KOZWavinar is the online conference series for the KOZWaves community.

Each KOZWavinar features three 30-minute talks followed by a 30-minute discussion session with the speakers (i.e. 2h in total).

To be added to the mailing list __send an email to kozwavinar@gmail.com with the subject "add me"__.

Visit the KOZWavinar YouTube channel.

Marie Graff, University of Auckland

Amin Chabchoub, Kyoto University

Luke Bennetts, University of Adelaide

Jörg Hennig, University of Otago, NZ

**Waves on the Schwarzschild black hole spacetime**

In General Relativity, gravitational waves are well-defined only at infinity. For a rigorous mathematical description, one can use the method of conformal compactification, which brings infinity to finite coordinate locations. As an illustrative example, we examine waves on the Schwarzschild spacetime, a simple exact solution describing a spherically-symmetric black hole. Specifically, we study the scalar, conformally invariant wave equation, which can be considered as a toy model for the full field equations. Although this wave equation is much simpler, it retains essential mathematical properties and challenges of the general problem. Our main interest is in a suitable treatment of a region called spatial infinity, which is represented as a cylinder. Firstly, we consider the Cauchy problem for the wave equation. We study a family of equations intrinsic to the cylinder, where the solutions turn out to have, in general, logarithmic singularities at infinitely many expansion orders. We derive regularity conditions that may be imposed on the initial data, in order to avoid the first singular terms. We then demonstrate that a fully pseudospectral time evolution scheme can be applied to solve the Cauchy problem numerically. Additionally, we investigate the characteristic initial value problem for the wave equation, highlighting key differences from the Cauchy problem.

Zuorui Lyu, Disaster Prevention Research Institute, Uji, Japan

**Freak Wave in a Two-Dimensional Directional Wavefield with Bottom Topography Change**

In the propagation and evolution of the sea waves, previous studies pointed out that the occurrence of the freak wave height is significantly related to the quasi-resonant four-wave interaction in the modulated waves and the nonlinear effect caused by the bathymetry change. To conduct a comprehensive analysis in a two-dimensional sea, we develop an evolution model for a directional random wavefield based on the modified Nonlinear Schrödinger equation, which considers the high-order nonlinear interactions, the wave shoaling in shallow water, wave refraction and crossing state. Through Monte Carlo simulation, we discuss the directional effect on the four-wave interaction in the wave train and the maximum wave height distribution from deep to shallow water with a slow varying slope. The numerical result indicates that the directional spreading has a dispersion effect on the occurrence of the freak wave height. In shallow water depth, this effect becomes weak, and the bottom topography change is the main influencing factor in the wave evolution.

Ying Wu, KAUST, Saudi Arabia

**Double-zero-index materials**

Wave propagating in a medium with one or more constitutive parameters vanishing does not accumulate any phase retardation and the medium is the so-called zero-index medium. Such a medium is not mathematically interesting but also bears unusual functionalities, such as wave front engineering, cloaking of objects and wave tunnelling.

In this seminar, I will report our progress in realizing double-zero-index materials for both electromagnetic and acoustic waves. Their unique feature in cloaking will be discussed. Then I will delve into our recent realization of a 2D double-zero-index material for air borne sound using space-coiling structures.

Yasushi Fujiwara, Kobe University

**Numerical study of turbulence production by attenuating surface waves**

Ocean surface waves are known to induce upper ocean turbulence, whereby influences the sea surface temperature. Therefore, a proper understanding and modeling of this process is essential for accurate climate prediction. Recent studies using direct numerical simulations of water waves and turbulence suggest that the viscous attenuation of waves and resulting Eulerian streaming induce near-surface turbulence even without wave breaking or wind stress, in a similar manner to the production of Langmuir circulations. Here, we further investigate this process using a newly developed numerical code that simulates the two-phase fluid motions, where the interfacial wave is freely propagated without external forcing. Compared to the water-only simulation, the resulting water-side turbulence was enhanced by the presence of the air-layer above. Energy budget analysis reveals that a large amount of energy is dissipated at the bottom boundary layer of air, where a strong shear is present. This enhances the production of Eulerian streaming at the upper water layer, which results in stronger Langmuir circulations. This process can be successfully reproduced with a Craik-Leibovich type wave-averaged simulation, where the Eulerian streaming is forced through the "virtual wave stress" of Longuet-Higgins [1969] with a modification for the enhanced wave attenuation due to air-side boundary layer.

