{"id":"https://openalex.org/W2000898555","doi":"https://doi.org/10.1121/1.1903274","title":"Closed-form solutions for SOFAR ray caustics in media with bilinear sound-speed profiles","display_name":"Closed-form solutions for SOFAR ray caustics in media with bilinear sound-speed profiles","publication_year":1974,"publication_date":"1974-08-01","ids":{"openalex":"https://openalex.org/W2000898555","doi":"https://doi.org/10.1121/1.1903274","mag":"2000898555"},"language":"en","primary_location":{"id":"doi:10.1121/1.1903274","doi":"https://doi.org/10.1121/1.1903274","is_oa":false,"landing_page_url":"https://doi.org/10.1121/1.1903274","pdf_url":null,"source":{"id":"https://openalex.org/S11296630","display_name":"The Journal of the Acoustical Society of America","issn_l":"0001-4966","issn":["0001-4966","1520-8524","1520-9024"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","ccf-b","cwts-core","dongbi-b","erih-plus","jufo-3","ki-jl-1","medline","norway-2","poland-100","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310320226","host_organization_name":"Acoustical Society of America","host_organization_lineage":["https://openalex.org/P4310320226"],"host_organization_lineage_names":["Acoustical Society of America"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Journal of the Acoustical Society of America","raw_type":"journal-article"},"volume":null,"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/A5044447773","display_name":"David T. Raphael","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"David Towns Raphael","raw_affiliation_strings":["Biomathematics Group, F361 Health Sciences Building, University of Washington, Seattle, Washington 98195"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Biomathematics Group, F361 Health Sciences Building, University of Washington, Seattle, Washington 98195","institution_ids":["https://openalex.org/I201448701"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5044447773"],"corresponding_institution_ids":["https://openalex.org/I201448701"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.15256046,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":93},"biblio":{"volume":"56","issue":"2","first_page":"416","last_page":"426"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11698","display_name":"Underwater Acoustics Research","score":0.9692000150680542,"subfield":{"id":"https://openalex.org/subfields/1910","display_name":"Oceanography"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11698","display_name":"Underwater Acoustics Research","score":0.9692000150680542,"subfield":{"id":"https://openalex.org/subfields/1910","display_name":"Oceanography"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10822","display_name":"Acoustic Wave Phenomena Research","score":0.014700000174343586,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical 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/T11757","display_name":"Seismic Waves and Analysis","score":0.006200000178068876,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/analytical-solution","display_name":"analytical solution","score":0.7950000166893005},{"id":"https://openalex.org/keywords/reciprocity","display_name":"reciprocity","score":0.8899999856948853},{"id":"https://openalex.org/keywords/underwater-acoustic-propagation","display_name":"underwater acoustic propagation","score":0.4180000126361847}],"concepts":[{"id":"https://openalex.org/C2779567845","wikidata":"https://www.wikidata.org/wiki/Q3492725","display_name":"Caustic (mathematics)","level":2,"score":0.7699682116508484},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.729506254196167},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5964955687522888},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5715484619140625},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5322126150131226},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.4834171533584595},{"id":"https://openalex.org/C65682993","wikidata":"https://www.wikidata.org/wiki/Q1056451","display_name":"Reflection (computer programming)","level":2,"score":0.4832633137702942},{"id":"https://openalex.org/C200351514","wikidata":"https://www.wikidata.org/wiki/Q124003","display_name":"Speed of sound","level":2,"score":0.4457915425300598},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.43106698989868164},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.41114112734794617},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.355258584022522},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.29211074113845825},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2720987796783447},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12348967790603638},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1121/1.1903274","doi":"https://doi.org/10.1121/1.1903274","is_oa":false,"landing_page_url":"https://doi.org/10.1121/1.1903274","pdf_url":null,"source":{"id":"https://openalex.org/S11296630","display_name":"The Journal of the Acoustical Society of America","issn_l":"0001-4966","issn":["0001-4966","1520-8524","1520-9024"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","ccf-b","cwts-core","dongbi-b","erih-plus","jufo-3","ki-jl-1","medline","norway-2","poland-100","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310320226","host_organization_name":"Acoustical Society of America","host_organization_lineage":["https://openalex.org/P4310320226"],"host_organization_lineage_names":["Acoustical Society of America"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Journal of the Acoustical Society of America","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1969833055","https://openalex.org/W2356926090","https://openalex.org/W2164991698","https://openalex.org/W1974557205","https://openalex.org/W2021894011","https://openalex.org/W1965503951","https://openalex.org/W2047880759","https://openalex.org/W1967478346","https://openalex.org/W3153693167","https://openalex.org/W2053286753"],"abstract_inverted_index":{"In":[0],"an":[1,16],"unbounded":[2],"medium":[3],"with":[4],"a":[5,140],"bilinear":[6],"sound-speed":[7],"profile":[8],"composed":[9],"of":[10,21,43,55,67,116,121],"two":[11],"constant-gradient":[12],"layers":[13],"separated":[14],"by":[15,119,125],"interface":[17,136],"at":[18],"the":[19,34,44,53,56,60,65,68,76,81,117,135],"depth":[20],"minimum":[22],"sound":[23],"speed,":[24],"SOFAR":[25],"rays":[26,132],"are":[27,30,92,112,145],"generated":[28],"which":[29,85,133],"continuously":[31],"refracted":[32],"towards":[33],"channel":[35],"axis.":[36],"The":[37],"existence":[38],"conditions":[39],"for":[40,114],"real":[41],"roots":[42,87],"acquiring-ray":[45],"cubic":[46,69],"and":[47,96,101,104,124],"quartic":[48,71],"equations":[49],"treated":[50],"herein":[51],"facilitate":[52],"determination":[54],"caustic":[57,77],"equations.":[58],"When":[59],"condition":[61],"is":[62],"imposed":[63],"that":[64],"discriminant":[66],"or":[70,128,142],"equation":[72,78],"be":[73],"zero":[74],"valued,":[75],"emerges":[79],"as":[80],"boundary":[82],"line":[83],"along":[84],"repeated":[86],"occur.":[88],"Two":[89],"general":[90],"cases":[91],"examined:":[93],"(a)":[94],"receiver":[95,103],"source":[97,105],"in":[98,106],"same":[99],"layer,":[100],"(b)":[102],"different":[107],"layers.":[108],"Explicit,":[109],"closed-form":[110],"solutions":[111],"obtained":[113],"most":[115],"caustics":[118],"exploitation":[120],"reciprocity":[122],"considerations":[123],"reduction-to-single":[126],"reflection":[127,144],"crossing":[129],"arguments.":[130],"Only":[131],"cross":[134],"without":[137],"undergoing":[138],"either":[139],"surface":[141],"bottom":[143],"treated.":[146]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-08-18T07:49:30.821534","created_date":"2016-06-24T00:00:00"}
