{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:36:36Z","timestamp":1787315796506,"version":"build-2736575974"},"reference-count":23,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2015,6,1]],"date-time":"2015-06-01T00:00:00Z","timestamp":1433116800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2015,6,12]]},"abstract":"<jats:label>Abstract<\/jats:label>\n                  <jats:p>\n                    We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">a<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    that is periodic with characteristic length scale\n                    <jats:italic toggle=\"yes\">\u03b5<\/jats:italic>\n                    ; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">u<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    well by a non-dispersive wave equation on fixed time intervals\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mn>0<\/mml:mn>\n                        <mml:mo mathvariant=\"normal\">,<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">T<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">w<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    describe\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">u<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    well on time intervals\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mn>0<\/mml:mn>\n                        <mml:mo mathvariant=\"normal\">,<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">T<\/mml:mi>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>\u2212<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . More precisely, we provide a norm and uniform error estimates of the form\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mo stretchy=\"false\">\u2225<\/mml:mo>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">u<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">t<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                        <mml:mo>\u2212<\/mml:mo>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">w<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">t<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                        <mml:mo stretchy=\"false\">\u2225<\/mml:mo>\n                        <mml:mo>\u2a7d<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">C<\/mml:mi>\n                        <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    for\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"italic\">t<\/mml:mi>\n                        <mml:mo>\u2208<\/mml:mo>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mn>0<\/mml:mn>\n                        <mml:mo mathvariant=\"normal\">,<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">T<\/mml:mi>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">\u03b5<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>\u2212<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an\n                    <jats:italic toggle=\"yes\">\u03b5<\/jats:italic>\n                    -independent equation of third order that describes dispersion along rays and we present numerical examples.\n                  <\/jats:p>","DOI":"10.3233\/asy-141280","type":"journal-article","created":{"date-parts":[[2015,6,16]],"date-time":"2015-06-16T05:25:13Z","timestamp":1434432313000},"page":"21-49","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":15,"title":["Dispersive homogenized models and coefficient formulas for waves in general periodic media"],"prefix":"10.1177","volume":"93","author":[{"given":"T.","family":"Dohnal","sequence":"first","affiliation":[{"name":"Technische Universit\u00e4t Dortmund","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Lamacz","sequence":"additional","affiliation":[{"name":"Technische Universit\u00e4t Dortmund","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B.","family":"Schweizer","sequence":"additional","affiliation":[{"name":"Technische Universit\u00e4t Dortmund","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2015,6]]},"reference":[{"key":"e_1_3_1_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2013.06.002"},{"key":"e_1_3_1_3_1","doi-asserted-by":"publisher","DOI":"10.5802\/afst.1055"},{"key":"e_1_3_1_4_1","doi-asserted-by":"crossref","unstructured":"[3]G.\u00a0Allaire C.\u00a0Conca and M.\u00a0Vanninathan The Bloch transform and applications in: Actes du 29\u00e8me Congr\u00e8s D\u2019Analyse Num\u00e9rique: CANum\u201997 Larnas 1997 ESAIM: Proc. 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