JSFHCap\303\255tulo 8 Equa\303\247\303\265es Diferenciais Parciais (EDPs)Introdu\303\247\303\243oO principal comando para resolver EDPs \303\251 pdsolve, cuja sintaxe \303\251 pdsolve(EDP) pdsolve(EDP, f, HINT=..., INTEGRATE, build)onde EDP \303\251 uma equa\303\247\303\243o diferencial parcialLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 o nome da fun\303\247\303\243o indeterminada (opcional) HINT=... \303\251 um par\303\242metro de sugest\303\265es dado pelo usu\303\241rio (opcional) INTEGRATE provoca a integra\303\247\303\243o autom\303\241tica das ODE's encontradas pelo m\303\251todo de separa\303\247\303\243o de vari\303\241veis da EDP (optional) build tenta construir uma express\303\243o expl\303\255cita para a fun\303\247\303\243o indeterminada seja qual for a generalidade da solu\303\247\303\243o encontrada (opcional)Os par\303\242metros opcionais podem ser usados em qualquer ordem. Se o comando pdsolve n\303\243o tiver sucesso, a op\303\247\303\243o HINT permite ao usu\303\241rio sugerir uma estrat\303\251gia para a resolu\303\247\303\243o da EDP. Solu\303\247\303\243o Geral e Solu\303\247\303\243o Quase-geralPdsolve tenta em primeiro lugar achar uma solu\303\247\303\243o geral da EDP. Por exemplo, considere a seguinte EDP.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Pdsolve encontra a solu\303\247\303\243o geral em termos da fun\303\247\303\243o arbitr\303\241ria LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkX0YxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnL0YzUSdub3JtYWxGJw==.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Por exemplo, a fun\303\247\303\243o arbitr\303\241ria LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkX0YxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnL0YzUSdub3JtYWxGJw== pode ser da seguinte forma: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 case a solu\303\247\303\243o fica com a seguinte cara:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkc29sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==Vamos testar se esta solu\303\247\303\243o est\303\241 correta: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Podemos fazer o mesmo teste atrav\303\251s do comando pdetest.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNigtRiw2JVEocGRldGVzdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNigtRiw2JVEkc29sRidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GLDYlUSRFRFBGJ0Y2RjkvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGREZARkBGYW9GZG9GQEYrRmFvRmRvRkBGK0Zhb0Zkb0ZAA solu\303\247\303\243o de uma EDP de ordem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= dependendo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= vari\303\241veis \303\251 geral quando \303\251 dada em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= fun\303\247\303\265es arbitr\303\241rias e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= vari\303\241veis. No exemplo acima, a ordem da EDP \303\251 1 (derivada de ordem mais alta) e o n\303\272mero de vari\303\241veis \303\251 3 (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY1LUYsNiVRInpGJ0YvRjJGOQ==). Podemos ver que a solu\303\247\303\243o \303\251 dada em termos de 1 fun\303\247\303\243o arbitr\303\241ria (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkX0YxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==) que depende de 2 vari\303\241veis.Se pdsolve n\303\243o consegue achar a solu\303\247\303\243o geral, a solu\303\247\303\243o \303\251 dada usando a estrutura (solu\303\247\303\243o em termos de _F1, _F2, ...) &where [{EDO's envolvendo _F1, _F2, ...}]Nesse caso, 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 n\303\243o s\303\243o fun\303\247\303\265es arbitr\303\241rias. Elas obedecem \303\240s EDO's descritas pelo segundo operando do operador &where. Por exemplo, a equa\303\247\303\243o de difus\303\243o (ou de propaga\303\247\303\243o de calor) em 1 dimens\303\243o \303\251 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A solu\303\247\303\243o \303\251 dada da 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkX2MxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnL0YzUSdub3JtYWxGJw== \303\251 a constante de separa\303\247\303\243o de vari\303\241veis. O argumento opcional build provoca a integra\303\247\303\243o das EDO's dadas no segundo operando de &where.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onde as constantes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSRfQzJGJ0YvRjJGNS1GLDYlUSRfQzNGJ0YvRjIvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGOQ== foram introduzidas na integra\303\247\303\243o de uma ODE de segunda ordem e uma ODE de primeira orderm. A op\303\247\303\243o INTEGRATE produz um resultado mantendo o operador &where.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 comando pdetest funciona com qualquer dessas formas de solu\303\247\303\243o.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\303\247\303\243o Num\303\251ricaSe os m\303\251todos anal\303\255ticos de resolu\303\247\303\243o de equa\303\247\303\265es diferenciais falharem, podemos usar m\303\251todos num\303\251ricos, cuja p\303\241gina de ajuda principal \303\251 pdsolve/numeric. Vejamos um exemplo:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=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A solu\303\247\303\243o \303\251 um m\303\263dulo (module). Para usar uma fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= dentro do m\303\263dulo de nome LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkc29sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==, devemos usar a nota\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkc29sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOi1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJmRidGL0YyRjk= onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEjOi1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGQ0Yv substitui o ponto na nota\303\247\303\243o usual das linguagens orientadas a objetos. Por exemplo, para usar a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para fazer o gr\303\241fico no instante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGOS1GIzYlLUZTNiRRIzEwRidGOUYvRjIvJS5saW5ldGhpY2tuZXNzR0ZVLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmluLyUpYmV2ZWxsZWRHRj1GOQ==: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O Pacote PDEtoolsO pacote PDEtools cont\303\251m fun\303\247\303\265es que complementam o comando pdesolve na tarefa de encontrar solu\303\247\303\265es para EDP's.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 buildO comando build constr\303\263i uma solu\303\247\303\243o expl\303\255cita para as fun\303\247\303\265es indeterminadas que obedecem as EDO's dentro do operador &where. O argumento do comando build deve ser uma solu\303\247\303\243o fornecido pelo comando pdsolve. Por exemplo, considere a EDP de Laplace com um termo n\303\243o homog\303\252neo.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A solu\303\247\303\243o \303\251 dada em termos de duas fun\303\247\303\265es indeterminadas.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Para construir uma solu\303\247\303\243o expl\303\255cita: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Comando dchangeO comando dchange serve para fazer mudan\303\247a de vari\303\241veis em EDP (e em outras express\303\265es alg\303\251bricas). A t\303\255tulo de exemplo vamos resolver a 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sem usar o comando pdsolve. Vamos considerar a seguinte tranforma\303\247\303\243o de vari\303\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e a partir dela determinar o valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJmFscGhhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGMg== que transforma a EDP numa ODE. Uma vez que essa tranforma\303\247\303\243o possui par\303\242metro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJmFscGhhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGMg==, vamos adicionar o terceiro argumento 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 no comando dchange.