{"id":"https://openalex.org/W2086299161","doi":"https://doi.org/10.1137/090761641","title":"Thin-Layer Solutions of the Helmholtz and Related Equations","display_name":"Thin-Layer Solutions of the Helmholtz and Related Equations","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2086299161","doi":"https://doi.org/10.1137/090761641","mag":"2086299161"},"language":"en","primary_location":{"id":"doi:10.1137/090761641","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090761641","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089772299","display_name":"J. R. Ockendon","orcid":"https://orcid.org/0000-0002-8125-9350"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"J. R. Ockendon","raw_affiliation_strings":["ock@maths.ox.ac.uk#TAB#"],"affiliations":[{"raw_affiliation_string":"ock@maths.ox.ac.uk#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019023769","display_name":"R. H. Tew","orcid":"https://orcid.org/0000-0003-3793-8513"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"R. H. Tew","raw_affiliation_strings":["richard.tew@nottingham.ac.uk#TAB#"],"affiliations":[{"raw_affiliation_string":"richard.tew@nottingham.ac.uk#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5089772299"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.8484,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.7405553,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"54","issue":"1","first_page":"3","last_page":"51"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11575","display_name":"Nonlinear Photonic Systems","score":0.9865000247955322,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/wkb-approximation","display_name":"WKB approximation","score":0.842940092086792},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.8388046026229858},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.6513470411300659},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.553617000579834},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5458292961120605},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5188392400741577},{"id":"https://openalex.org/keywords/heat-equation","display_name":"Heat equation","score":0.5187520980834961},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.508901834487915},{"id":"https://openalex.org/keywords/helmholtz-free-energy","display_name":"Helmholtz free energy","score":0.4326920807361603},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.30370718240737915},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.27887988090515137},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08507952094078064}],"concepts":[{"id":"https://openalex.org/C88030215","wikidata":"https://www.wikidata.org/wiki/Q907306","display_name":"WKB approximation","level":2,"score":0.842940092086792},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.8388046026229858},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6513470411300659},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.553617000579834},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5458292961120605},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5188392400741577},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.5187520980834961},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.508901834487915},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.4326920807361603},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.30370718240737915},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.27887988090515137},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08507952094078064}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/090761641","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090761641","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"},{"id":"pmh:oai:repository.kaust.edu.sa:10754/600010","is_oa":false,"landing_page_url":"http://hdl.handle.net/10754/600010","pdf_url":null,"source":{"id":"https://openalex.org/S4306401596","display_name":"King Abdullah University of Science and Technology Repository (King Abdullah University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I71920554","host_organization_name":"King Abdullah University of Science and