## 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 month, of the first Tuesday of the month 16.00-18.00 UK time. There will be three 25 minute talks (plus 5 min for questions) followed by optional informal discussion with one of the speakers.

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

**Recordings from the seminar series are available here**

.
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 this paper. 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,

** Please note this series will break for the summer, the next seminar will be 6th October 2020.**

### Future Seminars

*Breakout rooms available to have further discussion with each speaker.

**1 December 2020**

16.00-18:00 (BST)

Marion Darbas,(UPJV, France)Title:

Fast iterative BEM for high-frequency scattering problems in elastodynamicsAbstract: The numerical solution of high-frequency scattering problems of time-harmonic elastic waves by a three-dimensional obstacle is a challenging task. The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is inevitable to increase the size of the problems that can be considered. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. We propose to combine an approximate adjoint Dirichlet-to-Neumann (respectively Neumann-to-Dirichlet) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet (respectively Neumann) exterior scattering problems. We compare standard and preconditioned Combined Field Integral Equations (CFIEs).

Simon Chandler-Wilde,(Reading, UK)Title:

The complex-scaled half-space matching method and boundary integral equations in the complex domainAbstract: The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to PML or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.

This is joint work with: Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, and Christophe Hazard (CNRS-INRIA-ENSTA, Paris), Karl-Mikael Perfekt (Reading), and Yohanes Tjandrawidjaja (Technische Universitaet Dortmund)

Pawel Packo,(AGH, Poland)Title:

Guided wave propagation in nonlinear platesAbstract: Elastic waves in plates have been intensively studied over last decades due to their practical importance and wide applicability of thin-walled components in light-weight structures. Lamb waves display complex and multi-modal dispersion characteristics that make them attractive for various purposes, e.g. NDT and SHM, but more challenging in the context of signal analysis. Recently, nonlinear features of guided waves in plates have been investigated for their applications in high-resolution imaging, more accurate damage detection and various types of acoustic devices. In the following we propose a methodology for analyzing wave propagation in nonlinear plates i.e., a layer of nonlinear material bounded between two parallel traction-free boundaries. The proposed approach employs a perturbation-based strategy for sequentially solving the nonlinear elastodynamic set of equations. We show that the nonlinearity induces frequency shifts that depend on the primary wave amplitude, resulting in amplitude-dependent dispersion surfaces. It is also found that the nonlinear nature of the medium impacts other wave propagation phenomena, such as internal resonances.

*Breakout rooms available to have further discussion with each speaker.

### The seminar will take a break and return on 2 February 2021.

**2nd February 2021, **

16.00-18:00 (BST)

Peter Baddoo,(Imperial College London)Title:

TBCAbstract: TBC

Georgy Faranosov,(TsAGI, Russia)Title:

TBCAbstract: TBC

Andrew Gibbs,(University College London)Title:

TBCAbstract: TBC

*Breakout rooms available to have further discussion with each speaker.

** Previous seminars **

**3 November 2020**

16.00-18:00 (BST)

Professor Siyang Zhong ,(HKUST, China)Title:

Acoustic imaging of low-frequency sources with extended resolutionAbstract: Acoustic imaging is to identify the locations and strengths of the sound sources, based on which the analysis of sound generation mechanisms is possible. The resolution is often dependent on the frequency of the sound source. At low frequency, a practical issue is that various closely located sources may not be distinguishable due to the large wavelength, leading to challenges for correct interpretation of the acoustic physics. One key stage in our proposed extended-resolution acoustic imaging method is to introduce a high-frequency incident wave to interact with the target sources. The nonlinear interaction between different source components can induce sound waves at an alternative high frequency. The induced sound carries the information of the target low-frequency, which can be rigorously estimated base on the acoustic analogy theory. Also, the wavelength of the induced sound is relatively small (depends on the frequency of the incident wave) and is beneficial for the high-resolution imaging of the low-frequency sources. We will demonstrate that sources with a separation distance smaller than 1/10 of the wavelength can be identified.