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 fazer a seguinte substitui\303\247\303\243o para eliminar o par\303\242metro LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJmFscGhhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGMg== que infelizmente foi introduzido na fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJy9GM1Enbm9ybWFsRic=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiktRiw2JVEobmV3X0VEUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2KC1GLDYlUSVzdWJzRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYpRistRiM2KkYrLUYjNigtRiw2JVEiZkYnRjZGOUZlbi1GXG82JC1GIzYqLUYsNiVRInZGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0Zobi9GVVEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ3RidGNkY5Rl5wLUYsNiVRJyYjOTQ1O0YnL0Y3RkRGQC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0ZERkBGQEZbcUZecUZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYjNihGZG9GZW4tRlxvNiQtRiM2KEZbcEZecEZkcEZbcUZecUZARkBGW3FGXnFGQEYrRltxRl5xRkBGXnBGM0ZbcUZecUZARkBGW3FGXnFGQEYrRltxRl5xRkBGK0ZbcUZecUZARitGW3FGXnFGQA==Vamos determinar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJmFscGhhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGMg== de forma que o coeficiente do termo 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 seja zero.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Vejamos o formato da EDP agora.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUShuZXdfRURQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y4USdub3JtYWxGJ0YrRjpGPUZAEla pode ser resolvida como o comando dsolve. (em vez de pdsolve).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O \303\272ltimo passo \303\251 recuperar as vari\303\241veis originais.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 confirmar a solu\303\247\303\243o.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Comando PDEplotO comando PDEplot faz o gr\303\241fico da solu\303\247\303\243o de uma EDP de primeira ordem uma vez dadas as condi\303\247\303\265es iniciais. A EDP n\303\243o pode ter par\303\242metro livres. Como exemplo, considere a seguinte EDP.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiktRiw2JVEvTWFwbGVfbG9nb19FRFBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNilGKy1GIzYqRistRiM2Ji1JKG1mZW5jZWRHRiQ2JC1GIzYqRistRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ5RidGNkY5LUkjbW5HRiQ2JFEiMkYnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkAtRj02LVEiK0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZecC1GIzYkLUZebzYlLUYjNigtRiw2JVEiekYnRjZGOS1GPTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVGXXEtRmduNiQtRiM2KC1GLDYlUSJ4RidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GXHEvRlVRLDAuMzMzMzMzM2VtRidGYG8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGREZARkBGXHJGX3JGQEZjb0Znb0ZARmpvLUYjNiQtRl5vNiVGY3FGY29GZ29GQEYrRkBGQC1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPRlxxRl5xLUZnbjYkLUYjNiktSSZtZnJhY0dGJDYoLUYjNiYtRj02LVErJlBhcnRpYWxEO0YnRkBGQkZFRkdGSUZLRk1GT0ZccUZecUZcckZfckZALUYjNihGKy1GIzYnRmFzRmNxRlxyRl9yRkBGK0ZcckZfckZALyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZddC8lKWJldmVsbGVkR0ZERmVyRmRwRitGXHJGX3JGQEZARkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZdcEZfcC1GIzYqRmNvRmVyRmNxRmVyRmBvRmVyLUZnbjYkLUYjNiktRl1zNihGX3MtRiM2KEYrLUYjNidGYXNGYG9GXHJGX3JGQEYrRlxyRl9yRkBGaHNGW3RGXnRGYHRGZXJGZHBGK0ZcckZfckZARkBGQEZidC1GIzYoRmNvRmVyRmRwRmVyRmNxRkBGK0ZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUZkbzYkRmlvRkBGXHJGX3JGQEYrRlxyRl9yRkBGK0ZcckZfckZARitGXHJGX3JGQA==cuja solu\303\247\303\243o fornece o gr\303\241fico usado como logo da vers\303\243o 4 do Maple.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Exemplo 1 - Transmiss\303\243o de calorRef. 1: Wikipedia - Equa\303\247\303\243o de calorRef. 2: A. O. de Morais, Transfer\303\252ncia de calor no escoamento laminar n\303\243o Newtoniano com difusividade t\303\251rmica efetiva em tubos. Monografia, Universidade de S\303\243o Paulo, 2012.A equa\303\247\303\243o geral da transmiss\303\243o de calor \303\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onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= temperatura do fluidoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkdmVjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidkYnRi9GMi9GM1Enbm9ybWFsRidGPUY9 vetor velocidade do fluidoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy densidade do fluidoLUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJwRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn calor espec\303\255fico do fluidoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= condutividade t\303\251rmica do 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 gera\303\247\303\243o volum\303\251trica de calorProblema Considere uma barra de comprimento 85 cm e largura 15 cm inicialmente a 37 graus cent\303\255grados isolada do meio ambiente exceto pelas suas extremidades, que s\303\243o mantidas a 0 graus cent\303\255grados. Suponha que 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= para esta barra. Encontre a temperatura da barra.Solu\303\247\303\243o: A equa\303\247\303\243o que rege a evolu\303\247\303\243o da temperatura em fun\303\247\303\243o da vari\303\241vel expacial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e do tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\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pois LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbWZyYWNHRiQ2KC1JI21vR0YkNi1RKyZQYXJ0aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRi1GIzYmLUYvNi1GMUYyL0Y2USZmYWxzZUYnL0Y5Rk4vRjtGTi9GPUZOL0Y/Rk4vRkFGTi9GQ0ZORkRGRy1GLzYtUSJ+RidGMkZNRk9GUEZRRlJGU0ZURkRGRy1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGTkZVLUklbXN1YkdGJDYlLUZZNiVRInFGJ0ZmbkZpbi1GIzYkLUZZNiVRIlZGJ0ZmbkZpbkYyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82LVEiPUYnRjJGTUZPRlBGUUZSRlNGVC9GRVEsMC4yNzc3Nzc4ZW1GJy9GSEZncC1JI21uR0YkNiRGYnBGMkYy. Vamos supor que a barra vai de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= at\303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzg1RidGOUY5.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0ZFRj0=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As condi\303\247\303\265es de contorno e as condi\303\247\303\265es iniciais s\303\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A solu\303\247\303\243o fornecida pelo pdsolve \303\251 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Note que a solu\303\247\303\243o \303\251 uma s\303\251rie de Fourier. Vamos fazer a seguinte aproxima\303\247\303\243o. Vamos tomar um n\303\272mero finito (truncamento) de termos e substituir o \303\255ndice mudo do somat\303\263rio para a vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (por uma quest\303\243o de eleg\303\242ncia):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 ver que para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, teremos uma aproxima\303\247\303\243o da condi\303\247\303\243o inicial: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 aumentarmos o truncamento da s\303\251rie de Fourier, obteremos um aproxima\303\247\303\243o melhor. As oscila\303\247\303\265es da fun\303\247\303\243o representando a condi\303\247\303\243o inicial nas extremidades da barra s\303\243o chamadas de fen\303\264meno de Gibbs e refletem a dificuldade de aproximar fun\303\247\303\265es discont\303\255nuas por s\303\251ries de Fourier, que s\303\243o somas de fun\303\247\303\265es oscilat\303\263rias cont\303\255nuas.Podemos agora animar a solu\303\247\303\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 gr\303\241fico de densidade \303\251 bem mais elegante (Temperatura LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjVDBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn \303\251 vermelha e temperatura 0 graus cent\303\255grados \303\251 preta):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Para fazer a anima\303\247\303\243o usando o gr\303\241fico de densidade, \303\251 importante usar a op\303\247\303\243o scaletorange: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JSFHExemplo 2 - Propaga\303\247\303\243o de ondas em cordasRef.: Wikipedia - Equa\303\247\303\243o da ondaLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Vamos considerar uma corda