Classical Wave Research Articles

On the eigenvalues and eigendisplacement of the critical mode in horizontally layered media

Wave propagation in horizontally layered media is a classical problem in seismic-wave theory. In semi-infinite space, a nondispersive Rayleigh wave mode exists, and the eigendisplacement decays exponentially with depth. In a layered model with increasing layer velocity, the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer. If the phase velocity is the same as the P- or S-wave velocity of the layer, which is called the critical mode or critical phase velocity of surface waves, the general solution of the wave equation is not a homogeneous (expressed by trigonometric functions) or inhomogeneous (expressed by exponential functions) plane wave, but one whose amplitude changes linearly with depth (expressed by a linear function). Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode, owing to the singularity at the critical phase velocity. In this study, based on the classical framework of generalized reflection and transmission coefficients, the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity. Therefore, the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem. The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models. In contrast to the normal mode, the eigendisplacement associated with the critical phase velocity exhibits different characteristics. If the phase velocity is equal to the S-wave velocity in the bottom half-space, the eigendisplacement remains constant with increasing depth.

Water-Wave Vortices and Skyrmions.

Topological wave structures-phase vortices, skyrmions, merons, etc.-are attracting enormous attention in a variety of quantum and classical wave fields. Surprisingly, these structures have never been properly explored in the most obvious example of classical waves: water-surface (gravity-capillary) waves. Here, we fill this gap and describe (i)water-wave vortices of different orders carrying quantized angular momentum with orbital and spin contributions, (ii)skyrmion lattices formed by the instantaneous displacements of the water-surface particles in wave interference, and (iii)meron (half-skyrmion) lattices formed by the spin-density vectors, as well as (iv)spatiotemporal water-wave vortices and skyrmions. We show that all these topological entities can be readily generated in linear water-wave interference experiments. Our findings can find applications in microfluidics and show that water waves can be employed as an attainable playground for emulating universal topological wave phenomena.

Theory and simulation of shock waves freely propagating through monoatomic non-Boltzmann gas

The effect of non-Boltzmann energy distributions on the free propagation of shock waves through a monoatomic gas is investigated via theory and simulation. First, the non-Boltzmann heat capacity ratio γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, as a key property for describing shock waves, is derived from first principles via microcanonical integration. Second, atomistic molecular dynamics simulations resembling a shock tube setup are used to test the theory. The presented theory provides heat capacity ratios ranging from the well-known γ=5/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma = 5/3$$\\end{document} for Boltzmann energy-distributed gas to γ→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\rightarrow 1$$\\end{document} for delta energy-distributed gas. The molecular dynamics simulations of Boltzmann and non-Boltzmann driven gases suggest that the shock wave propagates about 9% slower through the non-Boltzmann driven gas, while the contact wave appears to be about 4% faster if it trails non-Boltzmann driven gas. The observed slowdown of the shock wave through applying a non-Boltzmann energy distribution was found to be consistent with the classical shock wave equations when applying the non-Boltzmann heat capacity ratio. These fundamental findings provide insights into the behavior of non-Boltzmann gases and might help to improve the understanding of gas dynamical phenomena.Graphical abstract

Stepless space-regulation of topological acoustic controller with high fault tolerance

In this paper, the stepless space-regulation of topological acoustic transmission channels with high fault tolerance is proposed through introducing structural defect dislocations into a topological acoustic controller. Due to the stability of topological order against local disturbance, the acoustic wave transmission is immune to dislocation boundaries with strong stability, and thus the topological acoustic controller has high fault tolerance. By continuous changing the dislocation, the position relationship between the outgoing and incident acoustic signals no longer limited to the integer multiple distance related to the lattice size, and can realize the efficient acoustic energy transmission without energy loss at the fractional multiple distance, that is, the topological controller can realize lossless acoustic energy transmission and reception in arbitrary position relationship. Furthermore, the coupling relationship between the defect dislocation and the topological acoustic channel is explored, which can realize the stepless space-regulation of the lossless channel in the wide band range. In addition, by further introducing multi-layer continuous dislocations, this high-fault-tolerant topological acoustic controller still has strong stability, and multiple error factors do not affect the transmission results, which greatly reduces the difficulty of manufacturing. Finally, the stepless space-regulation of topological acoustic channels and the high-fault-tolerant topological acoustic controller that are easy to manufacture are verified by our experiments. This research paves the way for the engineering applications of acoustic micro-control, micro-nano fabrication, remote acoustic energy transmission manipulation, acoustic measurement, weak signal processing, acoustic flexible control and other micro-shape and multi-functional acoustic devices, and will bring more inspiration to other classical wave communication fields such as light wave, electromagnetic wave and so on.

