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1. Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: transformation between Cartesian and ray-centred coordinates

   https://doi.org/10.1093/gji/ggab151  

   Iversen, Einar; Ursin, Bjørn; Saksala, Teemu; Ilmavirta, Joonas; de Hoop, Maarten V (May 2021, Geophysical Journal International)

   null (Ed.)

   SUMMARY Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we (1) visualize the validity limits of the ray-centred coordinate system, (2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates and (3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model. 

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2. Multivariable calculus textbook representation of non-cartesian coordinates: a misalignment between multivariable calculus textbook content and upper-division physics application

   https://doi.org/10.1088/1361-6404/ad2b98  

   Dalton, Chaelee; Farlow, Brian; Christensen, Warren M (March 2024, European Journal of Physics)

   Abstract Upper-division undergraduate physics coursework necessitates a firm grasp on and fluid use of mathematical knowledge, including an understanding of non-cartesian (specifically polar, spherical and cylindrical) coordinates and how to use them. A limited body of research into physics students’ thinking about coordinate systems suggests that even for upper-division students, understanding of coordinate system concepts is emergent. To more fully grasp upper-division physics students’ incoming understanding of non-cartesian coordinates, the prevalence of non-cartesian content in seven popular Calculus textbooks was studied. Using content analysis techniques, a coding scheme was developed to gain insight into the presentation of coordinate system content both quantitatively and qualitatively. An initial finding was that non-cartesian basis unit vectors were absent in all but one book. A deeper analysis of three of the calculus textbooks showed that cartesian coordinates comprise an overwhelming proportion of the textbooks’ content and that qualitatively the cartesian coordinate system is presented as the default coordinate system. Quantitative and qualitative results are presented with implications for how these results might impact physics teaching and research at the middle and upper-division. 

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3. Cartesian Operator Factorization Method for Hydrogen

   https://doi.org/10.3390/atoms10010014  

   Lyu, Xinliang; Daniel, Christina; Freericks, James K. (March 2022, Atoms)

   We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schrödinger factorizations. 

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4. Discretizing advection equations with rough velocity fields on non-Cartesian grids

   https://doi.org/10.1090/qam/1649  

   Jabin, Pierre-Emmanuel; Zhou, Datong (March 2023, Quarterly of Applied Mathematics)

   We investigate the properties of discretizations of advection equations on non-Cartesian grids and graphs in general. Advection equations discretized on non-Cartesian grids have remained a long-standing challenge as the structure of the grid can lead to strong oscillations in the solution, even for otherwise constant velocity fields. We introduce a new method to track oscillations of the solution for rough velocity fields on any graph. The method in particular highlights some inherent structural conditions on the mesh for propagating regularity on solutions. 

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5. Evidence for near-source nonlinear propagation of volcano infrasound from Strombolian explosions at Yasur Volcano, Vanuatu

   https://doi.org/10.1007/s00445-022-01552-w  

   Maher, Sean P.; Matoza, Robin S.; Jolly, Arthur; de Groot-Hedlin, Catherine; Gee, Kent L.; Fee, David; Iezzi, Alexandra M. (March 2022, Bulletin of Volcanology)

   Abstract Volcanic eruption source parameters may be estimated from acoustic pressure recordings dominant at infrasonic frequencies (< 20 Hz), yet uncertainties may be high due in part to poorly understood propagation dynamics. Linear acoustic propagation of volcano infrasound is commonly assumed, but nonlinear processes such as wave steepening may distort waveforms and obscure the sourcing process in recorded waveforms. Here we use a previously developed frequency-domain nonlinearity indicator to quantify spectral changes due to nonlinear propagation primarily in 80 signals from explosions at Yasur Volcano, Vanuatu. We find evidence for$$\le$$ $\le$10⁻³ dB/m spectral energy transfer in the band 3–9 Hz for signals with amplitude on the order of several hundred Pa at 200–400 m range. The clarity of the nonlinear spectral signature increases with waveform amplitude, suggesting stronger nonlinear changes for greater source pressures. We observe similar results in application to synthetics generated through finite-difference wavefield simulations of nonlinear propagation, although limitations of the model complicate direct comparison to the observations. Our results provide quantitative evidence for nonlinear propagation that confirms previous interpretations made on the basis of qualitative observations of asymmetric waveforms. 

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