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(Finora abbiamo studiato un moto noto sulla sorgente, non una forza generica). + +\begin{figure}[H] + \centering + % TODO: edit to remove little arrow at the bottom + \includegraphics[width=0.8\textwidth]{screenshots/2024-03-19-11-12-22.png} +\end{figure} + +Impostando l'equazione della dinamica per il moto dell'anello posto in \(x=0\) otteniamo: +\[ + \vec{F_e} + \vec{T} +\vec{R_v} = \mathrm{d}m \vec{a} \rightsquigarrow \begin{dcases} + T\cos \theta + R_v = \mathrm{d} m a_x\\ + F_y + T\sin \theta =\mathrm{d} m a_y \\ + \end{dcases} +\] + +Facendo tendere \(\mathrm{d} m \to 0\), i membri di destra tendono a zero. +\[ + \begin{dcases} + T\cos \theta + R_v = 0\\ + F_y + T\sin \theta = 0 \\ + \end{dcases} + \rightsquigarrow F_y = -T\sin \theta \thickapprox -T \tan \theta = -T \frac{\partial \xi }{\partial x} +\] +Di conseguenza \(F_y = -T \left. \frac{\partial \xi }{\partial x}\right\vert_x=0 \). In questo sistema si producono esclusivamente onde progressive (\(f(x-vt)\) ), se ci fosse un + al posto del - tratteremmo invece onde regressive. +\begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{screenshots/2024-03-19-11-22-13.png} +\end{figure} +Per quanto scritto sopra, +\[ + \frac{\partial \xi }{\partial x} = - \frac{1}{v} \frac{\partial \xi }{\partial t} \rightsquigarrow F_y(t) = \frac{T}{v}\frac{\partial \xi (0,t)}{\partial t} +\] +La derivata parziale di \(\xi \) calcolata nel punto 0 rappresenta la velocità verticale del punto sorgente, che verrà indicata con \(v_c(t)\). Inoltre possiamo definire l'impedenza meccanica della corda. +\begin{definition} + [Impedenza meccanica] + Si definisce impedenza meccanica: + \[ + Z=\frac{T}{v}= T \sqrt{\frac{\mu }{T}} = \sqrt{\mu T} + \] +\end{definition} +L'effetto della forzante è quindi quello di mettere in moto il primo punto della corda con una velocità proporzionale alla forza (di solito non è così!): +\[ + F_y(t) = Zv_c(t) +\] +Usare \(v\) e \(Z\) al posto di T e \(\mu \) ci permetterà di trovare risultati molto più generali. + +\paragraph{C'è una forza viscosa?} + +Sembra che nel sistema ci sia una forza viscosa che porta la forza a essere proporzionale alla velocità. Si possono interpretare le formule come se la corda fosse in grado di applicare una forza dipendente dalla velocità: \(F_v(t) = -Z v_c(t)\). L'energia non viene dissipata, è assorbita e utilizzata per mettere in moto punti sempre più lontani della corda. + +\begin{note} + \(v_c(t)\) è sempre in fase con la forzante, quindi siamo sistematicamente in una situazione di risonanza. +\end{note} + +\subsection{Energia e potenza} + +Il lavoro compiuto dalla forzante è \(\delta L = \vec{F} \cdot \mathrm{d} \vec{s} = F_y \mathrm{d} y = F_y \frac{\mathrm{d}\xi }{\mathrm{d}t} \mathrm{d} t = F_y v_c \mathrm{d} t \). Tuttavia, per quanto ricavato prima so che + +\begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{screenshots/2024-03-19-11-33-56.png} +\end{figure} + +E la potenza risulta quindi: +\[ + \mathcal{P} = \frac{\delta L}{\mathrm{d} t} = -T \left. \frac{\partial \xi }{\partial x} \right\vert_{x=0} \frac{\partial \xi (0,t)}{\partial t} +\] +Tuttavia la relazione è più generale, che vale per ogni punto della corda: +\[ + \mathcal{P} = -T \frac{\partial \xi }{\partial x} \frac{\partial \xi }{\partial t} +\] +Per le onde progressive sappiamo già che +\begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{screenshots/2024-03-19-11-37-15.png} + \caption{Relazione fra la derivata temporale e spaziale per le onde progressive.