Sommerfeld-type integrals for discrete diffraction problems
Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem...
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| Veröffentlicht in: | Wave motion 2020-09, Vol.97, p.102606, Article 102606 |
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| creator | Shanin, A.V. Korolkov, A.I. |
| description | Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).
•Three problems for a discrete analog of the Helmholtz equation are studied.•The problem with a point source on an entire plane is studied.•The problem of diffraction by a Dirichlet half-line is studied.•The problem of diffraction by a Dirichlet right angle is studied.•The plane wave decomposition and the Sommerfeld integral approach are applied.•The solution is obtained in terms of the algebraic functions. |
| doi_str_mv | 10.1016/j.wavemoti.2020.102606 |
| format | Article |
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•Three problems for a discrete analog of the Helmholtz equation are studied.•The problem with a point source on an entire plane is studied.•The problem of diffraction by a Dirichlet half-line is studied.•The problem of diffraction by a Dirichlet right angle is studied.•The plane wave decomposition and the Sommerfeld integral approach are applied.•The solution is obtained in terms of the algebraic functions.</description><identifier>ISSN: 0165-2125</identifier><identifier>EISSN: 1878-433X</identifier><identifier>DOI: 10.1016/j.wavemoti.2020.102606</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Algebra ; Canonical diffraction problem ; Diffraction ; Dirichlet problem ; Discrete element method ; Discrete green’s function ; Discrete Helmholtz equation ; Dispersion equation ; Elliptic functions ; Elliptic integrals ; Green's functions ; Helmholtz equations ; Integrals ; Mathematical analysis ; Mathematical functions ; Mathematical problems ; Plane waves ; Reflection method ; Sommerfeld integral ; Studies ; Toruses</subject><ispartof>Wave motion, 2020-09, Vol.97, p.102606, Article 102606</ispartof><rights>2020 Elsevier B.V.</rights><rights>Copyright Elsevier BV Sep 2020</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c340t-a409601cb17e13cb8fa50d952bd4568a40eeed47adda6909370bdf7c88674cf33</citedby><cites>FETCH-LOGICAL-c340t-a409601cb17e13cb8fa50d952bd4568a40eeed47adda6909370bdf7c88674cf33</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.wavemoti.2020.102606$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Shanin, A.V.</creatorcontrib><creatorcontrib>Korolkov, A.I.</creatorcontrib><title>Sommerfeld-type integrals for discrete diffraction problems</title><title>Wave motion</title><description>Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).
•Three problems for a discrete analog of the Helmholtz equation are studied.•The problem with a point source on an entire plane is studied.•The problem of diffraction by a Dirichlet half-line is studied.•The problem of diffraction by a Dirichlet right angle is studied.•The plane wave decomposition and the Sommerfeld integral approach are applied.•The solution is obtained in terms of the algebraic functions.</description><subject>Algebra</subject><subject>Canonical diffraction problem</subject><subject>Diffraction</subject><subject>Dirichlet problem</subject><subject>Discrete element method</subject><subject>Discrete green’s function</subject><subject>Discrete Helmholtz equation</subject><subject>Dispersion equation</subject><subject>Elliptic functions</subject><subject>Elliptic integrals</subject><subject>Green's functions</subject><subject>Helmholtz equations</subject><subject>Integrals</subject><subject>Mathematical analysis</subject><subject>Mathematical functions</subject><subject>Mathematical problems</subject><subject>Plane waves</subject><subject>Reflection method</subject><subject>Sommerfeld integral</subject><subject>Studies</subject><subject>Toruses</subject><issn>0165-2125</issn><issn>1878-433X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNqFkE1LxDAQhoMouK7-BSl47pqvJi1elMUvWPCggreQJhNJ2bZrkl3Zf2-W6tnTDDPvOy_zIHRJ8IJgIq67xbfeQT8mv6CYHoZUYHGEZqSWdckZ-zhGsyysSkpodYrOYuwwxkSyZoZuXse-h-Bgbcu030DhhwSfQa9j4cZQWB9NgAS5cS5ok_w4FJswtmvo4zk6cVkIF791jt4f7t-WT-Xq5fF5ebcqDeM4lZrjRmBiWiKBMNPWTlfYNhVtLa9EndcAYLnU1mrR4IZJ3FonTV0LyY1jbI6uprs5-GsLMalu3IYhRyrKuZCUEdJklZhUJowxBnBqE3yvw14RrA6gVKf-QKkDKDWBysbbyQj5h52HoKLxMBiwPoBJyo7-vxM_Syx1bA</recordid><startdate>202009</startdate><enddate>202009</enddate><creator>Shanin, A.V.</creator><creator>Korolkov, A.I.</creator><general>Elsevier B.V</general><general>Elsevier BV</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>KR7</scope></search><sort><creationdate>202009</creationdate><title>Sommerfeld-type integrals for discrete diffraction problems</title><author>Shanin, A.V. ; Korolkov, A.I.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c340t-a409601cb17e13cb8fa50d952bd4568a40eeed47adda6909370bdf7c88674cf33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algebra</topic><topic>Canonical diffraction problem</topic><topic>Diffraction</topic><topic>Dirichlet problem</topic><topic>Discrete element method</topic><topic>Discrete green’s function</topic><topic>Discrete Helmholtz equation</topic><topic>Dispersion equation</topic><topic>Elliptic functions</topic><topic>Elliptic integrals</topic><topic>Green's functions</topic><topic>Helmholtz equations</topic><topic>Integrals</topic><topic>Mathematical analysis</topic><topic>Mathematical functions</topic><topic>Mathematical problems</topic><topic>Plane waves</topic><topic>Reflection method</topic><topic>Sommerfeld integral</topic><topic>Studies</topic><topic>Toruses</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Shanin, A.V.</creatorcontrib><creatorcontrib>Korolkov, A.I.</creatorcontrib><collection>CrossRef</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Civil Engineering Abstracts</collection><jtitle>Wave motion</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shanin, A.V.</au><au>Korolkov, A.I.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Sommerfeld-type integrals for discrete diffraction problems</atitle><jtitle>Wave motion</jtitle><date>2020-09</date><risdate>2020</risdate><volume>97</volume><spage>102606</spage><pages>102606-</pages><artnum>102606</artnum><issn>0165-2125</issn><eissn>1878-433X</eissn><abstract>Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).
•Three problems for a discrete analog of the Helmholtz equation are studied.•The problem with a point source on an entire plane is studied.•The problem of diffraction by a Dirichlet half-line is studied.•The problem of diffraction by a Dirichlet right angle is studied.•The plane wave decomposition and the Sommerfeld integral approach are applied.•The solution is obtained in terms of the algebraic functions.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.wavemoti.2020.102606</doi></addata></record> |
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| subjects | Algebra Canonical diffraction problem Diffraction Dirichlet problem Discrete element method Discrete green’s function Discrete Helmholtz equation Dispersion equation Elliptic functions Elliptic integrals Green's functions Helmholtz equations Integrals Mathematical analysis Mathematical functions Mathematical problems Plane waves Reflection method Sommerfeld integral Studies Toruses |
| title | Sommerfeld-type integrals for discrete diffraction problems |
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