Lifespan estimates for 1d damped wave equation with zero moment initial data

In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case when the Fourier 0th moment of sum of initial position and speed is 0. Especially, it is shown that the behavior of lifespan changes with p=3/2 with respect to the size of the initial data.

Parallel-swath-array analysis of passive source surface waves based on beamforming

The detection of mined-out coal area remains a pivotal subject in geophysical prospecting and presents an urgent issue requiring resolution. A refined and effective approach to tackle this challenge involves conducting dispersion analysis on passive surface waves, which are derived from ambient noises. While classical passive surface wave methods, such as the Spatial Autocorrelation Method (SPAC), rely on spatial and time domain stationary assumptions, these can be difficult to fulfill. This is especially true when using ambient noises with frequencies >1 Hz, leading to a growing disparity between basic assumptions and actual scenario. To reduce this discrepancy, it's crucial to account for the non-stationary features of noise sources. In this study, we introduce a pioneering approach for parallel-swath-array analysis of passive source surface waves based on beamforming, making full use of non-stationary ambient noises. The parallel-swath array allows for efficient data collection and high-density spatial observation. The method's core concept involves: 1) The collection of noise data by a 2-D array arranged in three parallel lines; 2) The use of a plane-wave basis function to directly scan the original noise records in a time-segment mode within the time domain. These are conducive to unveil the azimuths and time-variant characteristics of the noise sources, resulting in accurate surface wave dispersion information. To ascertain the feasibility and effectiveness of this innovative method, we simulated passive surface wave records for a geological model, accommodating both time-invariant and time-variant distribution of noise sources. The results demonstrate that the method successfully reveals the time-invariant or time-variant characteristics of noise sources and the associated surface wave dispersion information. Simultaneously, we employ the preprocessed steepest-descent (PSD) algorithm to invert the dispersion curves, and the low-velocity layer in the numerical tests can be clearly revealed. In conclusion, we deployed this method in a coal mine situated in Northern China, which encompasses abandoned mined-out areas. This procedure effectively disclosed the intricate azimuthal properties of the noise sources, yielding high-resolution dispersion images. The inversion outcomes depicted a 2-D S-wave velocity section plunging to a depth of 60 m. By interpreting the low-velocity features of the mined-out area, we can accurately delineate its boundaries. The results of the interpretation align coherently with the findings from the drilling data, to verify the feasibility and effectiveness of parallel-swath-array analysis of passive source surface waves based on beamforming in detecting coal mined-out area under non-stationary ambient noises.

De Broglie, General Covariance and a Geometric Background to Quantum Mechanics

What is striking about de Broglie’s foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ2Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow.

Topological phase-dependent thermalization dynamics in radiative heat transfer: insights from a one-dimensional Su-Schrieffer-Heeger model.

Various unusual behavior of artificial materials is governed by their topological properties, among these, the edge state in classical and quantum wave systems has captured significant attention due to its widespread relevance and applications across various fields of study. Observation of such topological features has led researchers to extend the idea of band theory to diffusive systems. Inspired by the well-known Su-Schriefer-Heegar (SSH) model we employed the concept of band topology to explore the topological characteristics of radiative heat transfer in a one-dimensional chain consisting of an odd number of nanoparticles. We demonstrate the topological phase transition, and topological modes with edge as well as bulk states in an array of nanoparticles exchanging heat via radiation. The demonstrated topological features of radiative systems can find important applications in the future studies of heat transfer at the nanoscale.

Wave Propagation in High-Contrast Media: Periodic and Beyond

Abstract This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition and show a priori error estimates in suitably weighted norms. Numerical experiments illustrate the convergence rates in various settings.

Multiplicity of the large periodic solutions to a super-linear wave equation with general variable coefficient

<abstract><p>In this paper, we were concerned with the multiplicity of the large periodic solutions to a super-linear wave equation with a general variable coefficient. In general, the variable coefficient $ \rho(\cdot) $ needs to be satisfied $ \text{ess inf}\, \eta_\rho(\cdot) > 0 $ with $ \eta_\rho(\cdot) = \frac{1}{2}\frac{\rho''}{\rho}-\frac{1}{4}\big(\frac{\rho'}{\rho}\big)^2 $. Especially, the case $ \eta_\rho(\cdot) = 0 $ is presented as an open problem in [Trans. Amer. Math. 349: 2015-2048, 1997]. Here, without any restrictions on $ \eta_{\rho}(\cdot) $, we established the multiplicity of large periodic solutions for the Dirichlet-Neumann boundary condition and Dirichlet-Robin boundary condition when the period $ T = 2\pi\frac{2a-1}{b} $ with $ a, b \in \mathbb{N}^+ $. The key ingredient of the proof is the combination of the variational method and an approximation argument. Since the sign of $ \eta_\rho(\cdot) $ can change, our results can be applied to the classical wave equation.</p></abstract>

