[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85916-en":3,"doc-seo-85916-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85916,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Computing critical velocities in waveguides via multiparameter eigenvalue problems","Critical velocities in waveguide dynamics and moving-load systems mark the onset of strong vibration amplification, such as when high-speed trains approach characteristic wave speeds. For direction-invariant systems, critical points can be characterized by the coincidence of a propagating mode’s phase and group velocities, but locating them via dispersion-curve tracing is tedious and unreliable for multimodal settings with complex branching. A direct semi-analytical method is proposed, converting a wavenumber–frequency polynomial eigenvalue problem into a singular multiparameter eigenvalue problem. Linearization and standard algorithms yield all critical points at once, avoiding curve tracing.","arXiv :2607 . 10462v1 [math .NA] 11 Jul 2026  \nComputing critical velocities in waveguides via multiparameter eigenvalue  \nproblems  \nHauke Gravenkampa,∗, Xiang Liub  \na Institute of Materials, Technologies and Mechanics, Otto von Guericke University Magdeburg  \n39106 Magdeburg, Germany  \nb School of Traffic and & Transportation Engineering, Central South University  \nChangsha, China  \nAbstract  \nIn waveguide dynamics and moving-load problems (e.g., high-speed trains), critical velocities indicate the onset of strong vibration amplification. In systems that are invariant in the direction of motion, these velocities can be identified from dispersion relations as points where the phase and group velocities of a propagating mode coincide. Finding such points indirectly by tracing dispersion curves can be cumbersome and potentially unreliable for multimodal systems with complex branch interactions. We present a direct method for computing critical velocities in such scenarios, specifically in the context of semi-analytical methods. Starting from a polynomial parameter-dependent eigenvalue problem for the wavenumber–frequency relation, incorporating the additional condition of equal phase and group velocities yields a singular polynomial multiparameter eigenvalue problem that can be linearized and solved using established algorithms. The proposed approach enables the simultaneous computation of all critical points without requiring the tracing of dispersion curves. Its performance is demonstrated by several benchmark problems, confirming the accurate and robust identification of critical velocities.  \nKeywords: critical velocity; wave propagation; multiparameter eigenvalue problem; dispersion; high-speed trains  \n1. Introduction  \nCritical velocities are a central concept in high-speed transportation and, more generally, in movingload and waveguide dynamics [1, 2] . In railway engineering, the phenomenon is commonly asso-  \n∗ Corresponding author  \nEmail address: [hauke.gravenkamp@ovgu.de](hauke.gravenkamp@ovgu.de) (Hauke Gravenkamp)  \nciated with a rapid amplification of track and ground vibration when the train speed approaches a characteristic wave speed of the coupled track–embankment–soil system. Early theoretical work already linked this amplification to the generation of strong surface-wave radiation by superfast trains [3] . More recent studies have established that the relevant threshold is governed by the dispersive wave-propagation characteristics of the supporting system and, in particular, by minima of the phase-velocity spectrum rather than by a classical resonance of a finite structure [2, 4, 5] . Specifically, critical velocities occur when the phase velocity cp of a propagating mode equals its group velocity cg . The same physical idea reappears in overhead contact systems, where the so-called catenary barrier reflects the interaction between the contact-point speed and the wave-propagation properties of the tensioned cable system [6, 7] . It also extends naturally to high-speed magnetically levitated vehicles, where wave-induced instability becomes relevant once the operating speed enters the supercritical regime [8] . Beyond transportation, the broader mechanics community has studied how wave speeds can be tailored or even self-controlled in nonlinear media, which further underlines the relevance of robust tools for locating characteristic propagation thresholds [9] .  \nFrom a modeling perspective, many dynamic systems of interest are invariant, or at least locally periodic, in the direction of motion. Such structures include beams on elastic or viscoelastic foundations, layered soil profiles, periodically supported rails, overhead contact lines, and general prismatic structures. Such systems are naturally described in terms of dispersion relations between frequency and wavenumber. In the railway context, this viewpoint has been employed for criticalspeed prediction in coupled track–ground models [2, 4, 5] and wa","cbCairXtXEGuQJw2","https://ap.wps.com/l/cbCairXtXEGuQJw2","pdf",1092104,3,1,20,"English","en",105,"# Introduction\n## Motivation and physical meaning of critical velocities\n## Dispersion-relations viewpoint and challenges\n## Modeling frameworks and semi-analytical waveguide methods","[{\"question\":\"What physical condition defines a critical velocity for a propagating mode?\",\"answer\":\"Critical velocities occur when the phase velocity equals the group velocity of the same propagating mode.\"},{\"question\":\"Why is tracing dispersion curves often difficult for multimodal systems?\",\"answer\":\"Dispersion branches can interact and branch complexities make indirect tracing cumbersome and potentially unreliable.\"},{\"question\":\"How does the proposed method compute critical velocities directly?\",\"answer\":\"It starts from a polynomial parameter-dependent eigenvalue problem for the wavenumber–frequency relation and adds the equal phase/group velocity condition to form a singular multiparameter eigenvalue problem, which is linearized and solved with established algorithms to obtain all critical points simultaneously.\"}]",1784207160,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"computing-critical-velocities-in-waveguides-via-multiparameter-eigenvalue-problems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/computing-critical-velocities-in-waveguides-via-multiparameter-eigenvalue-problems/85916/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What physical condition defines a critical velocity for a propagating mode?","Question",{"text":75,"@type":76},"Critical velocities occur when the phase velocity equals the group velocity of the same propagating mode.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is tracing dispersion curves often difficult for multimodal systems?",{"text":80,"@type":76},"Dispersion branches can interact and branch complexities make indirect tracing cumbersome and potentially unreliable.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed method compute critical velocities directly?",{"text":84,"@type":76},"It starts from a polynomial parameter-dependent eigenvalue problem for the wavenumber–frequency relation and adds the equal phase/group velocity condition to form a singular multiparameter eigenvalue problem, which is linearized and solved with established algorithms to obtain all critical points simultaneously.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":22,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":22,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]