On Gaussian Beams Described by Jacobi's Equation
Gaussian beams describe the amplitude and phase of rays and are widely used to model acoustic propagation. This paper describes four new results in the theory of Gaussian beams. (1) A new version of the Červený equations for the amplitude and phase of Gaussian beams is developed by applying the equi...
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| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics
2014
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| Online Access: | http://hdl.handle.net/1721.1/92544 |
| _version_ | 1811074923237474304 |
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| author | Smith, Steven T. |
| author2 | Lincoln Laboratory |
| author_facet | Lincoln Laboratory Smith, Steven T. |
| author_sort | Smith, Steven T. |
| collection | MIT |
| description | Gaussian beams describe the amplitude and phase of rays and are widely used to model acoustic propagation. This paper describes four new results in the theory of Gaussian beams. (1) A new version of the Červený equations for the amplitude and phase of Gaussian beams is developed by applying the equivalence of Hamilton--Jacobi theory with Jacobi's equation that connects Riemannian curvature to geodesic flow. Thus the paper makes a fundamental connection between Gaussian beams and an acoustic channel's so-called intrinsic Gaussian curvature from differential geometry. (2) A new formula π([c over c''])[superscript 1 over 2] for the distance between convergence zones is derived and applied to the Munk and other well-known profiles. (3) A class of “model spaces” are introduced that connect the acoustics of ducting/divergence zones with the channel's Gaussian curvature K = cc''-(c')[superscript 2]. The model sound speed profiles (SSPs) yield constant Gaussian curvature in which the geometry of ducts corresponds to great circles on a sphere and convergence zones correspond to antipodes. The distance between caustics π([c over c''])[superscript 1 over 2] is equated with an ideal hyperbolic cosine SSP duct. (4) An intrinsic version of Červený's formulae for the amplitude and phase of Gaussian beams is derived that does not depend on an extrinsic, arbitrary choice of coordinates such as range and depth. Direct comparisons are made between the computational frameworks used by the three different approaches to Gaussian beams: Snell's law, the extrinsic Frenet--Serret formulae, and the intrinsic Jacobi methods presented here. The relationship of Gaussian beams to Riemannian curvature is explained with an overview of the modern covariant geometric methods that provide a general framework for application to other special cases. |
| first_indexed | 2024-09-23T09:57:24Z |
| format | Article |
| id | mit-1721.1/92544 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:57:24Z |
| publishDate | 2014 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/925442022-09-30T17:55:41Z On Gaussian Beams Described by Jacobi's Equation Smith, Steven T. Lincoln Laboratory Smith, Steven T. Gaussian beams describe the amplitude and phase of rays and are widely used to model acoustic propagation. This paper describes four new results in the theory of Gaussian beams. (1) A new version of the Červený equations for the amplitude and phase of Gaussian beams is developed by applying the equivalence of Hamilton--Jacobi theory with Jacobi's equation that connects Riemannian curvature to geodesic flow. Thus the paper makes a fundamental connection between Gaussian beams and an acoustic channel's so-called intrinsic Gaussian curvature from differential geometry. (2) A new formula π([c over c''])[superscript 1 over 2] for the distance between convergence zones is derived and applied to the Munk and other well-known profiles. (3) A class of “model spaces” are introduced that connect the acoustics of ducting/divergence zones with the channel's Gaussian curvature K = cc''-(c')[superscript 2]. The model sound speed profiles (SSPs) yield constant Gaussian curvature in which the geometry of ducts corresponds to great circles on a sphere and convergence zones correspond to antipodes. The distance between caustics π([c over c''])[superscript 1 over 2] is equated with an ideal hyperbolic cosine SSP duct. (4) An intrinsic version of Červený's formulae for the amplitude and phase of Gaussian beams is derived that does not depend on an extrinsic, arbitrary choice of coordinates such as range and depth. Direct comparisons are made between the computational frameworks used by the three different approaches to Gaussian beams: Snell's law, the extrinsic Frenet--Serret formulae, and the intrinsic Jacobi methods presented here. The relationship of Gaussian beams to Riemannian curvature is explained with an overview of the modern covariant geometric methods that provide a general framework for application to other special cases. United States. Dept. of the Navy (United States. Air Force. Contract FA8721-05-C-0002) 2014-12-29T22:10:58Z 2014-12-29T22:10:58Z 2014-10 2013-04 Article http://purl.org/eprint/type/JournalArticle 0036-1399 1095-712X http://hdl.handle.net/1721.1/92544 Smith, Steven T. “On Gaussian Beams Described by Jacobi’s Equation.” SIAM Journal on Applied Mathematics 74, no. 5 (January 2014): 1637–1656. © 2014 Society for Industrial and Applied Mathematics en_US http://dx.doi.org/10.1137/130915996 SIAM Journal on Applied Mathematics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial and Applied Mathematics Society for Industrial and Applied Mathematics |
| spellingShingle | Smith, Steven T. On Gaussian Beams Described by Jacobi's Equation |
| title | On Gaussian Beams Described by Jacobi's Equation |
| title_full | On Gaussian Beams Described by Jacobi's Equation |
| title_fullStr | On Gaussian Beams Described by Jacobi's Equation |
| title_full_unstemmed | On Gaussian Beams Described by Jacobi's Equation |
| title_short | On Gaussian Beams Described by Jacobi's Equation |
| title_sort | on gaussian beams described by jacobi s equation |
| url | http://hdl.handle.net/1721.1/92544 |
| work_keys_str_mv | AT smithstevent ongaussianbeamsdescribedbyjacobisequation |