Tristan Lawrie, University of Nottingham, UK

**A Quantum Graph Approach to Metamaterial Design - Reflection, Refraction, Cloaking and Fourier Filters**

Since the turn of the century, metamaterials have gained a large amount of attention due to their potential for possessing highly nontrivial and exotic properties—such as cloaking or perfect lensing. There has been a great push to create reliable mathematical models that accurately describe the required material composition. Here, we consider a quantum graph approach to metamaterial design. An infinite square periodic quantum graph acts as a paradigm for a 2D metamaterial. The metamaterial properties are understood and engineered by manipulating the band diagram of the periodic structure. The engineered properties are then demonstrated in terms of the reflection and transmission behaviour of Gaussian beam solutions at an interface between different metamaterials. To engineer the interfaces transmissive properties, the boundary is decorated with additional graph structures, constructing a Fourier filter. The metamaterial interface then acts as an angular filter, designed to yield perfect transmission at customizable and discrete angles of incidence.

Kate Lee, University of Auckland

**Calibrating approximate Bayesian credible intervals of gravitational-wave parameters**

Approximations are commonly employed in realistic applications of scientific Bayesian inference, often due to convenience if not necessity. In the field of gravitational-wave (GW) data analysis, fast-to-evaluate but approximate waveform models of astrophysical GW signals are sometimes used in lieu of more accurate models to infer properties of a true GW signal buried within detector noise. In addition, a Fisher-information-based normal approximation to the posterior distribution can also be used to conduct inference in bulk, without the need for extensive numerical calculations such as Markov chain Monte Carlo (MCMC) simulations. Such approximations can generally lead to an inaccurate posterior distribution with poor statistical coverage of the true posterior. In this article, we present a novel calibration procedure that calibrates the credible sets for a family of approximate posterior distributions, to ensure coverage of the true posterior at a level specified by the analyst. Tools such as autoencoders and artificial neural networks are used within our calibration model to compress the data (for efficiency) and to perform tasks such as logistic regression. As a proof of principle, we demonstrate our formalism on the GW signal from a high-mass binary black hole merger, a promising source for the near-future space-based GW observatory LISA.

Miro Erkintalo, University of Auckland

**Cavity solitons in externally driven optical resonators**

An optical resonator is a device in which light can circulate along a closed path. When externally driven with coherent laser light, strong build-up of light intensity can occur within the device, giving rise to rich nonlinear optical effects. Arguably the most celebrated of such effects is the spontaneous emergence of localized dissipative structures known as temporal cavity solitons: pulses of light that go round-and-round the resonator, maintaining constant shape and energy (theoretically indefinitely). At the resonator output, the solitons emanate as a periodic train of pulses, and are thus associated with a Fourier spectrum comprising of an array of sharp, equally-spaced lines – an optical frequency comb. Since their discovery about 10 years ago, soliton frequency combs have enabled numerous breakthroughs in applications ranging from telecommunications to astrophysics, and today correspond to one of the most vibrant topics of research in photonics.

Takahito Iida, Osaka University

**Anti-scattering cloaking against water waves using floating thin plate**

Offshore structures are often exposed to violent waves. To protect these structures from sea waves, a cloaking concept is utilized. A thin circular plate consisting of some horizontal layers is designed to cancel scattering waves. The plate is arranged on the water surface to surround the offshore structure. Since a wave drift force acts on the structure as a result of the scattering wave generation, almost zero wave drift force could be achieved by the anti-scattering condition. In addition, we also try to realize a calm water space inside the floating thin plate.