horizontal infinita uniformemente distentida por uma tra\303\247\303\243o T. Vamos supor que as oscila\303\247\303\265es da corda s\303\243o sempre na dire\303\247\303\243o vertical. Sob certas restri\303\247\303\265es, como por exemplo pequenas amplitudes, a equa\303\247\303\243o de onda que descreve o deslocamento vertical LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (dependente das vari\303\241vel espacial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e do tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) na aus\303\252ncia de for\303\247as externas \303\251 dada por: 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onde 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 a densidade de massa. Resolu\303\247\303\243o da equa\303\247\303\243o de ondaVamos atribuir a equa\303\247\303\243o diferencial deste problema \303\240 vari\303\241vel eq_de_onda:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=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 solu\303\247\303\243o geral pode ser encontrada pelo comando pdsolve: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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkX0YxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkX0YyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== s\303\243o fun\303\247\303\265es gen\303\251ricas. Um pulso senoidal se movendo para a direita e gerado da seguinte maneira: 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Note que a anima\303\247\303\243o dura 10 unidades de tempo (por exemplo: segundos) e o pulso foi do ponto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= at\303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOUY5 em unidades de dist\303\242ncia (por exemplo: cent\303\255metros). Portanto, a velocidade de onda \303\251 1 m/s. Como exerc\303\255cio, mude o valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e verifique que a velocidade do pulso muda (o intervalo para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= tamb\303\251m deve ser mudado).A superposi\303\247\303\243o de dois pulsos senoidais se movendo em dire\303\247\303\243o oposta e gerado da seguinte maneira: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Se a corda estiver presa nas extremidades (em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyRjk=), a equa\303\247\303\243o diferencial parcial pode ser resolvida pelo m\303\251todo de separa\303\247\303\243o de vari\303\241veis e a solu\303\247\303\243o geral pode ser expressa por uma s\303\251rie de Fourier. A corda de comprimento a presa nas extremidades \303\251 caracterizada pelas seguintes condi\303\247\303\265es de contorno: 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 e 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para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ger\303\251rico.Vamos resolver a equa\303\247\303\243o de onda usando o m\303\251todo de separa\303\247\303\243o de vari\303\241veis. Vamos supor que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 uma fun\303\247\303\243o separ\303\241vel nas vari\303\241veis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidEYnRi9GMkY5, isto \303\251, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJYRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEieEYnRi9GMi8lK2ZvcmVncm91bmRHUSpbMjU1LDAsMF1GJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjlGOS1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORmBvLUYsNiVRIlRGJ0YvRjItRlM2JC1GIzYnLUYsNiVRInRGJ0YvRjJGWkZnbkZpbkY5RjlGWkZnbkZpbkY5: 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Cada termo da express\303\243o acima deve ser constante, pois cada um \303\251 fun\303\247\303\243o de uma vari\303\241vel independente. Vamos selecionar o termo dependente de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e igualar a uma constante :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 equa\303\247\303\243o diferencial acima pode ter solu\303\247\303\265es oscilat\303\263rias, lineares ou exponenciais dependendo se a constante for negativa, zero ou positiva, respectivamente (exerc\303\255cio: resolva a EDO acima para 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. Para que as condi\303\247\303\265es de contorno sejam satisfeitas, isto \303\251, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVRistRjY2JC1GIzYkLUYsNiVRImFGJ0YvRjJGPkY+RkBGOkY+, temos que impor que a constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM5NTU7MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= seja negativa: 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 a condi\303\247\303\243o de contorno LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRj5GPi1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGOkZARj4= ser satisfeita, vamos impor a condi\303\247\303\243o de contorno 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. 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 que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSRfWjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnRitGOg== tem que ser um n\303\272mero inteiro.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 n \303\251 um inteiro positivo (no lugar de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkX1oxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==). Vamos eliminar a constante arbitr\303\241ria por enquanto: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A equa\303\247\303\243o diferencial para a fun\303\247\303\243o dependente da vari\303\241vel t \303\251 obtida da seguinte forma: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 solu\303\247\303\243o geral \303\251: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 colocar esta solu\303\247\303\243o numa forma mais conveniente: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 A e B s\303\243o constantes que podem depender dos autovalores m e n. Vamos definir a frequ\303\252ncia do modo normal <n>:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRIm5GJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1GNjYtUSgmc3JhcnI7RidGMkY5RjtGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLRlctRjY2LVEifkYnRjJGOUY7Rj1GP0ZBRkNGRUZWRlgtSSZtZnJhY0dGJDYoLUYjNiotRiw2JUZORi9GMi1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGMkY5RjtGPUY/RkFGQ0ZFRlZGWC1GLDYlUScmIzk2MDtGJ0YvRjJGWS1GLDYlUSJjRidGT0ZRLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHRjFGMi1GIzYmLUYsNiVRImFGJ0ZPRlFGZm9GaW9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZXAvJSliZXZlbGxlZEdGMUZmb0Zpb0Yye definir os modos normais como uma fun\303\247\303\243o de m e n. Existem duas possibilidades que s\303\243o: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 vez que a equa\303\247\303\243o diferencial do movimento da corda \303\251 linear e homog\303\252nea, a solu\303\247\303\243o geral \303\251 uma superposi\303\247\303\243o de todas as solu\303\247\303\265es linearmente independentes poss\303\255veis. Assim, tomaremos o somat\303\263rio sobre todos os valores de m e n: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As constantes arbitr\303\241rias 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 e 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 s\303\243o determinadas a partir das condi\303\247\303\265es iniciais. Geralmente s\303\243o elas a posi\303\247\303\243o inicial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjdTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiPUYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSJ1RidGL0YyRjVGPQ== e a velocidade inicial 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. Anima\303\247\303\243o dos Modos NormaisOs modos normais podem ser animados como a fun\303\247\303\243o animate. Vamos construir a fun\303\247\303\243o animate_mode, que anima o modo especificado pelos seus argumentos. Por exemplo, animate_mode(2) faz a anima\303\247\303\243o do modo n=2. 