## Virtual Seminar Series: Waves in Complex Continua (Wavinar)

The focus of this seminar series is on waves interaction with complex media, membranes and metamaterials. The applications include acoustics, aero- and hydro-acoustics, elastic, electromagnetic waves. Key challenge areas to be addressed include the analysis of propagation in complex domains (e.g. with multiscale structure and/or complicated geometries), the development of effective methods for high frequency problems (in particular, the study of canonical problems arising in the Geometrical Theory of Diffraction), and the rigorous analysis and validation of computational methods.

Seminars will be held once every 2 weeks. Timings will alternate between afternoons and mornings. There will be two 20 minute talks (plus 5 min for questions) separated by a five minute break. The first talk will start on the hour, and the second on the half hour.

The talks will be recorded (when permission is given) and made available (via links) on this webpage.

If you are interested in speaking, or would like to suggest a speaker, please contact Anastasia Kisil (anastasia.kisil@manchester.ac.uk).

*A quadrupole sound source scattering off multiple elastic plates spelling WAVINAR, computed using the method developed in [1]. Plate thickness is emphasised for visualisation*

The first seminar was held on **Tuesday 28th April**, 2020**.**

To register in advance for subsequent Tuesday afternoon meetings, use this link,

Tues 26/05: 16.00 - 17.30

Tues 23/06: 16.00 - 17.30

To register in advance for the Tuesday morning/lunchtime, use this link,

Tues 12/05: 11.00 - 12.30

Tues 09/06: 11.00 - 12.30

### Future Seminars

**Tuesday 26th May, 2020 **16.00-17.00

Matthew Colbrook(University of Cambridge)Title:

Scattering, Acoustic Black Holes and Mathieu Functions: A boundary spectral method for diffraction by multiple variable poro-elastic platesAbstract: Many fluid dynamical and acoustic problems can be modelled by PDEs on unbounded domains with complicated boundary conditions. Accurate and fast solutions of these systems is key to predicting the effect of physical variables/parameters and external forces, and thus crucial for providing insight into a wide range of problems. In this talk I will focus on the problem of Helmholtz scattering off multiple plates in arbitrary locations, with boundary conditions that model variable elasticity and porosity. Such boundary conditions present a considerable challenge to current methods.

The use of local Mathieu function expansions and their asymptotics leads to an efficient and robust high-order boundary spectral method, suitable for a wide range of frequencies, and tackling the ODE boundary conditions. The solution representation directly provides a sine series approximation of the far-field directivity and can be evaluated near or on the scatterers, meaning that the near field can be computed stably and efficiently. As an example, it is shown that a power-law decrease to zero in stiffness parameters gives rise to unexpected scattering and aeroacoustic effects similar to an acoustic black hole metamaterial.

This talk is based on joint work with Anastasia Kisil

The paper can be found here, and code can be found here,

Daniel Torrent Martí(Universitat Jaume I, Spain)Title:

Advanced Artificial Structures For The Control of Acoustic and Mechanic EnergyThe control of acoustic energy is a challenging problem with a broad range of applications. In this talk we will review some specific mechanism to achieve this control by means of artificial structures specially designed for this purpose. It will be shown how engineered gratings can be used to modulate the flow of acoustic and elastic energy towards specific directions, and a very efficient inverse design method will be presented. Some applications will be summarized, and the experimental realization of an acoustic carpet cloak based on these structures will also be presented.

**Tuesday 9th June, 2020 **11.00-12.00

Andrey Korolkov(Moscow State University)

Sommerfeld integral method for discrete diffraction problemsTwo diffraction problems on a 2D square lattice are studied: finding the Green’s function and scattering by a half-line. The governing equation in both cases is the Helmholtz equation with a standard discrete Laplacian. For the Green’s function of an entire plane, we build the field in the form of a plane wave decomposition. This decomposition is rewritten as a contour integral of an analytical differential 1-form on a complex manifold, which is the lattice dispersion diagram (the solution of the dispersion equation). The dispersion diagram of the lattice is, topologically, a torus, and this makes possible to develop an invariant representation of the field. For the diffraction problems, an analogue of the Sommerfeld integral is built. This is an integral of a Sommerfeld transformant (a two-valued function on the torus) along a system of appropriate integrals.

