Ondes by André Nunes https://padlet.com/a30nunes/s0dsy9gzr499 2P011 en-us 2018-11-09 14:45:45 UTC 2023-04-05 00:22:05 UTC hello@padlet.com https://padlet-assets.s3.amazonaws.com/icons/Thunder.png a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302584409  Lorsqu'une perturbation se propage à partir de sa source en fonction du temps, dans les différentes directions spatiales du système, sans que les constituants du milieu subissent un mouvement d’ensemble, il s’agit d’une onde.]]> 2018-11-09 14:48:15 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302584409 ONDES a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302584622 2018-11-09 14:48:36 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302584622 Équation d'Alembert a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302586503 \:c^2 \frac{\partial^2X}{\partial x^2} = \frac{\partial^2X}{\partial t} ]]> 2018-11-09 14:51:31 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302586503 Fonction d'onde a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302631064 Une fonction est celle d'une onde  si :
  • pour les x croissants
f\left(x,\:t\right)=f\left(x\:-\:ct\right)

  • pour les x décroissants
f\left(x,\:t\right)=f\left(x\:+\:ct\right)]]> 2018-11-09 16:05:27 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302631064 Onde sinusoïdale a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302635927 Une onde sinusoïdale a pour formule:f\left(x,\:t\right)=A.cos\left(\omega .t\:-\:k.x\:+\varphi\right)]]> 2018-11-09 16:13:51 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302635927 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302639611 f\left(x,\:t\right)=A.cos\left(\omega .t\:-\:k.x\:+\varphi\right)

\Leftrightarrow \:f\left(x,\:t\right)=A.cos\left(-\text{k}.\left(\text{x}\:-\:c.t\right)\:+\:\varphi \right)

]]> 2018-11-09 16:20:19 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302639611 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302647915 Si :f\left(x,\:t\right)=f\left(x\:-\:ct\right)
Alors :
\left(\:x_1-c.t_1\right)=\left(x_2-c.t_2\:\right)\:\Rightarrow \:x_1+c.\left(\:t_2\:-\:t_1\right)\:=\:x_2\:\:\Rightarrow \:\:x_2\:>x_1

x croissants
]]> 2018-11-09 16:34:54 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302647915 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302654143   ]]> 2018-11-09 16:44:59 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302654143 ω : Pulsation a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302655362 \omega \:=\:\frac{2\pi }{T}

\nu \:=\:\frac{1}{T}

\omega \:=\:2\pi \nu]]> 2018-11-09 16:47:27 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302655362 k : Nombre d'onde a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302656731 \:k \:=\:\frac{2\pi }{\lambda}

\lambda = \:\frac{2\pi}{\left|\:k\right|}]]> 2018-11-09 16:50:08 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302656731 φ : Déphasage a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302658683 2018-11-09 16:53:58 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302658683 A : Amplitude a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302659045 2018-11-09 16:54:38 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302659045 Relation de dispersion a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302661066 k^2\:=\:\frac{\omega ^2}{c^2}\:

\Rightarrow \:k\:=\pm \frac{w}{c}]]> 2018-11-09 16:58:11 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302661066 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302662401 L'équation d'Alembert est valable pour toutes le ondes.
Soit:f\left(x,\:t\right)=A.cos\left(\omega .t\:-\:k.x\:+\varphi\right)
On injecte f dans l'equation d'Alembert:

\:c^2 \frac{\partial^2f\left(x,\:t\right)}{\partial x^2} = \frac{\partial^2f\left(x,\:t\right)}{\partial t}

\Rightarrow \:- c^2 k^2 A.cos\left(\omega .t\:-\:k.x\:+\varphi\right) = \:- t^2 A.cos\left(\omega .t\:-\:k.x\:+\varphi\right)

\Rightarrow \:- c^2 k^2 f\left(x,\:t\right) = \:- t^2 f\left(x,\:t\right)

\Rightarrow k^2\:=\:\frac{\omega ^2}{c^2}\:]]> 2018-11-09 17:00:52 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302662401 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302670036
  • Onde progressive:
    • \:k > 0 \Rightarrow \:k\:= + \frac{w}{c}

    • Onde régressive:
    \:k > 0 \Rightarrow \:k\:= - \frac{w}{c}]]>     2018-11-09 17:14:40 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302670036     Phase Φ a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302672929 f\left(x,\:t\right)=A.cos\left(\omega .t\:-\:k.x\:+\varphi\right)

    f\left(\Phi\left(x,\:t\right)\right) = A.cos\left(\Phi\left(x,\:t\right)\right)
    Où:

    \Phi\left(x,\:t\right) = \omega .t\:-\:k.x\:+\varphi


    ]]>     2018-11-09 17:20:04 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302672929     a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302677939 \Phi\left(x,\:t\right) = \omega .t\:-\:k.x\:+\varphi