Technology","host_organization_lineage":["https://openalex.org/I71920554"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":57,"referenced_works":["https://openalex.org/W58044373","https://openalex.org/W583761747","https://openalex.org/W625517374","https://openalex.org/W1538143344","https://openalex.org/W1573761753","https://openalex.org/W1926440615","https://openalex.org/W1964082576","https://openalex.org/W1964928956","https://openalex.org/W1971892010","https://openalex.org/W1973818653","https://openalex.org/W1981162208","https://openalex.org/W1983349998","https://openalex.org/W1984302887","https://openalex.org/W1992590796","https://openalex.org/W1992899169","https://openalex.org/W1998241491","https://openalex.org/W2001431479","https://openalex.org/W2003798820","https://openalex.org/W2008004348","https://openalex.org/W2009090147","https://openalex.org/W2012723463","https://openalex.org/W2022724749","https://openalex.org/W2024895678","https://openalex.org/W2036233227","https://openalex.org/W2039343706","https://openalex.org/W2041363811","https://openalex.org/W2053574666","https://openalex.org/W2053946421","https://openalex.org/W2054313643","https://openalex.org/W2057059545","https://openalex.org/W2060071796","https://openalex.org/W2061007110","https://openalex.org/W2061491118","https://openalex.org/W2062002669","https://openalex.org/W2063864535","https://openalex.org/W2071338011","https://openalex.org/W2074683093","https://openalex.org/W2075409146","https://openalex.org/W2081446976","https://openalex.org/W2086271398","https://openalex.org/W2091931476","https://openalex.org/W2094046716","https://openalex.org/W2100118866","https://openalex.org/W2105116556","https://openalex.org/W2122536615","https://openalex.org/W2130927234","https://openalex.org/W2138420827","https://openalex.org/W2139131426","https://openalex.org/W2146034485","https://openalex.org/W2160108567","https://openalex.org/W2169164332","https://openalex.org/W2170031024","https://openalex.org/W2312410307","https://openalex.org/W2314882304","https://openalex.org/W2318995732","https://openalex.org/W2788277701","https://openalex.org/W4242505850"],"related_works":["https://openalex.org/W2079455266","https://openalex.org/W4365420316","https://openalex.org/W3193896743","https://openalex.org/W3176862315","https://openalex.org/W3120022531","https://openalex.org/W2593412144","https://openalex.org/W2045397465","https://openalex.org/W2122095750","https://openalex.org/W2010309086","https://openalex.org/W2286968803"],"abstract_inverted_index":{"This":[0],"paper":[1],"concerns":[2],"a":[3,133,147],"certain":[4],"class":[5,38,134],"of":[6,39,64,99,135,140,169,172,188,217],"two-dimensional":[7],"solutions":[8,82,163,171],"to":[9,68,132],"four":[10,190],"generic":[11,191],"partial":[12],"differential":[13],"equations\u2014the":[14],"Helmholtz,":[15,17],"modified":[16],"and":[18,21,80,114,117,119,221],"convection\u2013diffusion":[19],"equations,":[20,139],"the":[22,27,30,57,62,65,71,77,81,86,100,142,158,173,189,201,212,215],"heat":[23,143],"conduction":[24,144],"equation":[25,145,162],"in":[26,47,56,83,91,97,151],"frequency":[28],"domain\u2014and":[29],"connections":[31],"between":[32,214],"these":[33,103],"equations":[34,192],"for":[35],"this":[36,75],"particular":[37],"solutions.":[40],"Specifically,":[41],"we":[42,154],"consider":[43],"\u201cthin-layer\u201d":[44],"solutions,":[45],"valid":[46,96],"narrow":[48],"regions":[49,101],"across":[50],"which":[51,141],"there":[52],"is":[53,76,146],"rapid":[54],"variation,":[55],"singularly":[58],"perturbed":[59],"limit":[60,79],"as":[61,167],"coefficient":[63],"Laplacian":[66],"tends":[67],"zero.":[69],"For":[70],"well-studied":[72],"Helmholtz":[73,174],"equation,":[74],"high-frequency":[78],"question":[84],"underpin":[85],"conventional":[87],"ray":[88],"theory/WKB":[89],"approach":[90],"that":[92,125,157],"they":[93],"provide":[94],"descriptions":[95],"some":[98,209],"where":[102],"classical":[104],"techniques":[105],"fail.":[106],"Examples":[107],"are":[108],"caustics,":[109],"shadow":[110],"boundaries,":[111],"whispering":[112],"gallery,":[113],"creeping":[115],"waves":[116],"focusing":[118],"bouncing":[120],"ball":[121],"modes.":[122],"It":[123],"transpires":[124],"virtually":[126],"all":[127],"such":[128],"thin-layer":[129,182],"models":[130],"reduce":[131],"generalized":[136],"parabolic":[137,160],"wave":[138,161],"special":[148],"case.":[149],"Moreover,":[150],"most":[152],"situations,":[153],"will":[155],"find":[156],"appropriate":[159],"can":[164],"be":[165],"derived":[166],"limits":[168],"exact":[170],"equation.":[175],"We":[176],"also":[177,207],"show":[178],"how":[179],"reasonably":[180],"well-understood":[181],"phenomena":[183],"associated":[184,199],"with":[185,200],"any":[186],"one":[187],"may":[193],"translate":[194],"into":[195],"less":[196],"well-known":[197],"effects":[198],"others.":[202],"In":[203],"addition,":[204],"our":[205],"considerations":[206],"shed":[208],"light":[210],"on":[211],"relationship":[213],"methods":[216],"matched":[218],"asymptotic,":[219],"WKB,":[220],"multiple-scales":[222],"expansions.":[223]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