} + \label{fig:dt-dx-onda-prog} +\end{figure} +Di conseguenza la potenza può essere riscritta: +\begin{gather*} + \mathcal{P} (x,t) = \frac{T}{v}\left(\frac{\partial \xi (x,t)}{\partial t}\right)^{2} = Tv\left(\frac{\partial \xi (x,t)}{\partial x} \right)^{2} \\ + \mathcal{P} (x,t) = Z \left(\frac{\partial \xi (x,t)}{\partial t}\right)^{2} +\end{gather*} +Si vede subito che la potenza è sempre maggiore o uguale a zero. Quanto ricavato è valido in generale per tutte le onde meccaniche. + +\paragraph{Energia cinetica} +Consideriamo un tratto di corda infinitesima in moto con velocità \(v=\frac{\partial \xi }{\partial t} \). +\begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{screenshots/2024-03-19-11-44-58.png} + \caption{L'immagine di riferimento per lo studio di energia cinetica e potenziale sulla corda.} + \label{fig:energia-corda} +\end{figure} +Ottengo che +\[ + \mathrm{d} K = \frac{1}{2} \mathrm{d} m v ^{2} = \frac{1}{2} \mu \left( \frac{\partial \xi }{\partial t} \right) ^{2} \mathrm{d} x +\] +Da cui posso definire una densità lineare di energia cinetica: +\[ + u_K(x,t) = \frac{\mathrm{d}K}{\mathrm{d}x} = \frac{1}{2} \mu \left( \frac{\partial \xi }{\partial t} \right) ^{2} +\] + +\paragraph{Energia potenziale} + +Si fa riferimento sempre alla figura \ref{fig:energia-corda}. Il tratto infinitesimo di corda viene allungato facendo lavoro contro la tensione della corda: \(\delta L = -T (\mathrm{d}l - \mathrm{d} x ) = -T(\sqrt{\mathrm{d}x^{2} + \mathrm{d} y ^{2} } - \mathrm{d} x )\). Applicando l'approssimazione delle piccole oscillazioni otteniamo che +\begin{gather*} + \delta L = -T \mathrm{d} x \left( \sqrt{1+ \left( \frac{\partial \xi }{\partial x} \right)^{2} } -1 \right) \thickapprox -T \mathrm{d} x \left( 1+\frac{1}{2} \left( \frac{\partial \xi }{\partial x} \right)^{2} + \cdots - 1 \right) \thickapprox \\ + \thickapprox -\frac{1}{2}T \left( \frac{\partial \xi }{\partial x} \right)^{2} \mathrm{d} x\\ + u_P(x,t) = -\frac{\delta L}{\mathrm{d} x} = \frac{1}{2}T\left( \frac{\partial \xi }{\partial x} \right) ^{2} +\end{gather*} +Dove nell'ultima riga abbiamo definito la densità di energia potenziale. + +\paragraph{Energia meccanica} +Possiamo definire la densità di energia meccanica: +\begin{definition} + [Densità lineare di energia meccanica] + \[ + u(x,t) = \frac{1}{2} \mu \left( \frac{\partial \xi }{\partial t} \right) ^{2} + \frac{1}{2}T\left( \frac{\partial \xi }{\partial x} \right) ^{2} + \] +\end{definition} +Nel caso di onde progressive abbiamo le formule già viste che collegano la derivata in x alla derivata in t (Fig. \ref{fig:dt-dx-onda-prog}), quindi possiamo ricavare la seguente uguaglianza: +\begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{screenshots/2024-03-19-11-56-45.png} +\end{figure} +L'energia potenziale e l'energia cinetica sono uguali istante per istante in ogni punto! Quindi l'energia totale è il doppio dell'energia cinetica e il doppio dell'energia potenziale. La potenza e l'energia trasmessa sono inoltre proporzionali: +\begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{screenshots/2024-03-19-11-58-19.png} +\end{figure} \ No newline at end of file diff --git a/tex/waves/waves.tex b/tex/waves/waves.tex index 2f72c5e..f6f0f56 100644 --- a/tex/waves/waves.tex +++ b/tex/waves/waves.tex @@ -49,6 +49,6 @@ \newpage -\lec{3}{6} +\lec{3}{7} \end{document} \ No newline at end of file