Getting the most out of maths: How to coordinate mathematical modelling research to support a pandemic, lessons learnt from three initiatives that were part of the COVID-19 response in the UK.

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits.

Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays.

We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials.

Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays.

We present a novel multipole formulation for computing the band structures of two-dimensional arrays of cylindrical Helmholtz resonators. This formulation is derived by combining existing multipole methods for arrays of ideal cylinders with the method of matched asymptotic expansions. We construct asymptotically close representations for the dispersion equations of the first band surface, correcting and extending an established lowest-order (isotropic) result in the literature for thin-walled resonator arrays.

The Wiener-Hopf technique, its generalizations and applications: constructive and approximate methods.

This paper reviews the modern state of the Wiener-Hopf factorization method and its generalizations. The main constructive results for matrix Wiener-Hopf problems are presented, approximate methods are outlined and the main areas of applications are mentioned. The aim of the paper is to offer an overview of the development of this method, and demonstrate the importance of bringing together pure and applied analysis to effectively employ the Wiener-Hopf technique.

A Recruitment Model of Tendon Viscoelasticity That Incorporates Fibril Creep and Explains Strain-Dependent Relaxation.

Soft tissues exhibit complex viscoelastic behavior, including strain-rate dependence, hysteresis, and strain-dependent relaxation. In this paper, a model for soft tissue viscoelasticity is developed that captures all of these features and is based upon collagen recruitment, whereby fibrils contribute to tissue stiffness only when taut. We build upon existing recruitment models by additionally accounting for fibril creep and by explicitly modeling the contribution of the matrix to the overall tissue viscoelasticity.

On the asymptotic properties of a canonical diffraction integral.

We introduce and study a new canonical integral, denoted , depending on two complex parameters and . It arises from the problem of wave diffraction by a quarter-plane and is heuristically constructed to capture the complex field near the tip and edges. We establish some region of analyticity of this integral in , and derive its rich asymptotic behaviour as | | and | | tend to infinity.

A proof that multiple waves propagate in ensemble-averaged particulate materials.

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterize wave propagation through an inhomogeneous material, the most crucial parameter is the . For this reason, there are many published studies on how to calculate a single effective wavenumber.

Thermo-viscous damping of acoustic waves in narrow channels: A comparison of effects in air and water.

Recent work in the acoustic metamaterial literature has focused on the design of metasurfaces that are capable of absorbing sound almost perfectly in narrow frequency ranges by coupling resonant effects to visco-thermal damping within their microstructure. Understanding acoustic attenuation mechanisms in narrow, viscous-fluid-filled channels is of fundamental importance in such applications. Motivated by recent work on acoustic propagation in narrow, air-filled channels, a theoretical framework is presented that demonstrates the controlling mechanisms of acoustic propagation in arbitrary Newtonian fluids, focusing on attenuation in air and water.

Correction to 'Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media'.

[This corrects the article DOI: 10.1098/rspa.2015.

Reflection from a multi-species material and its transmitted effective wavenumber.

We formally deduce closed-form expressions for the transmitted effective wavenumber of a material comprising multiple types of inclusions or particles (multi-species), dispersed in a uniform background medium. The expressions, derived here for the first time, are valid for moderate volume fractions and without restriction on the frequency. We show that the multi-species effective wavenumber is not a straightforward extension of expressions for a single species.

An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems.

In Parnell & Abrahams (2008 , 1461-1482. (doi:10.1098/rspa.

Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media.

The effect of a longitudinal stretch and a pressure-induced inhomogeneous radial deformation on the scattering of antiplane elastic waves from a cylindrical cavity is determined. Three popular nonlinear strain energy functions are considered: the neo-Hookean, the Mooney-Rivlin and a two-term Arruda-Boyce model. A new method is developed to analyse and solve the governing wave equations.

On nonlinear viscoelastic deformations: a reappraisal of Fung's quasi-linear viscoelastic model.

This paper offers a reappraisal of Fung's model for quasi-linear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models.

On the existence of flexural edge waves on thin orthotropic plates.

This paper is concerned with an investigation into the existence of waves propagating along a free edge of an orthotropic plate, where the edge is inclined at arbitrary angle to a principal direction of the material. After deriving the governing equation and edge conditions, an edge wave ansatz is substituted into this system to reduce it to a set of algebraic equations for the edge wave wave number and wave vector. These are solved numerically for several typical composite materials although analytic expressions can be obtained in the case of special values of the material parameters and inclination angle.