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 fazer a anima\303\247\303\243o de um determinado modo, temos que dar valores num\303\251ricos para os par\303\242metros a, b e c. Vamos supor que o comprimento \303\251 a=1, a largura \303\251 b=1 e a constante c=1 em um certo sistema de unidades:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiY0YnRi9GMi1GNjYtUSomY29sb25lcTtGJ0Y5RjsvRj9GPUZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GWC1JI21uR0YkNiRRIjFGJ0Y5RjVGWi8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0Y9Rjk=Vamos fazer a anima\303\247\303\243o do modo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=: 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Vejamos a anima\303\247\303\243o do modo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjVGJ0Y5Rjk=:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEsYW5pbWF0ZW1vZGVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRIjVGJy9GM1Enbm9ybWFsRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPkY+RkBGQ0Y+Os pontos que ficam im\303\263veis s\303\243o chamados pontos nodais.Anima\303\247\303\243o com Condi\303\247\303\265es IniciaisEncontramos anteriormente a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= que descreve o movimento vertical da corda de comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= presa nas extremidades. Esta solu\303\247\303\243o \303\251 uma s\303\251rie de Fourier. Uma vez que os modos normais com pequenos valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= dominam sobre os que tem grandes valores, podemos truncar o somat\303\263rio. O ponto de truncagem satisfat\303\263rio depende das condi\303\247\303\265es iniciais do problema. Vamos converter o somat\303\263rio infinito num somat\303\263rio finito com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiNRIUYnRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn termos e definir a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkdV9wRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNigtRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidEYnRi9GMkY9LUYsNiVRInBGJ0YvRjJGQUZBRkE= da seguinte forma: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As constantes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE=, que aparecem na fun\303\247\303\243o 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Butkov, F\303\255sica Matem\303\241tica, cap. 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de fazer a anima\303\247\303\243o com condi\303\247\303\265es iniciais, vamos calcular as contantes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE= para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= inteiros positivos gen\303\251ricos. \303\211 mais eficiente calcular estas constantes para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= gen\303\251ricos e depois substituir valores num\303\251ricos, do que calcular diretamente para diversos valores num\303\251ricos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Vamos usar as seguintes condi\303\247\303\265es iniciais: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 calcular LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIm5GJ0YvRjIvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJy9GM1Enbm9ybWFsRidGQEY9RkA= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIm5GJ0YvRjIvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJy9GM1Enbm9ybWFsRidGQEY9RkA= para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw== inteiro positivo gen\303\251rico, vamos usar a fun\303\247\303\243o assume. Desta forma, o comando simplify das fun\303\247\303\265es LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIm5GJ0YvRjIvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJy9GM1Enbm9ybWFsRidGQEY9RkA= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIm5GJ0YvRjIvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJy9GM1Enbm9ybWFsRidGQEY9RkA= poder\303\241 fazer as simplifica\303\247\303\265es pertinentes: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Vamos fazer o gr\303\241fico da posi\303\247\303\243o inicial da corda desta vez descrita pela fun\303\247\303\243o 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, pois a partir dela podemos achar um bom valor para a constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw==, por compara\303\247\303\243o com o gr\303\241fico obtido diretamente de 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. Observe que para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOS8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRjk= a s\303\251rie de Fourier truncada aproxima razoavelmente bem a posi\303\247\303\243o inicial da corda: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Assim, o valor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSMxMEYnRjlGOQ== \303\251 satisfat\303\263rio para esse tipo de condi\303\247\303\265es iniciais. Note exite uma ondula\303\247\303\243o na parte superior do gr\303\241fico. Isto \303\251 consqu\303\252ncia do truncamento da s\303\251rie de Fourier dupla. Esta ondula\303\247\303\243o s\303\263 desparece a rigor, no limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GIzYkLUYsNiVRKCYjODczNDtGJ0YvRjJGOS1GLDYjUSFGJ0Y5. Outro detalhe que podemos observar, \303\251 que a s\303\251rie truncada obedece \303\240s condic\303\265es de contorno, enquanto que a condi\303\247\303\243o inicial n\303\243o obedece (veja a express\303\243o de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjdTBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGL0YyL0YzUSdub3JtYWxGJ0Y9Rj0=). Esta discrep\303\242ncia tamb\303\251m s\303\263 desaparece quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GIzYkLUYsNiVRKCYjODczNDtGJ0YvRjJGOS1GLDYjUSFGJ0Y5. As oscila\303\247\303\265es da fun\303\247\303\243o representando a condi\303\247\303\243o inicial nas extremidades da barra s\303\243o chamadas de fen\303\264meno de Gibbs e refletem a dificuldade de aproximar fun\303\247\303\265es discont\303\255nuas por s\303\251ries de Fourier, que s\303\243o somas de fun\303\247\303\265es oscilat\303\263rias cont\303\255nuas.A fun\303\247\303\243o a seguir faz a anima\303\247\303\243o do movimento da corda descrita pela fun\303\247\303\243o 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. O primeiro argumento \303\251 o tempo de anima\303\247\303\243o e o segundo \303\251 o n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= que representa a truncagem do somat\303\263rio duplo: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Para as condi\303\247\303\265es iniciais descritas acima, a anima\303\247\303\243o do movimento da corda com duas oscila\303\247\303\265es dominantes e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzMwRidGOUY5 \303\251:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEvYW5pbWF0ZWluaWNvbmRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ki1JI21uR0YkNiRRIzEwRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjs2JFEkMzAwRidGPkZALUY7NiRRIzIwRidGPi8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0ZGRj5GPi1GQTYtUSJ+RidGPkZEL0ZIRkZGSUZLRk1GT0ZRRlMvRldGVUZeby1GQTYtUSIjRidGPkZERmFvRklGS0ZNRk9GUUZTRmJvLUYsNiVRSn50ZW1wb35kZX5hbmltYSYjMjMxOyYjMjI3O28sfnZhbG9yfmRlfnB+RidGL0YyLUY2NiQtRiM2Li1GLDYlUSx0cnVuY2FtZW50b0YnRi9GMkZeby1GLDYlUSNkYUYnRi9GMkZeby1GLDYlUStzJiMyMzM7cmllRidGL0YyRl5vLUYsNiVRI2RlRidGL0YyRl5vLUYsNiVRKEZvdXJpZXJGJ0YvRjJGaW5GXG9GPkY+RkBGXm9GXm8tRiw2JVEsbiYjMjUwO21lcm9GJ0YvRjJGXm9GZnBGXm8tRiw2JVEnZnJhbWVzRidGL0YyRmluRlxvRj4=Vamos fazer a anima\303\247\303\243o no caso em que a corda est\303\241 parada na posi\303\247\303\243o horizontal, e sofre um impulso de cima para baixo em seu centro: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 antes, vamos calcular LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE= com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= inteiros positivos gen\303\251ricos:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm5GJ0YvRjIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkNGPUZARkM=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Vamos fazer a anima\303\247\303\243o durante um tempo 5 e truncar o somat\303\263rio em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzMwRidGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORldGOQ==com 40 frames: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Exerc\303\255cios1) Verifique que normalmode, normalmode1 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= satisfazem a equa\303\247\303\243o de onda.2) a) Resolva a equa\303\247\303\243o de onda para uma corda