{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T00:22:01Z","timestamp":1778890921711,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,2]],"date-time":"2021-06-02T00:00:00Z","timestamp":1622592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2016YFC1401800"],"award-info":[{"award-number":["2016YFC1401800"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61972406"],"award-info":[{"award-number":["61972406"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51709267"],"award-info":[{"award-number":["51709267"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev-Tau and Chebyshev-Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs are developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After the eigenvalues and eigenvectors are obtained, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre-Galerkin spectral method proposed previously.<\/jats:p>","DOI":"10.3390\/e23060705","type":"journal-article","created":{"date-parts":[[2021,6,2]],"date-time":"2021-06-02T10:38:39Z","timestamp":1622630319000},"page":"705","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6752-0436","authenticated-orcid":false,"given":"Yongxian","family":"Wang","sequence":"first","affiliation":[{"name":"College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1006-3932","authenticated-orcid":false,"given":"Houwang","family":"Tu","sequence":"additional","affiliation":[{"name":"College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenbin","family":"Xiao","sequence":"additional","affiliation":[{"name":"College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiang","family":"Lan","sequence":"additional","affiliation":[{"name":"College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Salomons, E.M. (2001). Computational Atmospheric Acoustics, Springer Science Business Media.","DOI":"10.1007\/978-94-010-0660-6"},{"key":"ref_2","unstructured":"Yang, X. (2015). Computational Atmospheric Acoustics, Science Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1121\/1.397587","article-title":"Application of the parabolic equation to sound propagation in a refracting atmosphere","volume":"85","author":"Gilbert","year":"1989","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2343","DOI":"10.1121\/1.407454","article-title":"A fast Green\u2019s function method for one-way sound propagation in the atmosphere","volume":"94","author":"Gilbert","year":"1993","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jensen, F.B., Kuperman, W.A., Porter, M.B., and Schmidt, H. (2011). Computational Ocean Acoustics, Springer.","DOI":"10.1007\/978-1-4419-8678-8"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1121\/1.391906","article-title":"A fast-field program for sound propagation in a layered atmosphere above an impedance ground","volume":"77","author":"Gilbert","year":"1985","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_7","unstructured":"Porter, M.B. (2021, June 02). Scooter: A Finite Element FFP Code. Available online: https:\/\/oalib-acoustics.org\/AcousticsToolbox\/index_at.html."},{"key":"ref_8","unstructured":"Pierce, A.D. (1991). Acoustics: An Introduction to Its Physical Principles and Applications, American Institute of Physics."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2352","DOI":"10.1121\/1.4777890","article-title":"An eigenfunction expansion for low-frequency acoustic propagation in a downward-refracting stratified medium over a complex impedance plane","volume":"111","author":"Waxler","year":"2002","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2540","DOI":"10.1121\/1.1514930","article-title":"A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane","volume":"112","author":"Waxler","year":"2002","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_11","unstructured":"Dzieciuch, M. (2021, June 02). A Matlab Code for Computing Normal Modes Based on Chebyshev Approximations. Available online: https:\/\/oalib-acoustics.org\/Modes\/aw."},{"key":"ref_12","unstructured":"Evans, R.B. (2021, June 02). A Legendre-Galerkin Technique for Differential Eigenvalue Problems with Complex and Discontinuous Coefficients, Arising in Underwater Acoustics. Available online: https:\/\/oalib-acoustics.org\/Modes\/rimLG\/."},{"key":"ref_13","first-page":"1","article-title":"A Chebyshev Spectral Method for Normal Mode and Parabolic Equation Models in Underwater Acoustics","volume":"2020","author":"Tu","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_14","first-page":"1","article-title":"A Chebyshev-Tau spectral method for normal modes of underwater sound propagation with a layered marine environment","volume":"2020","author":"Tu","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Tu, H. (2021, June 02). A Chebyshev-Tau Spectral Method for Normal Modes of Underwater Sound Propagation with a Layered Marine Environment in Matlab and Fortran. Available online: https:\/\/oalib-acoustics.org\/Modes\/NM-CT.","DOI":"10.1016\/j.jsv.2020.115784"},{"key":"ref_16","unstructured":"Tu, H., Wang, Y., Lan, Q., Liu, W., Xiao, W., and Ma, S. Applying a Legendre collocation method based on domain decomposition to calculate underwater sound propagation in a horizontally stratified environment. J. Sound Vib., (under review)."},{"key":"ref_17","first-page":"1","article-title":"Application of Chebyshev collocation method to solve parabolic equation model of underwater acoustic propagation","volume":"2021","author":"Wang","year":"2021","journal-title":"Acoust. Aust."},{"key":"ref_18","unstructured":"Tu, H., Wang, Y., Ma, X., and Zhu, X. Applying Chebyshev-Tau spectral method to solve the parabolic equation model of wide-angle rational approximation in ocean acoustics. J. Theor. Comput. Acoust., (under review)."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gottlieb, D., and Orszag, S.A. (1977). Numerical Analysis of Spectral Methods, Theory and Applications, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611970425"},{"key":"ref_20","unstructured":"Boyd, J.P. (2001). Chebyshev and Fourier Spectral Methods, Dover. [2nd ed.]."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. (2006). Spectral Methods Fundamentals in Single Domains, Spring.","DOI":"10.1007\/978-3-540-30726-6"},{"key":"ref_22","unstructured":"Jie, S., Tao, T., and Lilian, W. (2011). Spectral Methods Algorithms, Analysis and Applications, Springer."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Evans, R.B., Di, X., and Gilbert, K.E. (2017). A Legendre-Galerkin Technique for Finding Atmospheric Acoustic Normal Modes, Acoustical Society of America.","DOI":"10.1121\/2.0000617"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3595","DOI":"10.1121\/1.5040481","article-title":"A Legendre-Galerkin spectral method for constructing atmospheric acoustic normal modes","volume":"143","author":"Evans","year":"2018","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2050002","DOI":"10.1142\/S2591728520500024","article-title":"The Convergence of the Legendre\u2013Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes","volume":"28","author":"Evans","year":"2020","journal-title":"J. Theor. Comput. Acoust."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"464","DOI":"10.3813\/AAA.919328","article-title":"A Multi-Domain Collocation Method for the Accurate Computation of Normal Modes in Open Oceanic and Atmospheric Waveguides","volume":"105","author":"Sabatini","year":"2019","journal-title":"Acta Acust. United Acust."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/S0024-3795(99)00063-4","article-title":"Backward error and condition of polynomial eigenvalue problems","volume":"309","author":"Tisseur","year":"2000","journal-title":"Linear Algebra Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1137\/S0036144500381988","article-title":"The quadratic eigenvalue problem","volume":"43","author":"Francoise","year":"2001","journal-title":"Siam Rev."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/S0024-3795(01)00423-2","article-title":"Perturbation theory for homogeneous polynomial eigenvalue problems","volume":"358","author":"Pierre","year":"2003","journal-title":"Linear Algebra Appl."},{"key":"ref_30","unstructured":"Pichon, A.L., and Hauchecorne, E.B.A. (2019). Infrasound Monitoring for Atmospheric Studies, Challenges in Middle Atmosphere Dynamiacs and Societal Benefits, Springer Nature."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF01396332","article-title":"A Jacobi Type Method for Complex Symmetric Matrices","volume":"25","author":"Anderson","year":"1976","journal-title":"Numer. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/705\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:10:13Z","timestamp":1760163013000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/705"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,2]]},"references-count":31,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["e23060705"],"URL":"https:\/\/doi.org\/10.3390\/e23060705","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,2]]}}}