As a result, we obtain new representations of the discrete wave fields. Potentially, these methods can be applied to a wider class of problems.

Stéphanie Chaillat-Loseille(ENSTA Paris, France)

**Tuesday 23rd June, 2020**

Anne-Sophie Bonnet-Ben Dhia(ENSTA Paris)

** Previous seminars **

Recordings from the seminar series are available here.

**Tuesday 12th May, 2020 ** 11.00 -12.30

Michael H Meylan(University of Newcastle, Australia)Title:

Lax-Phillips Scattering Theory for Simple Wave ScatteringAbstract: Lax-Philips scattering theory is a method to solve for scattering as an expansion over the singularities of the analytic extension of the scattering problem to complex frequencies. I will show how a complete theory can be developed in the case of simple scattering problems. Even for the simplest case, it requires a non-trivial generalised eigenfunction transformation to project into the space of analytic functions on the real line. The scattering operator in this space is simply the complex exponential. I will illustrate how this theory can be used to find a numerical solution, and I will demonstrate the method by applying it to the vibration of ice shelves.

Richard Wiltshaw(Imperial College London)Title:

Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterersWe study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on topical examples from topological wave physics.

**28th April, 2020 **

Georg Maierhofer(University of Cambridge)Title:

Extended Filon quadrature methods for high-frequency wave scatteringRecording of presentation is available here.

Abstract: Motivated by collocation methods for wave scattering problems in 2D, we study the efficient approximation of highly oscillatory integrals in the presence of singularities and stationary points arising from the Green's function of the Helmholtz equation. From asymptotic theory we know that, for large frequencies, these integrals are dominated by contributions at corners, singular and stationary points. We exploit the understanding of this asymptotic behaviour to design bespoke quadrature methods based on the classical Filon method for oscillatory integrals. These methods allow for numerical approximation at uniform cost both for small and for very large frequencies. We demonstrate how the required Chebyshev moments can be stably computed based on a duality to a spectral method applied to Bessel’s equation. Our design for this algorithm has significant potential for further

generalisations that would allow Filon methods to be constructed for a wide range of integrals involving special functions.In this talk we will place special emphasis on the application of these methods to numerical wave scattering. They provide a flexible and frequency-independent way of assembling the collocation matrix for hybrid methods in high-frequency wave scattering problems on convex polygonal shapes, and we will demonstrate this favourable performance on

several numerical examples.

Rodolfo Brandao(Imperial College London)Title:

Asymptotic modelling of micro-structured acoustic devices including thermoviscous effectsRecording of presentation is available here.

Abstract: There is renewed interest in using structured acoustic devices to filter, guide and absorb sound, especially on small scales and incorporating new ideas originating in the field of electromagnetic metamaterials. Analytical modelling of such devices, however, has often relied on heuristics and the use of `fudge' parameters to fit experiments — especially when including thermoviscous effects, which play a singular role in many acoustic metamaterials and are key to sound absorption. I will present several new analytical models, derived using matched asymptotics with thermoviscous effects systematically accounted for. I will first revisit the classical problem of calculating the acoustic impedance of a cylindrical orifice. Building on this, I will present a detailed asymptotic model of a three-dimensional Helmholtz resonator embedded in a wall, as well as acoustic metasurfaces formed of finite and infinite arrays of such resonators. Lastly, I will also discuss the important role of thin thermoviscous boundary layers in experiments of extraordinary acoustic transmission through narrow slits. (Joint work with Ory Schnitzer and Jacob Holley.)

**This seminar series is supported as part of the ICMS/INI Online Mathematical Sciences Seminars. **

[1] Matthew Colbrook, Anastasia Kisil “A Mathieu function boundary spectral method for diffraction by multiple variable poro-elastic plates, with applications to metamaterials and acoustics”, in preparation. Code developed by Matthew is available at https://github.com/MColbrook/MathieuFunctionCollocation.