    ]]>     2018-11-09 17:29:56 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302677939     a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302679879 \lambda = c.T]]> 2018-11-09 17:34:06 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302679879 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302681408  


    ]]>     2018-11-09 17:36:31 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302681408     Vitesse de phase a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302682413 C'est  la vitesse à laquelle se « déplace » la phase .

    v_{\varphi }=\frac{\omega }{\left|\:k\right|\:}\:=c]]>     2018-11-09 17:38:05 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302682413     a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302684654 \Phi\left(x,\:t\right) = \omega .t\:-\:k.x\:+\varphi


    \Phi\left(x,\:t\right) = \varphi_{0}


    \omega .t\:-\:k.x\:+\varphi = \varphi_{0}


    x = \frac{\omega}{\:k}.t + \frac{\varphi - \varphi_{0}}{\:k}


    v_{\varphi } =\frac{d}{dt}\left(\frac{\omega }{\:k}.t\:+\:\frac{\varphi \:-\:\varphi _0}{\:k}\right)\:=\:\frac{\omega \:}{\:k}    ]]>     2018-11-09 17:42:08 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302684654     Fonction d'onde complexe a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302755918 \widetilde{f}\left(x,\:t\right)=Ãe^{i\left(\omega .t\:-\:k.x\right)}

    avec \:Ã = A.e^{i\varphi_{0}} ]]>     2018-11-09 20:05:45 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302755918     Principe de Superposition a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302764802 Une combinaison linéaire de deux ondes est aussi une onde.]]> 2018-11-09 20:29:37 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302764802 Équation à 3 dimentions générale a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302777905 \:c^2\:\left(\frac{\partial \:^2f\left(\vec{x},\:t\right)}{\partial \:\:x^2}\:+\frac{\partial \:^2f\left(\vec{x},\:t\right)}{\partial \:\:y^2}\:+\frac{\partial \:^2f\left(\vec{x}\:,\:t\right)}{\partial \:z^2}\right)=\:\frac{\partial ^2f\left(\vec{x}\:,\:t\right)}{\partial \:t}]]> 2018-11-09 21:20:44 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302777905 Équation Ondes Planes a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302778784 Une onde plane ne dépend que de x

    \:c^2 \frac{\partial^2X}{\partial x^2} = \frac{\partial^2X}{\partial t}

    ]]>     2018-11-09 21:24:51 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302778784     Fonction d'onde plane sinusoïdale complexe a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302780209 \widetilde{f}\left(\vec{x},\:t\right)=Ãe^{i\left(\omega .t\:-\:\vec{k}.\vec{x}\right)}

    avec \:Ã = A.e^{i\varphi_{0}} ]]>     2018-11-09 21:31:43 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302780209     Relation de dispersion a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302782137 \:\frac{\omega ^2}{c^2} = \|\vec{k}\|^2]]> 2018-11-09 21:41:23 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302782137 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302783421 Le plan orthogonal au vecteur k est appelé le plan d'onde.

    ]]>     2018-11-09 21:47:15 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302783421     Fonction d'onde sphérique sinusoïdale complexe a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302789866 \widetilde{f}\left(\vec{r},\:t\right)=\frac{1}{r}Ãe^{i\left(\omega .t\:-\:\vec{k}.\vec{r}\right)}

    avec \:Ã = A.e^{i\varphi_{0}} ]]>     2018-11-09 22:21:08 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302789866     a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302796774 Les surfaces d'onde sont des sphères]]> 2018-11-09 22:38:43 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302796774 Aproximation d'une onde sphérique par une onde plane a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302798417 \partial \:l \ll \sqrt{\lambda.\:R}]]> 2018-11-09 22:43:53 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302798417 a30nunes https://padlet.com/a30nunes/s0dsy9gzr499/wish/302802021 La phase au point M, à une distance R \Phi_{S}\left(M \:, t\right) = \omega.t - k.R + \varphi

    Si on se deplace de ∂l vers le haut, on se trouve à une distance R +∂R de la source.

    \:R^2 + \left(\partial \:l \right)^2 = \left(R + \partial \:R \right)^2 \approx R^2 + 2.R.\partial R

    \Rightarrow \:R^2 + \left(\partial \:l \right)^2 \approx R^2 + 2.R.\partial R

    \Rightarrow \partial R \approx \frac{\left(\partial l\right)^2}{2.R}

    La difference de phase est:
    \partial \Phi = \Phi_{S}\left(M' \:, t\right) - \Phi_{S}\left(M \:, t\right) = -k.\partial R \approx - \frac{\pi.\left(\partial l \right)^2}{\lambda .R}

    Et
    \frac{\pi.\left(\partial l \right)^2}{\lambda .R} \ll \pi

    \Rightarrow \partial \:l \ll \sqrt{\lambda.\:R}]]>     2018-11-09 23:15:52 UTC https://padlet.com/a30nunes/s0dsy9gzr499/wish/302802021