Dislocation defect states in acoustic quadrupole topological insulators

Quadrupole topological insulator (QTI) is the first proposed higher-order topological phase of matter with quantized quadrupole moment but zero dipole moment. The QTI has expanded widely the traditional bulk-boundary correspondence, thereby the lower-dimensional topological boundary state can be observed. The recent interest has turned to the bulk-dislocation correspondence, which dominates the topological states localized to disclinations, and links the reciprocal-space topology of lattices with the appearance of dislocation states. Recently, many research groups have turned the studies of dislocation defects to classical wave systems. In these researches, the method of inducing dislocation defects is to remove a portion of the lattices of topological insulator and then rearrange the remaining lattices of the topological insulator. Through such a method, the micro structure of the lattices is changed, but it is difficult to realize in the actual operation. In this work, we study the dislocation defect states in acoustic QTIs. The acoustic QTI is designed by reversing the magnitude of the intracellular and extracellular coupling in the system, and the bulk energy bands and topological corner states are studied. Subsequently, by introducing partial trivial lattices into acoustic QTI structure, the dislocation bound states are generated in the corner formed by two different topological phases, which can be characterized by a 1/2 quantized fractional charge. The robustness of the topological dislocation states is verified by introducing the imperfection inside the system. Further, it is demonstrated that the dislocation positions can be designed at will. Without changing the microstructure of the lattice, we successfully modulate the line dislocation states and bulk dislocation states. The topological dislocation states studied in this work broaden the types of higher-order topological states in artificial structures, and provide new insights into the acoustic applications of higher-order topological insulators, such as sensing and high-performance energy harvesting.

Acoustic Modeling and Analysis of Automotive Air-Filters

Air filters are placed upstream of internal combustion engines to remove dust particles in the suction air. In this article, we examine the acoustic characterization of an air-filtering unit in the absence of mean flow. For this purpose, a circular air-filtering unit with an axial inlet and side outlet widely used in the automobile industry is considered. The air-filter element is modeled as an equivalent acoustic fluid with finite flow resistivity. The flow resistivity of the region is estimated from the permeability of the filter paper and the geometrical arrangement of the paper folds (pleats) inside the air-filter element using an electrical analogy. A numerical model based on the finite element method (FEM) and an analytical model using classical one-dimensional plane wave analysis (1-D PWA) was developed. Experiments were carried out using an impedance tube to estimate transmission loss. A good correlation is observed between the FEM model and the experiments. The results obtained from the 1-D PWA are in reasonable agreement with those obtained from the other two methods.

Classical Wave Equation: A Gross Error in Mathematics and Physics

A detailed proof of the incorrectness of the classical wave equation is proposed. The correct methodological basis for the proof is the unity of formal logic and rational dialectics. The proof leads to the following irrefutable statement: the classical wave equation and the derivation of the classical wave equation are a gross error in mathematics and physics. The proof of this statement is based on the following irrefutable main results: (1) The first gross error is the following approximate relationship: “sine of angle is approximately equal to tangent of angle; cosine of angle is approximately equal to 1”. This relationship means (implies) that the quantity of the angle and the right-angled triangle do not exist. Consequently, the relationship between the tangent of the angle and the derivative of the displacement with respect to coordinate does not exist; (2) The second gross error is that the second-order derivative of the displacement with respect to coordinate does not exist, because the dimensions of displacement and coordinate are “meter”, the first-order derivative of the displacement with respect to coordinate is dimensionless quantity (i.e. the first-order derivative of the displacement respect to coordinate has no the dimension “meter”); (3) The third gross error is that the first-order derivative of the displacement with respect to coordinate cannot be expanded into the Taylor series, because the second-order derivative of the displacement with respect to coordinate does not exist; (4) The fourth gross error is as follows: (a) the left side of the wave equation contains the condition that time is a variable quantity, and the coordinate is a constant quantity; (b) the right side of the wave equation contains the condition that time is a constant, and the coordinate is a variable. This means (implies) that the equation contains contradictory conditions (propositions). Therefore, the equation represents a violation of the formal-logical law of the lack (absence) of contradiction; (5) The fifth gross error is that the standard derivation of the equation relies on the following false theories: negative number theory, complex numbers theory, trigonometry, vector calculus, differential calculus, and Newton's second law. Thus, the classical wave equation does not satisfy the criterion of truth and is not a scientific achievement.

