discussed. This is joint work with Matthew Nethercote, Matthew Riding and Raphael Assier.

9 February 2021

**Leonid Slepyan**, Tel Aviv University -* Waves and Latent Energy*

Situations are discussed, where the energy conservation law does not hold in a macrolevel model, such as a continuous elastic medium. Under certain conditions, the latent energy as the macro-to-micro energy flux plays a considerable (and possibly decisive) role in the process. Here we consider the cases where this happens for waves in a material of the hardening type stress- strain relation, and in (linear and nonlinear) transition waves. We discuss how this phenomenon manifests itself in fracture and in fast moving elastic contacts, where the energy releases through moving singular points. Also, we consider recently developed models of a homogeneous material with uniformly distributed dynamic `microstructure' elevated on the macrolevel.

23 February 2021

**Georgios Sarris**, Imperial College - *The effect of roughness on the attenuation of surface waves*

Rayleigh waves are well known to attenuate due to scattering when they propagate over a rough surface. In the literature, three scattering regimes have been identified and studied analytically - the Rayleigh (short wavelength), stochastic (short to medium wavelength) and geometric (long wavelength) regimes. This study uses two-dimensional finite element (FE) modelling to validate and extend existing theory through a unified approach of studying the problem of Rayleigh wave scattering from rough surfaces. Very good agreement is found between the theory and the FE results in all scattering regimes. Additionally, the results provide useful insight in verifying the three-dimensional theory, since the method used for its derivation is analogous.

9 March 2021

**Richard Porter,** University of Bristol* - Plate arrays as water wave metamaterials*

This talk is intended to highlight the different effects that plate arrays (closely-spaced periodic arrays of thin vertical plates) can have on the propagation of surface water waves. It will focus on examples of plate arrays in a number of settings some of which will have analogues in two-dimensional acoustics or electromagnetics. We will start by showing how plate arrays extending through the fluid depth can be used as an all-frequency prefectly-transmitting negative refraction medium. We will include examples of the broadbanded absorption of wave energy by plate arrays which include dissipative effects as metasurfaces or as absorbing cavities in waveguides. We shall also demonstrate the remarkable potential for ocean wave energy absorption by cylindrical structures occupied by plate arrays. Finally, we shall look at how to use submerged bathymetric plate arrays as devices for creating anisotropic depth effects which can be applied for the bespoke steering of water waves as required, for example, in cloaking. If all goes to plan the talk will be light on mathematical detail and heavy on examples.

23 March 2021

**Jonathan Mestel, **Imperial College - From double-glazing to dynamos: ﬂows linear in one coordinate

From double-glazing to dynamos: flows linear in one coordinate Self-similar structures are often exploited to reduce the order of a PDE system. Many advection-diffusion systems, such as the Navier-Stokes, heat flow and magnetic induction equations, involve Laplacian and Jacobian terms. If the solution fields are linear in one coordinate, say x, then these terms will also be linear in x. x may then scale out of the problem, which will nevertheless retain some residual three-dimensional structure. A few problems of this ilk are considered, and some new Navier-Stokes solutions are presented. The resulting flow fields are shown to be able to drive dynamos with a related similarity structure, which extend exactly into the nonlinear regime. One of these is Von K´arm´an-like flow between two differentially rotating discs, whose relevance to laboratory dynamo experiments is discussed.