This talk is based on a joint work with W. Q. Chen and X. Huang at Peking University, China.

Stanislav. L. Denisov,(TsAGI, Russia)Title:

Application of Uniform Geometric Theory of Diffraction (UGTD) to Investigations of Aircraft Power Plant Noise Reduction by Means of Shielding Effect.Abstract: In this talk the results of aircraft power plant noise reduction by means of shielding effect will be presented. Based on the Uniform Geometric Theory of Diffraction (UGTD) expressions for the diffraction of sound on an infinite half-plane for point dipole and quadrupole sources are obtained. The obtained expressions were used to calculate the shielding efficiency for the round single-stream jet noise and azimuthal modes radiated from a cylindrical channel. The jet noise correlation theory was used as a model which describing the jet noise radiation, and the azimuthal modes radiation from a cylindrical channel was described based on the exact Weinstein solution. The calculations were performed for a flat acoustically absolutely rigid rectangular and polygonal shields which modeling the Blended-Wing-Body (BWB) airframe. The comparison of the calculated and experimental data showed a good agreement which made it possible to develop a method for calculating external noise reduction for aircraft that implement the shielding effect.

This talk is based on a joint work with Viktor. F. Kopiev, Nikolay. N. Ostrikov.

Matthew Nethercote,(University of Manchester, UK)Title:

Edge Diffraction of Acoustic Waves by Periodic Composite Metamaterials: The Hollow WedgeAbstract: Generally in metamaterial research, some form of infinite periodicity is assumed which allows us to restrict the study to a small portion known as the unit cell. This has led to many studies which increase the complexity of the unit cell and reconstruct the global scattering using the periodicity of the metamaterial. An alternative approach looks into the case where infinite periodicity is no longer assumed. This means that the metamaterial will have well-defined boundaries that can symbolise many different interfaces such as edges and corners. In this presentation, the scattering of an acoustic pressure wave by a hollowed out wedge is studied where, for simplicity, the unit cells will be sound-soft cylinders with infinite height and a small radius. This configuration can also be viewed as two separate semi-infinite gratings with two sets of scattering coefficients to determine. We will construct an iterative scheme from the resulting infinite system of equations and find a solution using the discrete Wiener-Hopf technique. We shall also discuss some tools that are useful for computations of the Wiener-Hopf kernel such as tail-end asymptotics and rational approximations. Comparison to COMSOL simulations are made and resonant cases are identified.

This is joint work with Anastasia Kisil and Raphael Assier.

**6 October 2020**

16.00-18:00 (BST)

Marie Touboul,(LMA, France)Title:

High-frequency homogenisation in 1D periodic media with imperfect interfaces of the spring-mass typeAbstract:In this work, the concept of high-frequency homogenisation is extended to the case of one-dimensional periodic media that have an imperfect interface at the edges of the periodic cell. Indeed, when considering the propagation of elastic waves, displacement and stress discontinuities of the spring-mass types are allowed across the borders of the periodic cell. The high-frequency homogenisation of such media is carried out about the edges of the Brillouin zone that do not correspong to angular frequencies that are small, contrary to the classical low-frequency homogenization framework. At these edges, the dispersion diagram displays band-gaps, i.e. regions in the angular frequency space where waves do not propagate. A two-scale asymptotic expansion method is applied in order to approximate how the dispersion relation will behave around these zones. Asymptotic expansions are thus provided for the higher branches of the dispersion diagram and the resulting wavefield. The limiting case of two branches of the dispersion diagram that intersect with a non-zero slope (Dirac point) is also studied. The intermediate case of narrow bandgaps is also considered in order to obtain a uniform approximation that remains valid in the Dirac point limit. The examples of a monolayerd and a bilayered material are treated by this homogenised approach and a Bloch-Floquet approach in order to illustrate the validity of the method presented.