obedecendo \303\240s seguintes condi\303\247\303\265es de contorno: Uma extremidade da corda est\303\241 presa e a outra est\303\241 livre. [Sug.: Quando a extremidade est\303\241 livre, a derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= deve se anular.]b) Obtenha os modos normais e a solu\303\247\303\243o geral obedecendo \303\240s condi\303\247\303\265es iniciais gen\303\251ricas. c) Fa\303\247a a anima\303\247\303\243o dos modos normais.d) Fa\303\247a a anima\303\247\303\243o com diversas condi\303\247\303\265es iniciais.Exemplo 3 - Oscila\303\247\303\243o de membranas usando s\303\251ries de FourierRef.: R. Portugal, L. Golebiowski, D. Frenkel. Oscillation of membranes using computer algebra. American Journal of Physics 67 (6), 534-537, 1999.JSFHVamos considerar uma membrana horizontal uniformemente distentida em todas as dire\303\247\303\265es por uma tra\303\247\303\243o T. A forma da membrana ser\303\241 especificada pelas condi\303\247\303\265es de contorno. Qualquer que seja a forma, vamos analisar o caso em que toda a borda da membrana est\303\241 fixa. As oscila\303\247\303\265es s\303\243o sempre na dire\303\247\303\243o vertical. Sob certas restri\303\247\303\265es, como por exemplo pequenas oscila\303\247\303\265es, a equa\303\247\303\243o de onda que descreve o deslocamento vertical LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (dependente das vari\303\241veis espaciais e do tempo) na aus\303\252ncia de for\303\247as externas \303\251 dada por: 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 a densidade de massa. Esta equa\303\247\303\243o diferencial parcial pode ser resolvida pelo m\303\251todo de separa\303\247\303\243o de vari\303\241veis e a solu\303\247\303\243o geral pode ser expressa por uma s\303\251rie de Fourier generalizada. No caso da membrana retangular, a solu\303\247\303\243o \303\251 uma s\303\251rie de Fourier tripla usual e no caso da membrana circular axialmente sim\303\251trica, a solu\303\247\303\243o \303\251 uma s\303\251rie de Fourier-Bessel na parte espacial e Fourier usual na parte temporal. Vamos analisar o movimento oscilat\303\263rio destes destes dois tipos de membranas.Membrana RetangularResolu\303\247\303\243o da equa\303\247\303\243o de onda A membrana retangular de comprimento a e largura b presa nas bordas \303\251 caracterizada pelas seguintes condi\303\247\303\265es de contorno: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQC1GLDYlUSJ0RidGL0YyRj5GPi1GQTYtUSI9RidGPkZEL0ZIRkZGSUZLRk1GT0ZRL0ZUUSwwLjI3Nzc3NzhlbUYnL0ZXRl5vRjpGPg== e 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJiZsZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEiYkYnRi9GMkY5 e 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 e 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJiZsZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEiYUYnRi9GMi8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRjk=. Vamos atribuir a equa\303\247\303\243o diferencial deste problema \303\240 vari\303\241vel eq_de_onda:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic=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Vamos resolver essa equa\303\247\303\243o usando o m\303\251todo de separa\303\247\303\243o de vari\303\241veis. Vamos supor que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= \303\251 uma fun\303\247\303\243o separ\303\241vel nas vari\303\241veis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY1LUYsNiVRInRGJ0YvRjJGOQ==, isto \303\251, 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: 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Cada termo da express\303\243o acima deve ser constante, pois cada um \303\251 fun\303\247\303\243o de uma vari\303\241vel independente. Vamos selecionar o termo dependente de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, e igualar a uma constante :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A equa\303\247\303\243o diferencial acima pode ter solu\303\247\303\265es oscilat\303\263rias, lineares ou exponenciais dependendo se a constante for negativa, zero ou positiva, respectivamente. Para que as condi\303\247\303\265es de contorno sejam satisfeitas, isto \303\251, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVRistRjY2JC1GIzYkLUYsNiVRImFGJ0YvRjJGPkY+RkBGOkY+, temos que impor que a constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM5NTU7MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= seja negativa e que ela assuma valores discretos dados por: 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 m \303\251 um inteiro positivo. A solu\303\247\303\243o geral da equa\303\247\303\243o da parte \303\251:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIidGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5RJjAuMGVtRidGK0ZPLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHRj1GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiktRiw2JVEmc29sX1hGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNigtRiw2JVEnZHNvbHZlRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUZcbzYmLUYjNiktRiw2JVEkZXF4RidGNkY5LUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiM2KUYrLUYjNigtRiw2JVEiWEYnRjZGOUZlbi1GXG82JC1GIzYmLUkjbW5HRiQ2JFEiMEYnRkAvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGREZARkBGXHFGX3FGQC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPRlFGVEZocEZccUZfcUZARitGXHFGX3FGQEZALyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0YrRlxxRl9xRkBGQEZccUZfcUZARitGXHFGX3FGQEYrRlxxRl9xRkBGK0ZccUZfcUZAVamos eliminar a constante arbitr\303\241ria por enquanto: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Vamos repetir o mesmo procedimento para o termo dependente de y: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 equa\303\247\303\243o diferencial para a fun\303\247\303\243o dependente da vari\303\241vel t \303\251 obtida da seguinte forma: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cuja solu\303\247\303\243o geral \303\251: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 colocar esta solu\303\247\303\243o numa forma mais conveniente: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onde A e B s\303\243o constantes que podem depender dos autovalores m e n. Vamos definir a frequ\303\252ncia do modo normal <>: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e definir os modos normais como uma fun\303\247\303\243o de m e n. Existem duas possibilidades que s\303\243o: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Uma vez que a equa\303\247\303\243o diferencial do movimento da membrana \303\251 linear e homog\303\252nea, a solu\303\247\303\243o geral \303\251 uma superposi\303\247\303\243o de todas as solu\303\247\303\265es linearmente independentes poss\303\255veis. Assim, tomaremos o somat\303\263rio sobre todos os valores de m e n: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As constantes arbitr\303\241rias 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 e 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 s\303\243o determinadas a partir das condi\303\247\303\265es iniciais. Geralmente s\303\243o elas a posi\303\247\303\243o inicial 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 e a velocidade inicial 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. Anima\303\247\303\243o dos Modos NormaisOs modos normais podem ser animados como a fun\303\247\303\243o animate3d. Vamos construir a fun\303\247\303\243o animate_mode, que anima o modo especificado pelos seus argumentos. Por exemplo, animate_mode(1,2) faz a anima\303\247\303\243o do modo m=1 e n=2. 