{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:40:20Z","timestamp":1759970420628,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,21]],"date-time":"2025-01-21T00:00:00Z","timestamp":1737417600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The minimalist approach in the study of perturbations in fluid dynamics and magnetohydrodynamics involves describing their evolution in the linear regime using a single first-order ordinary differential equation, dubbed the principal equation.The dispersion relation is determined by requiring that the solution of the principal equation be continuous and satisfy specific boundary conditions for each problem. The formalism is presented for flows in Cartesian geometry and applied to classical cases such as the magnetosonic and gravity waves, the Rayleigh\u2013Taylor instability, and the Kelvin\u2013Helmholtz instability. For the latter, we discuss the influence of compressibility and the magnetic field, and also derive analytical expressions for the growth rates and the range of instability in the case of two fluids with the same characteristics.<\/jats:p>","DOI":"10.3390\/sym17020150","type":"journal-article","created":{"date-parts":[[2025,1,21]],"date-time":"2025-01-21T05:47:42Z","timestamp":1737438462000},"page":"150","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Classical Waves and Instabilities Using the Minimalist Approach"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8913-5176","authenticated-orcid":false,"given":"Nektarios","family":"Vlahakis","sequence":"first","affiliation":[{"name":"Department of Physics, National and Kapodistrian University of Athens, University Campus, Zografos, GR-157 84 Athens, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,21]]},"reference":[{"key":"ref_1","unstructured":"Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability, Clarendon Press."},{"key":"ref_2","unstructured":"Bateman, G. (1978). MHD Instabilities, MIT Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Boyd, T.J.M., and Sanderson, J.J. (2003). The Physics of Plasmas, Cambridge University Press.","DOI":"10.1017\/CBO9780511755750"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Freidberg, J.P. (2014). Ideal MHD, Cambridge University Press.","DOI":"10.1017\/CBO9780511795046"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Goedbloed, H., Keppens, R., and Poedts, S. (2019). Magnetohydrodynamics of Laboratory and Astrophysical Plasmas, Cambridge University Press.","DOI":"10.1017\/9781316403679"},{"key":"ref_6","first-page":"43","article-title":"Relativistic Kelvin-Helmholtz instabilities in extragalactic radio sources","volume":"64","author":"Ferrari","year":"1978","journal-title":"Astron. Astrophys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1086\/166818","article-title":"Spatial Stability of the Slab Jet. I. Linearized Stability Analysis","volume":"334","author":"Hardee","year":"1988","journal-title":"Astrophys. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1051\/0004-6361:200809605","article-title":"On the linear theory of Kelvin-Helmholtz instabilities of relativistic magnetohydrodynamic planar flows","volume":"490","author":"Osmanov","year":"2008","journal-title":"Astron. Astrophys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1093\/mnras\/stx2012","article-title":"Linear theory of the Rayleigh-Taylor instability at a discontinuous surface of a relativistic flow","volume":"472","author":"Matsumoto","year":"2017","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2325","DOI":"10.1093\/mnras\/sty3167","article-title":"The magnetic Rayleigh-Taylor instability around astrophysical black holes","volume":"483","author":"Papadopoulos","year":"2019","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"L125","DOI":"10.1093\/mnrasl\/sly016","article-title":"Relativistic centrifugal instability","volume":"475","author":"Gourgouliatos","year":"2018","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1093\/mnras\/stad1833","article-title":"Linear analysis of the Kelvin-Helmholtz instability in relativistic magnetized symmetric flows","volume":"524","author":"Chow","year":"2023","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"A80","DOI":"10.1051\/0004-6361\/202348972","article-title":"Kelvin-Helmholtz instability and heating in oscillating loops perturbed by power-law transverse wave drivers","volume":"688","author":"Karampelas","year":"2024","journal-title":"Astron. Astrophys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"e2024GL110477","DOI":"10.1029\/2024GL110477","article-title":"Giant Kelvin-Helmholtz (KH) Waves at the Boundary Layer of the Coronal Mass Ejections (CMEs) Responsible for the Largest Geomagnetic Storm in 20 Years","volume":"51","author":"Nykyri","year":"2024","journal-title":"Geophys. Res. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Vlahakis, N. (2024). Linear Stability Analysis of Relativistic Magnetized Jets: The Minimalist Approach. Universe, 10.","DOI":"10.3390\/universe10040183"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Vlahakis, N. (2024). The Schwarzian Approach in Sturm-Liouville Problems. Symmetry, 16.","DOI":"10.3390\/sym16060648"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Rayleigh (1882). Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density. Proc. Lond. Math. Soc., s1-14, 170\u2013177.","DOI":"10.1112\/plms\/s1-14.1.170"},{"key":"ref_18","first-page":"192","article-title":"The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I","volume":"201","author":"Taylor","year":"1950","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1080\/14786447108640585","article-title":"XLVI. Hydrokinetic solutions and observations","volume":"42","author":"Thomson","year":"1871","journal-title":"Lond. Edinb. Dublin Philos. Mag. J. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Helmholtz (1868). XLIII. On discontinuous movements of fluids. Lond. Edinb. Dublin Philos. Mag. J. Sci., 36, 337\u2013346.","DOI":"10.1080\/14786446808640073"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/150\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:32:50Z","timestamp":1759919570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/150"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,21]]},"references-count":20,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["sym17020150"],"URL":"https:\/\/doi.org\/10.3390\/sym17020150","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,1,21]]}}}