Ory Schnitzer,(Imperial, UK)Title:

Asymptotic solutions of the plasmonic eigenvalue problem.Abstract:A flat interface between a metal and a dielectric supports surface electromagnetic waves, typically in the visible range. Such surface waves have the special property that their wavelength can be squeezed down to nanometric scales, small in comparison to the wavelength of light in free space. For this reason, subwavelength metallic nanoparticles support quasi-static “plasmonic” modes, governed by a purely geometric “plasmonic eigenvalue problem,” whose resonant excitation by incident radiation results in enhanced absorption and scattering. In particular, many applications of plasmonic resonance, such as targeted heating and bio-sensing based on nonlinear optics, employ multiple-scale particle geometries in order to enhance and localise electric fields on nanometric scales. Motivated by this, in this talk I will discuss the asymptotic properties of multiple-scale plasmonic particles including close-to-touching cylinders and spheres, and slender particles (work on slender particles is joint with Matias Ruiz). Time permitting, I will briefly comment on other asymptotic results in plasmonics, including the accumulation of modes at the surface-plasmon frequency and nonlocal effects.

Jean-François Mercier,(ENSTA Paris, France)Title:

Homogenization of an array of Helmholtz resonators: application to perfect absorptionAbstract: We inspect the influence of the spacing on the resonance of a periodic arrangement of Helmholtz resonators. An effective problem is used which captures accurately the properties of the resonant array within a large range of frequency. It is shown that the strength of the resonance is enhanced when the array becomes sparser. This degree of freedom on the radiative damping is of particular interest since it does not affect the resonance frequency nor the damping due to losses within each resonator. We show that it can be used for the design of a perfect absorbing walls.

**Tuesday 21st July, 2020**

16.00-17.30 (BST)

Andrey Shanin(Moscow State University)Title:

Exchange pulse corresponding to phase synchronism in a flexible plate loaded by gasAuthors: A.V.Shanin (presenting), M.A. Mironov, A.I.Korolkov, K.S.Kniazeva

Abstract:A problem of excitation of a sonic pulse in a system comprised by an elastic plate and a gas is studied. The source is localized in time and space, so the problem is non-stationary. It is known that such a system can have a phase synchronism point in the spectral domain, i.e. the values of frequency and wavenumber belonging both to the dispersion diagram of the plate and the gas. We demonstrate that such a point leads to appearing of an ``exchange pulse'', that is a quasi-monochromatic long pulse in the gas. In the talk, we write down an integral representation of the sonic field and discuss a method of asymptotic evaluation of the 2D Fourier integral. We demonstrate the term responsible for the exchange pulse. The work is motivated by an experiment made by one of the authors, who recorded sound produced by kicking of a thin (about 3 cm) layer of ice on a pond. The ice plays the role of the plate, and the gas is the air. The experimental signals generally support the theory presented in the talk. The work is supported by RFBR grant 19-29-06048 MK.

This talk is based on the following paper,

Artur L Gower(University of Sheffield)Title:

Ensemble average waves in random materials of any geometryAbstract: 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.

Suppose 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.

**Tuesday 7th July, 2020**

Lorna Ayton(University of Cambridge)Title:

Streamwise varying porosity and trailing-edge noiseAbstract: Whilst it is known that porosity can significantly reduce the aerodynamic noise scattered by sharp edges, previous studies focus only on structures which are uniformly porous, or have an impermeable section and uniformly permeable section. The air flow resistance through birds wings (related to the porosity) however varies along the chord, and chordwise variations from barn owl wings and common buzzard wings are measurably different. Unsuprisingly, the owl's distribution of porosity is predicted to produce less trailing edge noise. By considering similar monotonic variations of porosity, we illustrate that it is not only what the edge values of porosity are, but how you get between them that impacts the total scattered noise.

