Numerical Analysis of Dispersion and Wave Propagation in Spatiotemporal Metamaterials

Numerical Analysis of Dispersion and Wave Propagation in Spatiotemporal Metamaterials | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Numerical Analysis of Dispersion and Wave Propagation in Spatiotemporal Metamaterials Shih-Chun Liao, Chi-Chieh Ko, I-Ling Chang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7106190/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 15 Mar, 2026 Read the published version in Scientific Reports → Version 1 posted 12 You are reading this latest preprint version Abstract This study presents a novel numerical framework for analyzing wave propagation in spatiotemporally modulated metamaterials. A one-dimensional periodic spring-mass system is considered, wherein the spring stiffness varies in both space and time. To extract the dispersion relations of such systems, we propose a spatial perturbation method combined with two-dimensional Fourier transformation. The method leverages random initial velocity perturbations—analogous to thermal excitations in molecular dynamics—to excite all possible wave modes without requiring complex derivations. The computed dispersion curves exhibit excellent agreement with analytical solutions derived under the Bloch wave assumption for spatial, temporal, and spatiotemporal metamaterials. In addition to confirming the method’s accuracy, we demonstrate its general applicability to systems with arbitrary unit cell configurations. Transient wave propagation simulations reveal distinct behaviors across different metamaterial types. Spatial systems exhibit symmetric bidirectional wave propagation, while temporally modulated systems generate secondary wave components at frequencies \(\:{\omega\:}\text{ₑ}\:\pm\:\:\text{n}{\Omega\:}\) due to modulation-induced frequency conversion. In spatiotemporal systems, asymmetric dispersion relations result in directionally dependent energy propagation, though strictly non-reciprocal behavior is not observed due to the absence of directional bandgaps. The proposed method provides a robust and efficient tool for analyzing complex periodic structures, enabling the accelerated design and optimization of spatiotemporal metamaterials for advanced wave manipulation. Physical sciences/Mathematics and computing Physical sciences/Optics and photonics Physical sciences/Physics spatial perturbation method spatiotemporal metamaterial dispersion relation wave propagation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 15 Mar, 2026 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 24 Oct, 2025 Reviews received at journal 23 Oct, 2025 Reviews received at journal 20 Oct, 2025 Reviewers agreed at journal 29 Sep, 2025 Reviewers agreed at journal 28 Sep, 2025 Reviews received at journal 24 Sep, 2025 Reviewers agreed at journal 09 Sep, 2025 Reviewers invited by journal 09 Sep, 2025 Editor assigned by journal 23 Jul, 2025 Editor invited by journal 23 Jul, 2025 Submission checks completed at journal 22 Jul, 2025 First submitted to journal 22 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7106190","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":514841111,"identity":"0cd3e7a3-13b6-48d3-ba81-0306dc094dfb","order_by":0,"name":"Shih-Chun Liao","email":"","orcid":"","institution":"National Cheng Kung University","correspondingAuthor":false,"prefix":"","firstName":"Shih-Chun","middleName":"","lastName":"Liao","suffix":""},{"id":514841112,"identity":"852a91b6-187b-4331-bbee-55a938159300","order_by":1,"name":"Chi-Chieh Ko","email":"","orcid":"","institution":"National Cheng Kung 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method, spatiotemporal metamaterial, dispersion relation, wave propagation","lastPublishedDoi":"10.21203/rs.3.rs-7106190/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7106190/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study presents a novel numerical framework for analyzing wave propagation in spatiotemporally modulated metamaterials. A one-dimensional periodic spring-mass system is considered, wherein the spring stiffness varies in both space and time. To extract the dispersion relations of such systems, we propose a spatial perturbation method combined with two-dimensional Fourier transformation. The method leverages random initial velocity perturbations\u0026mdash;analogous to thermal excitations in molecular dynamics\u0026mdash;to excite all possible wave modes without requiring complex derivations. The computed dispersion curves exhibit excellent agreement with analytical solutions derived under the Bloch wave assumption for spatial, temporal, and spatiotemporal metamaterials.\u003c/p\u003e\u003cp\u003eIn addition to confirming the method\u0026rsquo;s accuracy, we demonstrate its general applicability to systems with arbitrary unit cell configurations. Transient wave propagation simulations reveal distinct behaviors across different metamaterial types. Spatial systems exhibit symmetric bidirectional wave propagation, while temporally modulated systems generate secondary wave components at frequencies \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}\\text{ₑ}\\:\\pm\\:\\:\\text{n}{\\Omega\\:}\\)\u003c/span\u003e\u003c/span\u003e due to modulation-induced frequency conversion. In spatiotemporal systems, asymmetric dispersion relations result in directionally dependent energy propagation, though strictly non-reciprocal behavior is not observed due to the absence of directional bandgaps. The proposed method provides a robust and efficient tool for analyzing complex periodic structures, enabling the accelerated design and optimization of spatiotemporal metamaterials for advanced wave manipulation.\u003c/p\u003e","manuscriptTitle":"Numerical Analysis of Dispersion and Wave Propagation in Spatiotemporal Metamaterials","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-16 12:17:18","doi":"10.21203/rs.3.rs-7106190/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-10-24T04:01:49+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-23T21:41:48+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-20T11:15:13+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"298672448981817767736156538101594105924","date":"2025-09-29T21:14:58+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"129441538182971888123796779511035990636","date":"2025-09-29T02:07:17+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-09-25T02:55:18+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"282784228734305783915462554830737587731","date":"2025-09-10T00:38:24+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-09-09T13:30:47+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-07-24T01:25:32+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-07-23T18:20:29+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-07-22T08:29:47+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-07-22T08:26:12+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"96b50a74-68e9-47c7-ae6f-055d6d1fafb9","owner":[],"postedDate":"September 16th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":54687189,"name":"Physical sciences/Mathematics and computing"},{"id":54687190,"name":"Physical sciences/Optics and photonics"},{"id":54687191,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-03-16T16:04:34+00:00","versionOfRecord":{"articleIdentity":"rs-7106190","link":"https://doi.org/10.1038/s41598-026-42208-5","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2026-03-15 15:58:41","publishedOnDateReadable":"March 15th, 2026"},"versionCreatedAt":"2025-09-16 12:17:18","video":"","vorDoi":"10.1038/s41598-026-42208-5","vorDoiUrl":"https://doi.org/10.1038/s41598-026-42208-5","workflowStages":[]},"version":"v1","identity":"rs-7106190","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7106190","identity":"rs-7106190","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}