Nicole Kessissoglou, University of New South Wales

**Sound radiation from a locally resonant coated shell**

This work investigates sound radiation from a locally resonant coated cylindrical shell submerged in water. The shell is externally coated with a viscoelastic material with an impedance similar to the surrounding water. The coating is embedded with layers of cavities or hard inclusions which predominantly exhibit monopole and dipole resonance scattering, respectively. The coating is treated as a multilayered equivalent fluid, in which the layers of inclusions are modelled as homogenised layers with effective material and geometric properties. The vibroacoustic response of the coated shell is derived by assembling and solving continuity and kinematic conditions at the interfaces between the cylindrical shell, the various coating layers, and the exterior acoustic domain. Coatings with inclusions tuned to different resonance frequencies are examined, and physical mechanisms governing the radiated sound from the coated shell are described.

Nicole Jones, University of Western Australia

**The dynamics of bottom boundary layers forced by tidal and nonlinear internal waves**

The interaction between ocean currents and the seabed generates a turbulent bottom boundary layer (BBL). Despite the BBL often being thin with respect to the total ocean depth, the dynamics of this layer are of great physical, biological, and engineering importance. In a stratified water column, the BBL produces a bottom mixing-layer (BML) capped by a strongly stratified pycnocline, which confines bottom-generated turbulence close to the seabed. The BBL plays a dominant role in dissipating kinetic energy, it contributes significantly to net mixing across the global ocean, it facilitates the exchange of sediments, nutrients, and organic matter between the seabed and the ocean interior, and it dictates the loading, undermining (i.e., scour), and embedment of seabed infrastructure. Here we use a combination of field observations and idealised large-eddy simulations to explore the stratified BBL under both tidal and nonlinear internal wave forcing.

We found that the BML height depends upon ambient stratification, tidal frequency and turbulence intensity, and can be approximated by a steady-state empirical model for a stratified Ekman layer, substituting the inertial frequency with the tidal frequency. The transport of turbulence, rather than local shear production, was the primary source of turbulence in the pycnocline. This has implications for the growth rate of the BML; as the BML height increases, turbulence transport to the pycnocline diminishes and the BML growth rate decreases. Nonlinear internal waves and internal tides compressed and expanded the BML, impacting on both the mean velocity profile and the turbulent flow in this layer.

Our predictive model of the BML height and pycnocline characteristics can be used to refine BBL parameterisations and inform grid resolution requirements for ocean models, inform the estimation of hydrodynamic loads on sub-sea engineering infrastructure and guide sampling for future observational studies.

co-authors: Andrew Zulberti, Madi Rosevear, Matthew Rayson and Gregory Ivey

Kei Matsushima, University of Tokyo

**Tailoring acoustic and electromagnetic waves based on structural optimization**

Recent studies have discovered novel physics behind the classical wave propagation and scattering. For example, some scattering systems exhibit resonant states that are completely confined in the vicinity of the scatterer despite of the continuous radiation spectrum. This phenomenon, bound state in the continuum, is expected to open up new possibilities for realizing innovative sensors, filters, and lasers. In this talk, we discuss some structural optimization-based techniques to realize such anomalous wave phenomena in acoustics and electromagnetics. First, we introduce the concept of structural optimization and explain its basic algorithm. The optimization algorithm is then incorporated into finite/boundary element software, which solves exterior time-harmonic scattering problems numerically. We show that the proposed method can realize the following three properties: (1) bound state in the continuum in open waveguide systems (2) unidirectional invisibility (cloaking) in a parity-time symmetric system (3) omnidirectional acoustic cloaking induced by a locally resonant sonic material.

Harald Schwefel, University of Otago

**Waves and nonlinear wave mixing in high quality optical resonators**

Waves bounded by convex boundaries can from a phenomenon called Whispering Gallery Modes. The name stems from acoustic waves creeping along a convex wall, such as found in Temple of Heaven or the St Paul’s Cathedral, first described mathematically by Lord Rayleigh. We use a similar phenomenon in circularly shaped disks of dielectric materials, where the electromagnetic radiation is trapped via total internal reflection close to the rim of the disk. As total internal reflection in a transparent structure much larger than the wavelength is nearly lossless, the confinement can be very high, leading to very localized and high electromagnetic fields.