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 fazer a anima\303\247\303\243o de um determinado modo, temos que dar valores num\303\251ricos para os par\303\242metros a, b e c. Vamos supor que o comprimento \303\251 a=1, a largura \303\251 b=1 e a constante c=1 em um certo sistema de unidades: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Vamos fazer a anima\303\247\303\243o do modo <1,2> 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As linhas nodais s\303\243o curvas cujos pontos se mant\303\252m em repouso durante todo instante. Por exemplo, vejamos a anima\303\247\303\243o do modo <2,5>: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 ver que h\303\241 4 retas nodais paralelas ao eixo y e 1 reta nodal paralela ao eixo x. Esta caracter\303\255stica pode ser generalizada para qualquer modo de vibra\303\247\303\243o. O modo <m,n> tem m-1 retas nodais paralelas ao eixo y e n-1 retas nodais paralelas ao eixo x. Em geral, as linhas nodais nem sempre s\303\243o retas. Abaixo, mostraremos um exemplo de uma linha nodal fechada aproximadamente circular em uma membrana quadrada. Isso ocorre quando existe degeneresc\303\252ncia.An\303\241lise da Degeneresc\303\252nciaNum modo normal de vibra\303\247\303\243o, todos os pontos da membrana vibram com a mesma frequ\303\252ncia; cada modo normal tem um frequ\303\252ncia caracter\303\255stica 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. Se a membrana for quadrada, dois modos normais diferentes podem oscilar com a mesma frequ\303\252ncia. Por exemplo, para a membrana quadrada de largura de uma unidade, os modos <1,2> e <2,1> t\303\252m a mesma frequ\303\252ncia: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 caso, qualquer combina\303\247\303\243o linear destes modos de vibra\303\247\303\243o tamb\303\251m vai oscilar na mesma frequ\303\252ncia. Este fen\303\264meno \303\251 chamado de degeneresc\303\252ncia. Isto ocorre sempre que um autovalor (a frequ\303\252ncia) \303\251 degenerado, possuindo mais de um autovetor associado (o modo normal). A fun\303\247\303\243o a seguir faz a anima\303\247\303\243o da combina\303\247\303\243o linear 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. 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 considerar uma membrana quadrada: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Vamos fazer a anima\303\247\303\243o da combina\303\247\303\243o linear 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, onde podemos verificar que a linha nodal \303\251 uma curva fechada: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 A combina\303\247\303\243o linear 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 tem como linha nodal uma curva fechada e uma reta (uma das diagonais da membrana):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A linha nodal \303\251 descrita pela equa\303\247\303\243o 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. Esta equa\303\247\303\243o depende das vari\303\241veis x e y. O fator temporal \303\251 comum aos dois termos e portanto \303\251 cancelado. Atrav\303\251s do comando implicitplot, podemos fazer o gr\303\241fico da curva y contra x mesmo sem resolver explicitamente a equa\303\247\303\243o. A fun\303\247\303\243o nodal_line a seguir faz o desenho da linha nodal. Os argumentos desta fun\303\247\303\243o s\303\243o iguais ao do fun\303\247\303\243o LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVErYW5pbWF0ZV9sY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==. 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 linha nodal dos dois \303\272ltimos exemplos que vimos acima s\303\243o: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Anima\303\247\303\243o com Condi\303\247\303\265es IniciaisEncontramos anteriormente a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGPS1GLDYlUSJ0RidGL0YyRkFGQUZB que descreve o movimento vertical da membrana retangular de largura LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= presa nas bordas. Esta solu\303\247\303\243o \303\251 uma s\303\251rie de Fourier dupla. Uma vez que os modos normais com pequenos valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= dominam sobre os que tem grandes valores, podemos truncar o somat\303\263rio duplo. O ponto de truncagem satisfat\303\263rio depende das condi\303\247\303\265es iniciais do problema. Vamos converter o somat\303\263rio infinito num somat\303\263rio finito com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= termos e definir a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkdV9wRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiotRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY9LUYsNiVRInRGJ0YvRjJGPS1GLDYlUSJwRidGL0YyRkFGQUZB da seguinte forma: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LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFEI35jfi1+dGhlfmNvbnN0YW50fm9mfndhdmV+ZXF1YXRpb25GJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFBI35tfmFuZH5ufi1+dGhlfm9zY2lsbGF0aW9ufm1vZGVGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFPI354LHksdH4tfnRoZX5zcGF0aWFsfmFuZH50ZW1wb3JhbH5jb29yZGluYXRlc0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFPI35BfmFuZH5Cfi1+dGhlfmNvbnN0YW50c35vZn5Gb3VyaWVyfmV4cGFuc2lvbkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFQI351MH4tfnRoZX5pbml0aWFsfmRpc3BsYWNlbWVudH5vZn50aGV+bWVtYnJhbmVGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFMI352MH4tfnRoZX5pbml0aWFsfnZlbG9jaXR5fm9mfnRoZX5tZW1icmFuZUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==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As constantes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE=, que aparecem na fun\303\247\303\243o 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, podem ser obtidas a partir das condi\303\247\303\265es iniciais posi\303\247\303\243o inicial 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 e velocidade inicial 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 usando as seguintes f\303\263rmulas: 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e 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de fazer a anima\303\247\303\243o com condi\303\247\303\265es iniciais, vamos calcular as contantes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm1GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIm5GJ0YvRjJGQUZBRkE= e 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Vamos usar as seguintes condi\303\247\303\265es iniciais: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O gr\303\241fico da posi\303\247\303\243o inicial \303\251: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Para calcular 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 e 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 para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw== inteiros positivos gen\303\251ricos, vamos usar a fun\303\247\303\243o assume. Desta forma, o comando simplify das fun\303\247\303\265es 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 e 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 poder\303\241 fazer as simplifica\303\247\303\265es pertinentes: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Vamos fazer o gr\303\241fico da posi\303\247\303\243o inicial da membrana desta vez descrita pela fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkdV9wRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNistRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY9LUYsNiVRInRGJ0YvRjJGPS1GLDYlUSJwRidGL0YyLyUwZm9udF9zdHlsZV9uYW1lR1EnTm9ybWFsRidGQUZBRltvRkE=, pois a partir dela podemos achar um bom valor para a constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw==, por compara\303\247\303\243o com o gr\303\241fico obtido diretamente de 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. Observe que para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOS8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRjk= a s\303\251rie de Fourier truncada aproxima razoavelmente bem a posi\303\247\303\243o inicial da membrana: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, o valor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSMxMEYnRjlGOQ== \303\251 satisfat\303\263rio para esse tipo de condi\303\247\303\265es iniciais. Note exite uma ondula\303\247\303\243o na parte superior do gr\303\241fico. Isto \303\251 consqu\303\252ncia do truncamento da s\303\251rie de Fourier dupla. Esta ondula\303\247\303\243o s\303\263 desparece a rigor, no limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GIzYkLUYsNiVRKCYjODczNDtGJ0YvRjJGOS1GLDYjUSFGJ0Y5. Outro detalhe que podemos observar, \303\251 que a s\303\251rie truncada obedece as condic\303\265es de contorno, enquanto que a condi\303\247\303\243o inicial n\303\243o obedece. Esta discrep\303\242ncia tamb\303\251m s\303\263 desaparece quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GIzYkLUYsNiVRKCYjODczNDtGJ0YvRjJGOS1GLDYjUSFGJ0Y5.A