*Feb 09 2021*

16:00 - 17:15

**University of Liverpool Research Centre in Mathematics and Modelling - Wave Scattering and Solid Mechanics**

Organised by Stewart Haslinger, University of Liverpool. The seminars take place bi-weekly on Tuesdays at 16:00 – 17:15 (GMT).

To sign up to participate in these seminars, please complete this registration form

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

**9 February 2021**

Leonid Slepyan (Tel Aviv University): Waves and Latent Energy

**23 February 2021**

Georgios Sarris (Imperial College): The effect of roughness on the attenuation of surface waves

**9 March 2021**

Richard Porter (University of Bristol): Water wave metamaterials

**23 March 2021 **Jonathan Mestel (Imperial College)

**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.

### Previous Seminars

10th 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.

Click here to view the recording.

24th 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.

Click here to view the recording.

Tuesday 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.

Click here to view the recording.