Thin-Layer Solutions of the Helmholtz and Related Equations

Handle URI:
http://hdl.handle.net/10754/600010
Title:
Thin-Layer Solutions of the Helmholtz and Related Equations
Authors:
Ockendon, J. R.; Tew, R. H.
Abstract:
This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations-the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain-and the connections between these equations for this particular class of solutions.S pecifically, we consider thin-layer solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero.F or the wellstudied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail.E xamples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes.It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation.W e also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others.In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions. © 2012 Society for Industrial and Applied Mathematics.
Citation:
Ockendon JR, Tew RH (2012) Thin-Layer Solutions of the Helmholtz and Related Equations. SIAM Review 54: 3–51. Available: http://dx.doi.org/10.1137/090761641.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Review
KAUST Grant Number:
KUK–C1–013–04
Issue Date:
Jan-2012
DOI:
10.1137/090761641
Type:
Article
ISSN:
0036-1445; 1095-7200
Sponsors:
This work was based on research supported in part byaward KUK–C1–013–04 from King Abdullah University of Science and Technology.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorOckendon, J. R.en
dc.contributor.authorTew, R. H.en
dc.date.accessioned2016-02-28T06:34:18Zen
dc.date.available2016-02-28T06:34:18Zen
dc.date.issued2012-01en
dc.identifier.citationOckendon JR, Tew RH (2012) Thin-Layer Solutions of the Helmholtz and Related Equations. SIAM Review 54: 3–51. Available: http://dx.doi.org/10.1137/090761641.en
dc.identifier.issn0036-1445en
dc.identifier.issn1095-7200en
dc.identifier.doi10.1137/090761641en
dc.identifier.urihttp://hdl.handle.net/10754/600010en
dc.description.abstractThis paper concerns a certain class of two-dimensional solutions to four generic partial differential equations-the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain-and the connections between these equations for this particular class of solutions.S pecifically, we consider thin-layer solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero.F or the wellstudied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail.E xamples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes.It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation.W e also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others.In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions. © 2012 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThis work was based on research supported in part byaward KUK–C1–013–04 from King Abdullah University of Science and Technology.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectConvection-diffusion equationen
dc.subjectGeometrical theory of diffractionen
dc.subjectHeat conduction equationen
dc.subjectHelmholtz equationen
dc.subjectModified Helmholtz equationen
dc.subjectParabolic wave equationsen
dc.titleThin-Layer Solutions of the Helmholtz and Related Equationsen
dc.typeArticleen
dc.identifier.journalSIAM Reviewen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
dc.contributor.institutionUniversity of Nottingham, Nottingham, United Kingdomen
kaust.grant.numberKUK–C1–013–04en
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