fun\303\247\303\243o a seguir faz a anima\303\247\303\243o do movimento da membrana descrita pela fun\303\247\303\243o 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. O primeiro argumento \303\251 o tempo de anima\303\247\303\243o e o segundo \303\251 o n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= que representa a truncagem do somat\303\263rio duplo: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Para 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fazer a anima\303\247\303\243o no caso em que a membrana est\303\241 parada na posi\303\247\303\243o horizontal, e sofre um impulso de cima para baixo em seu 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fazer a anima\303\247\303\243o durante um tempo 5 e truncar o somat\303\263rio em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzMwRidGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORldGOQ==(900 termos) com 20 frames:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEvYW5pbWF0ZWluaWNvbmRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ki1JI21uR0YkNiRRIjVGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOzYkUSMzMEYnRj5GQC1GOzYkUSMyMEYnRj4vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGRkY+Rj5GaW5GXG9GPg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Exerc\303\255cios1) Verifique que normalmode, normalmode1 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGPS1GLDYlUSJ0RidGL0YyRkFGQUZB satisfazem a equa\303\247\303\243o de onda.2) Use o comando pdsolve para encontrar a solu\303\247\303\243o 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 O comando dsolve n\303\243o deve ser usado em nenhum momento. 3) a) Resolva a equa\303\247\303\243o de onda para uma membrana obedecendo \303\240s seguintes condi\303\247\303\265es de contorno: Duas extremidades opostas da membrana est\303\243o presas, e as outras duas est\303\243o livres. [Sug.: Quando a extremidade est\303\241 livre, a derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGPS1GLDYlUSJ0RidGL0YyRkFGQUZB deve se anular.]b) Obtenha os modos normais e a solu\303\247\303\243o geral obedecendo \303\240s condi\303\247\303\265es iniciais gen\303\251ricas. c) Fa\303\247a a anima\303\247\303\243o dos modos normais.d) Fa\303\247a uma an\303\241lise da degeneresc\303\252ncia.e) Fa\303\247a a anima\303\247\303\243o com diversas condi\303\247\303\265es iniciais.Membrana Circular Resolu\303\247\303\243o da Equa\303\247\303\243o de Onda A membrana circular de raio a presa na borda \303\251 caracterizada pela seguinte condi\303\247\303\243o de contorno em coordenadas cil\303\255ndricas: 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, para 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. Vamos atribuir a equa\303\247\303\243o diferencial deste problema \303\240 vari\303\241vel wave_eq: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 resolver essa equa\303\247\303\243o usando o m\303\251todo de separa\303\247\303\243o de vari\303\241veis. Substituindo 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 na vari\303\241vel LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEod2F2ZV9lcUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== obtemos: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Vamos fazer agora a separa\303\247\303\243o de vari\303\241veis: 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 condi\303\247\303\243o de contorno 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 imp\303\265e que a constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy seja negativa pois quando ela \303\251 positiva ou nula, a equa\303\247\303\243o Eqr n\303\243o adimite solu\303\247\303\265es que se anulem para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyRjk=. Al\303\251m disso, a vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 peri\303\263dica. Assim, vamos assumir que: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiktRiw2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIzo9RidGOS8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlAtRiM2KC1GPTYtUSomdW1pbnVzMDtGJ0Y5RkBGQkZERkZGSEZKRkwvRk9RLDAuMjIyMjIyMmVtRicvRlJGWS1GIzYkLUklbXN1cEdGJDYlLUYjNigtRiw2JVEia0YnL0Y3USV0cnVlRicvRjpRJ2l0YWxpY0YnLUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EmMC4wZW1GJy9GUkZnby1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRIm1GJ0Zfb0Zhby1GPTYtUSIsRidGOUZAL0ZDRmBvRkRGRkZIRkpGTEZmby9GUlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJuRidGX29GYW8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGOEY5RjlGanBGXXFGOS1JI21uR0YkNiRRIjJGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y5RitGanBGXXFGOUYrRmpwRl1xRjlGK0ZqcEZdcUY5RitGanBGXXFGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiktRiw2JVEoJiM5NTU7MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2KC1GPTYtUSomdW1pbnVzMDtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGZm4tRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJtRidGNkY5LUkjbW5HRiQ2JFEiMkYnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkBGKy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0ZERkBGK0Znb0Zqb0ZARitGZ29Gam9GQEYrRmdvRmpvRkA=A solu\303\247\303\243o geral para Eqr a menos de uma constante multiplicativa \303\251: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A solu\303\247\303\243o geral para 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 \303\251: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A solu\303\247\303\243o geral para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== \303\251: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C\303\241lculo dos Modos NormaisVamos agora analisar os modos normais da membrana circular. Os modos com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= n\303\243o dependem da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, isto \303\251, eles s\303\243o axialmente sim\303\251tricos. Os modos com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= dependem da vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy e em fun\303\247\303\243o disto veremos exemplos dos dois casos. A frequ\303\252ncia do modo normal \303\251: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Para que as condi\303\247\303\265es de contorno sejam satisfeitas, temos que impor que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiMEYnRj1GPQ== ou seja 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. Assim, 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 deve assumir valores que anulem a fun\303\247\303\243o de Bessel de ordem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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 fun\303\247\303\243o LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEtQmVzc2VsSlplcm9zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== calcula o n-\303\251simo zero da fun\303\247\303\243o de Bessel de ordem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Vamos agora definir os modos normais em fun\303\247ao de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= existe uma degeneresc\303\252ncia de quarta ordem, isto \303\251, podemos definir os seguintes modos normais: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Os modos normais axialmente sim\303\251tricos s\303\243o os modos com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. Neste caso, a degeneresc\303\252ncia \303\251 dupla, j\303\241 que os modos LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEtbm9ybWFsX21vZGUyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEtbm9ybWFsX21vZGUzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMg== acima s\303\243o nulos. A solu\303\247\303\243o geral \303\251 a superposi\303\247\303\243o de todos os modos normais. A vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode assumir qualquer valor inteiro de 0 a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSgmIzg3MzQ7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= de 1 a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSgmIzg3MzQ7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo=. Por\303\251m, para as condic\303\265es iniciais que estamos tratando aqui, a s\303\251rie de Fourier converge rapidamente, de forma que podemos truncar o somat\303\263rio. As