We use such high quality whispering gallery modes to harness non-linear effects within the dielectric, in particular the second order nonlinear effects such as sum- and difference frequency generation, where two electromagnetic fields of different frequency can produce fields at either the sum or the difference frequency. The talk will provide the background and show application of these resonator for the generation of optical frequency combs and for quantum transduction.

Ross McPhedran, University of Sydney

**The Kronig-Penney Model: Applications to Wave Scattering**

The Kronig-Penney Model (published in 1931) is one of the most influential early works linking quantum mechanics to the properties of the solid state materials. It consisted in the application of the scalar Schrödinger equation to a periodic structure having a piecewise potential function taking alternately two different values, and its solution underpinned the development of the energy band concept for solids.

Fifty years later, an adaptation of the model to a periodic electromagnetic system was developed by the author and collaborators, and almost simultaneously by Ping Sheng and a collaborator. It consisted of a lamellar grating structure, having a dielectric constant varying periodically between two values, with the electromagnetic fields obeying the Helmholtz equation and a quasi-periodicity (Bloch/Floquet) condition. Within the grating, modes were expressed in trigonometric form, and the dispersion equation from the boundary conditions and quasiperiodicity again taking a trigonometric form. A complete set of modes could be obtained from numerical solution of the dispersion equation, with their inner products and normalisation factors being analytic. Scattering problems then could be solved by matching between plane waves and modes.

For lossless dielectrics, the modes are self-adjoint, but for lossy dielectrics and metals, adjoint modes are needed as well. The scattering problem in electromagnetism is strongly polarisation dependent, with diffraction anomalies being particularly strong for the polarisation supporting surface waves.

This work was well cited, and extended in various ways by later authors. It forms an elegant method for solving scattering by a particular type of scatterer, and the analytic nature of the expressions permits much insight to be gained into strong scattering phenomena.

Ludmila Adam, University of Auckland

**Sensing for changes in the Earth with waves**

Elastic (seismic) waves are all around us. These can be naturally generated via geological or environmental processes or human-made. The way seismic waves interact with a medium makes them highly sensitive to rock type, the fluids within and the physical conditions they experience (e.g., pressure and temperature). The sensitivity of all these parameters makes seismic waves ideal to detect small changes in the subsurface. Monitoring these changes are the building blocks for current and future early warning systems in geological hazards such as earthquakes and volcanic eruptions. I’ll discuss how we monitor volcanic processes with waves in the laboratory from three approaches: elastic wave signatures generated during volcanic unrest, monitoring the development of the seal cap within a volcano and the high sensitivity of linear and non-linear wave propagation of waves to small variations in temperature. The talk will focus on the nature and properties of waves, complemented with micro images of the medium (rocks) these waves sample to understand the implications at the field scale.

Hidetaka Houtani, University of Tokyo

**On the crest height amplification of unstable Stokes wave trains in deep water**

Rogue waves in the ocean can cause devastating structural damage to ships and offshore platforms. Modulational instability due to nonlinear quasi-resonant interaction is one of the possible causes of such rogue-wave formation. The modulational instability phenomenon is first found by Benjamin and Feir (1967). They discovered that Stokes wave is unstable to modulated perturbations. The present study focuses on the maximum crest height of unstable Stokes wave trains. The Akhmediev breather (AB) solution of the nonlinear Schrödinger equation is well known to describe the long-term nonlinear evolution of unstable Stokes wave trains. However, several fully nonlinear numerical simulations indicated that the maximum crest height of unstable Stokes wave trains exceeds the AB prediction. To clarify the physical mechanism of the crest height amplification, we performed a numerical simulation of unstable Stokes wave trains based on the higher-order spectral method (HOSM) and corresponding tank experiment. HOSM is capable of wave trains with higher nonlinearity and broader spectral bandwidth beyond the AB regime. The tank experiment demonstrated that the crest height amplification of the unstable Stokes wave trains was larger than the AB prediction at high initial wave steepness. Moreover, from the HOSM results, we will discuss the physics behind the crest-height amplification from the perspective of spectral broadening, bound-wave production, and phase-convergence.

KOZWavinar is inspired by the UK Wavinar.