On the operator origins of classical and quantum wave functions

We investigate operator algebraic origins of the classical Koopman–von Neumann wave function ψKvN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{KvN}$$\\end{document} as well as the quantum-mechanical one ψQM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{QM}$$\\end{document}. We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, ψKvN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{KvN}$$\\end{document} and ψQM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{QM}$$\\end{document} are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {E}}}$$\\end{document} follows from this. ψKvN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{KvN}$$\\end{document} and ψQM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{QM}$$\\end{document} are shown to be sections in E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {E}}}$$\\end{document}. The difference between ψKvN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{KvN}$$\\end{document} and ψQM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi _{QM}$$\\end{document} originates from a quantization map interpreted as “twisting” of sections over E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {E}}}$$\\end{document}. We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.

The quantum optics of gravitational waves

By utilizing quantum optics techniques, we examine the characteristics of a quantum gravitational wave (GW) signature at interferometers. In particular, we study the problem by analyzing the equations of motion of a GW interacting with an idealized interferometer. Using this method, we reconstruct the classical GW signal from a representation of the quantum version of an almost classical monochromatic wave (a single-mode coherent state), then we discuss the experimental signatures of some specific, more general quantum states. We calculate the observables that could be used at future interferometers to probe possible quantum states carried by the GWs.

Higher-order topological states in dual-band valley sonic crystals

As a quantum state of frequency extrema in the momentum space of acoustic systems, sonic valley pseudospin provides a new degree of freedom for controlling acoustic waves. Higher-order topological insulators (HOTIs) have extended the traditional bulk-edge correspondence principle and are a crucial concept for classic wave regulation. However, HOTIs in valley sonic crystals (VSCs) only appear in a single bandgap, which limits the multi-frequency selectivity of the corner state and is not conducive to the design of multi-frequency acoustic communication devices. Here, we demonstrate “Y-shaped” acoustic crystals with C3 symmetry that form a double-band VSC, and the topological phase transitions in both low- and high-frequency band gaps coincide. We realize theoretically and experimentally higher-order states in dual-band valley sonic crystals. Our work enriches the application of HOTIs in acoustic multi-frequency regulatory systems and provides different avenues for designing of multi-band acoustic devices.

Acoustic real second-order nodal-loop semimetal and non-Hermitian modulation

The unique features of spinless time-reversal symmetry and tunable ℤ2 gauge fields in artificial systems facilitate the emergence of topological properties in the landscape, such as the recently explored Möbius-twisted phase and real second-order nodal-loop semimetals. However, these properties have predominantly been proposed only in theoretical frameworks. In this study, we present a cunningly designed blueprint for realizing an acoustic real second-order nodal-loop semimetal through the incorporation of projective translation symmetry into a three-dimensional stacked acoustic graphitic lattice. Additionally, we introduce non-Hermitian modulation to the topologically protected propagation of degenerate drumhead surface and hinge states, which depend on the specific on-site gain and loss textures. It should be emphasized that this demonstration can be extended to other classical wave systems, thereby potentially opening up opportunities for the design of functional topological devices.

Orbital topological edge states and phase transitions in one-dimensional acoustic resonator chains

Topological phases of matter have attracted significant attention in recent years, due to the unusual robustness of their response to defects and disorder. Various research efforts have been exploring classical and quantum topological wave phenomena in engineered materials, in which different degrees of freedom (DoFs) – for the most part based on broken crystal symmetries associated with pseudo-spins – induce synthetic gauge fields that support topological phases and unveil distinct forms of wave propagation. However, spin is not the only viable option to induce topological effects. Intrinsic orbital DoFs in spinless systems may offer a powerful alternative platform, mostly unexplored to date. Here we reveal orbital-selective wave-matter interactions in acoustic systems supporting multiple orbital DoFs, and report the experimental demonstration of disorder-immune orbital-induced topological edge states in a zigzag acoustic 1D spinless lattice. This work expands the study of topological phases based on orbitals, paving the way to explore other orbital-dependent phenomena in spinless systems.

Bohmian mechanics of the three-slit experiment in the linear potential

We report on a three-slit experiment in the presence of a linear potential with surface gravity water waves. For these classical waves, we reconstruct the Bohm trajectories as well as the corresponding quantum potentials.

Symplectic time-domain finite element method (STD-FEM) for room acoustic modeling

A new, extended symplectic time-domain finite element method (STD-FEM) is proposed for room acoustic modeling. As a mathematical model for sound propagation and reflection, the classical time-domain wave equations are used, which can be extended with air absorption. Frequency-dependent locally reactive boundary conditions are also introduced to the model. Spatially, a third-order tensor product type spectral element discretization is applied, which allows us to use explicit time steps. For this purpose, a partitioned Runge-Kutta method is employed, which is an extension of a third-order symplectic time-discretization. The method is validated on a number of examples and the performance and stability benefits are presented. We present the implementation of the method in the soundy.ai application.