vari\303\241veis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn especificam o truncamento em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, respectivamente. Vamos separar os termos com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= pois estes coeficientes seguem uma regra diferente dos coeficientes com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= diferente de zero: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 coeficientes 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, 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjQjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2KS1GLDYlUSJtRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJuRidGL0YyLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUrZXhlY3V0YWJsZUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGQUZBRmVuRmhuRmpuRkE= e 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 s\303\243o constantes que dependem dos par\303\242metros LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Quando as condi\303\247\303\265es iniciais s\303\243o axialmente sim\303\251tricas, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInJGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJyYjOTUyO0YnL0YwRkVGQUY9LUYsNiVRInRGJ0YvRjJGQUZBRkE= n\303\243o depende de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. Neste caso, temos que tomar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, o que \303\251 equivalente a tomar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5. A express\303\243o de 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 se reduz a: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Anima\303\247\303\243o dos Modos NormaisVamos construir uma fun\303\247\303\243o para a membrana circular da mesma forma que foi constru\303\255do para a membrana retangular: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Para fazer a anima\303\247\303\243o de um determinado modo normal, temos que dar valores num\303\251ricos para os par\303\242metros LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Vamos supor que o raio \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= em um certo sistema de unidades: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Primeiro, vamos analisar o movimento do modos normais axialmente sim\303\251ticos (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=). Vamos fazer a anima\303\247\303\243o do modo <0,1>: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 primeiro modo axialmente sim\303\251trico n\303\243o possui linha nodal, como era de se esperar. O segundo modo deve ter uma linha nodal: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 linha nodal \303\251 um c\303\255rculo cujo raio pode ser calculado da seguinte maneira. Observe que um corte vertical passando pelo centro do gr\303\241fico acima tem a seguinte forma: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A posi\303\247\303\243o inicial do modo <0,2> \303\251 a figura formada pela rota\303\247\303\243o do gr\303\241fico acima em torno do eixo vertical. \303\211 f\303\241cil ver pelo gr\303\241fico que, de maneira geral, a primeira linha nodal do modo <0,n> tem o raio 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. No caso acima obtemos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJzAuNDM1NkYnRjlGOQ==.O primeiro modo normal n\303\243o axialmente sim\303\251trico \303\251 o modo <1,1>: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Neste caso, a linha nodal \303\251 uma reta passando pelo centro cujo \303\242ngulo \303\251 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. Os modos normais com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= s\303\243o de dif\303\255cil visualiza\303\247\303\243o sem recursos de anima\303\247\303\243o. Vejamos o modo <1,2>: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 maneira geral, o modo <LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEibkYnRi9GMkY5> tem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= retas nodais passando pelo centro separadas entre si pelo \303\242ngulo 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= c\303\255rculos de raios 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, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= vai de 1 at\303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=. As linhas nodais s\303\243o mostradas pelo seguinte procedimento: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 exemplo, o modo <2,3> tem as seguintes linha nodais: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Observe que o c\303\255rculo mais externo n\303\243o deve ser contado como uma linha nodal, pois representa a borda da membrana. Anima\303\247\303\243o com Condi\303\247\303\265es 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LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFEI35jfi1+dGhlfmNvbnN0YW50fm9mfndhdmV+ZXF1YXRpb25GJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFQI351MH4tfnRoZX5pbml0aWFsfmRpc3BsYWNlbWVudH5vZn50aGV+bWVtYnJhbmVGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFFMI352MH4tfnRoZX5pbml0aWFsfnZlbG9jaXR5fm9mfnRoZX5tZW1icmFuZUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==As constantes 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, 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, 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 e 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 que aparecem na fun\303\247\303\243o 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 definida anteriormente, podem ser obtidas a partir das express\303\265es te\303\263ricas derivadas a partir da invers\303\243o da s\303\251rie de Fourier: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fun\303\247\303\243o a seguir faz a anima\303\247\303\243o do movimento da membrana descrita pela fun\303\247\303\243o 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. O primeiro argumento do procedimento representa o tempo de oscila\303\247\303\243o, o seegundo argumento \303\251 o n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn que representa a truncagem do somat\303\263rio na vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, o terceiro argumento \303\251 o n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn que representa a truncagem do somat\303\263rio na vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e o quarto \303\251 o n\303\272mero de frames na anima\303\247\303\243o.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Vamos ver dois exemplos com condi\303\247\303\265es iniciais axialmente sim\303\251tricas. O primeiro com velocidade inicial nula e o segundo com delocamento inicial nulo. Vamos tomar as seguintes condi\303\247\303\265es iniciais: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 seja, a membrana est\303\241 repouso na posi\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=. Tomaremos o raio da membrana igual a 5: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 coeficientes 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, 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, 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 e 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s\303\243o: 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 fazer a anima\303\247\303\243o com duas oscila\303\247\303\265es dominantes e com . Uma vez que a condi\303\247\303\243o inicial \303\251 axialmente sim\303\251trica, vamos tomar :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Podemos ver pela anima\303\247\303\243o que a onda reflete na borda da membrana e se superp\303\265e no centro, produzindo um propaga\303\247\303\243o t\303\255pica de ondas, como deveria ser. A reflex\303\243o na borda inverte o deslocamento da onda, j\303\241 que a extremidade \303\251 fixa.Vamos fazer a anima\303\247\303\243o do movimento da membrana no caso em que ela est\303\241 na posi\303\247\303\243o horizontal e sofre um impulso de cima para baixo na regi\303\243o central. As condi\303\247\303\265es iniciais s\303\243o: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A anima\303\247\303\243o \303\251 obtida da atrav\303\251s do comando: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=