Martin Richter(Nottinham University)Title:

Convergence Properties of Dynamical Energy Analysis - A Ray-based Method Using Transfer OperatorsAbstract:Describing the distribution of vibrational energy in real world applications is challenging, especially in the mid-to-high frequency regime. A very promising way is using densities of rays and creating a transfer operator T by a method called Dynamical Energy Analysis. This has been used in a range of real-world scenarios ranging from gear boxes over car bodies to whole ship hulls, see for example [1]. The role of this operator T is to propagate the intensity of the vibrational excitation across a given structure. Using local properties of the structure, this propagation can account for different material properties and geometrical aspects. For thin shells for example, it describes the vibrations in terms of pressure, shear, and bending wave modes. In order to compare predictions with experimental data this tool has to be used numerically. This makes it necessary to represent T in terms of a finite basis. To have an efficient implementation with a good convergence and small errors it is necessary to choose an optimal set of basis functions. The talk will cover convergence properties of this method as well comparing them with a simple analytical model. [1] Hartmann, Morita, Tanner, Chappell Wave Motion, 87, 2019, 132-150.

**Tuesday 23rd June, 2020 **

Anne-Sophie Bonnet-Ben Dhia(ENSTA Paris)

Title:Complex-scaling method for the plasmonic resonances of a 2D subwavelength particle with cornersAbstract:It is well-known that a metallic particle can support surface plasmonic modes. For a subwavelength particle, these modes correspond to negative values of the permittivity, which are solutions of a self-adjoint eigenvalue problem. In this work, we are interested in the finite element computation of plasmonic modes in the case of a 2D particle whose boundary is smooth except for one corner. While a smooth particle has a discrete sequence of plasmonic eigenvalues, the corner leads to the presence of an essential spectrum, due to the existence of hyper-oscillating waves at the corner, the so called black-hole waves. Following our previous works, we introduce a complex scaling at the corner, and solve the complex-scaled eigenvalue problem. Numerical results show that the complex scaling deforms the essential spectrum, so as to unveil both embedded eigenvalues and complex plasmonic resonances. The later are analogous to scattering resonances with the local behavior at the corner playing the role of the behavior at infinity. We illustrate in particular the study of Li and Shipman (J. Integral Equations and Appl., 31(4), 2019), which proved the existence of embedded eigenvalues for the Neumann-Poincaré operator for a geometry with reflectional symmetry.

Erik Garcia(University of Manchester)Title:

Thermo-Visco-Elastic effects in Wave PropagationAbstract: Recent work in the metamaterial literature has shown the importance of taking into account the presence of visco-thermal losses through dissipation for the accurate description of acoustic fields in a variety of metamaterial designs of practical interest. In general, the background fluid in which the material is aimed to operate plays an important role in the design and associated mathematical modelling. This is for example illustrated in underwater structures which are typically subject to high hydrostatic pressure loads, and hence fluid-structure interaction (FSI) effects must be taken into account, as opposed to standard in-air scenarios where FSI is often negligible. Many interesting solid metamaterials (such as membrane type media), are of a (thermo-) viscoelastic nature. It would therefore be convenient to develop a framework that allows the study of the acoustics of thermo- viscoelastic continua, including both solids and fluids. The derivation of such a model will be introduced and put into practice by considering the canonical problem of the interaction of two thermo-viscoelastic media separated by an interface. In particular, differences with conventional acoustic models will be highlighted. This is a joint work with A. L. Gower (University of Sheffield), P. A. Cotterill, R. C. Assier, W. J. Parnell (University of Manchester) and D. Nigro (Thales UK).

**Tuesday 9th June, 2020 ** 1

Andrey Korolkov(Moscow State University)

Title:Sommerfeld integral method for discrete diffraction problemsAbstract: Two 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)Title:

Recent advances on the preconditioning of 3D fast Boundary Element Solvers for 3D acoustics and elastodynamicsAbstract:Recent works in the Boundary Element Method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. I will present some recent works on analytic and algebraic preconditioners for fast BEMs.

**Tuesday 26th May, 2020 **

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 12th May, 2020 **

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 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.