From 1700806d442992ef174a0f30ff9a75f50785885e Mon Sep 17 00:00:00 2001 From: JBR Date: Sun, 29 Mar 2026 23:05:02 +0200 Subject: [PATCH] =?UTF-8?q?Onde=20=C3=A8=20esploso:=20sono=20in=20pari,=20?= =?UTF-8?q?Per=20farlo=20ho=20costruito=20tool=20per=20fare=20speed=20up.?= =?UTF-8?q?=20Sistemati=20tutti=20gli=20errori=20di=20compilazione=20dati?= =?UTF-8?q?=20dalla=20libreria.=20Ora=20si=20compila=20senza=20warning=20(?= =?UTF-8?q?nuovo=20standard)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- jacbr.sty | 12 +- onde/onde.pdf | Bin 243710 -> 414923 bytes onde/onde.tex | 1826 +++++++++++++++++++++++++++++++++++++++++++++++-- 3 files changed, 1772 insertions(+), 66 deletions(-) diff --git a/jacbr.sty b/jacbr.sty index fd571e6..324e6e3 100644 --- a/jacbr.sty +++ b/jacbr.sty @@ -1,10 +1,11 @@ %!! ffmpeg -i logo.svg -vf scale=64:64 favicon.png -\ProvidesPackage{jacbr} +% \ProvidesPackage{jacbr} +\ProvidesPackage{./../jacbr} \RequirePackage[most]{tcolorbox} \RequirePackage{subcaption} -\usepackage[utf8]{inputenc} +% \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[a4paper,margin=2cm,nohead,foot=0.5cm]{geometry} \usepackage{xcolor} @@ -77,9 +78,10 @@ %fare siusetup % Header -\setlength{\headheight}{14pt} +\setlength{\headheight}{14.5pt} \setlength{\headsep}{10pt} -\addtolength{\topmargin}{-5pt} +\addtolength{\topmargin}{-5.5pt} +\setlength{\footskip}{14.5pt} \fancyhf{} \fancyhead[L]{\textbf{\thetitle} --- \monthname{\the\month} \the\year} @@ -120,6 +122,8 @@ \newcommand{\derx}[1]{\frac{d #1}{dx}} \newcommand{\derxx}[1]{\frac{d^2 #1}{dx^2}} \newcommand{\der}[2]{\frac{d #1}{d #2}} +\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}} + %per le parentesi \newcommand{\8}{\left(} \newcommand{\9}{\right)} diff --git a/onde/onde.pdf b/onde/onde.pdf index 7c4f119a37a2ce93a183e55a453d45f8e29ddd30..03d6292e48242f113f68a5e71169c3e38fb082ab 100644 GIT binary patch literal 414923 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z)t5Bt_D8UmR;bvR?VFUvH*Z`SQ%J4%e$}JUTrNwB96hgT@AA{4Q`i={_@y{>n`6au zSfQw}Vo^@aWEI4W=Ht(JncwqNP|o@@61^@7er?Qt393gY=A(=dv+a8NN@umX6ujX2 i%ATP8ok#8Lb!_dOzil><^o(?@^bFr!wV<>R 0 \imp \gamma^2 > \omega_p^2 \] @@ -366,13 +366,13 @@ Per questa equazione è però necessario applicare un metodo più sistematico e \] Scritta in questo modo si nota perfettamente che sono 2 esponenziali che tenendo a 0 senza aver compiuto un oscillazione vera e propria. - Questo sistema si chiama: SOVRASMORZATO. + Questo sistema si chiama: SOVRA-SMORZATO. \end{dimJ} -\begin{dimJ}{Oscillatore armonico smorzato: SMORZAMENTO CRITICO}{dim:SMORZAMENTOCRITICO} +\begin{dimJ}{Oscillatore armonico smorzato: SMORZAMENTO CRITICO}{dim:SMORZAMENTO-CRITICO} Passiamo alla seconda possibilità: \[ - \Delta = 0 + \Delta = 0 \] Questo ci crea non pochi problemi perché ora abbiamo una sola equazione e non 2 soluzioni particolari. @@ -385,7 +385,7 @@ Per questa equazione è però necessario applicare un metodo più sistematico e \[ x_2(t) = t e^{-\gamma t} \] - + La nostra combinazione lineare è quindi: \[ x(t) = e^{-\gamma t} \8 C_1 + C_2 t \9 @@ -394,12 +394,12 @@ Per questa equazione è però necessario applicare un metodo più sistematico e Questo sistema viene chiamato SMORZAMENTO CRITICO. Questo è quello che si vuole ottenere in sistemi meccanici reali come i pistoni di una macchina. \end{dimJ} -\begin{dimJ}{Oscillatore armonico smorzato: SOTTOSMORZATO}{dim:SOTTOSMORZATO} +\begin{dimJ}{Oscillatore armonico smorzato: SOTTO-SMORZATO}{dim:SOTTO-SMORZATO} Ora abbiamo l'ultimo caso: \[ - \Delta < 0 + \Delta < 0 \] - + Come sappiamo le soluzioni sono complesse coniugate: \begin{align*} \lambda_{1,2} = -\gamma \pm i \sqrt{\omega_p^2 - \gamma^2}\\ @@ -441,7 +441,7 @@ Per questa equazione è però necessario applicare un metodo più sistematico e x(t) = A e^{- \gamma t} \cos (\omega t + \varphi) \] - In questo caso l'oscillatore è detto SOTTOSMORZATO o DEBOLMENTE SMORZATO + In questo caso l'oscillatore è detto SOTTO-SMORZATO o DEBOLMENTE SMORZATO \end{dimJ} \begin{dimJ}{Trasformazione dell'equazione}{dim:trasformazioneInc} @@ -501,7 +501,7 @@ Per questa equazione è però necessario applicare un metodo più sistematico e \subsection{Condizioni iniziali} -Come fatto per l'oscillatore armonico semplice vogliamo trovare un legame tra le condizioni iniziali e i paramentri dell'equazione. +Come fatto per l'oscillatore armonico semplice vogliamo trovare un legame tra le condizioni iniziali e i parametri dell'equazione. \[ x(t) = A e^{-\gamma t} \cos (\omega t + \varphi) \] @@ -510,7 +510,7 @@ Come prima valutiamo sia la posizione che la velocità in 0. \begin{align*} align* x(0) = A \cos \varphi\\ \dert{x} = - \gamma A e^{-\gamma t} \cos (\omega t + \varphi) - \omega A e^{-\gamma t} \sin (\omega t + \varphi)\\ - \dert{x} (0) = - \gamma A \cos \varphi + - \omega A \sin \varphi + \dert{x} (0) = - \gamma A \cos \varphi + - \omega A \sin \varphi \end{align*} Si nota subito che questa non è una soluzione lineare, ma possiamo fare alcune semplificazioni del nostro sistema: @@ -538,7 +538,7 @@ Si nota subito che questa non è una soluzione lineare, ma possiamo fare alcune Ora sommandole otteniamo: \[ - A^2 = x^2(0) + \8 \frac{v(0) + \gamma x(0)}{\omega} \9^2 + A^2 = x^2(0) + \8 \frac{v(0) + \gamma x(0)}{\omega} \9^2 \] Mentre il rapporto fatto prima di quadrarle ci restituisce: @@ -573,7 +573,7 @@ Con questa ipotesi vale: $\gamma \cos(\dots) << \omega \sin(\dots)$. K \simeq \frac 1 2 m A^2 e^{- 2 \gamma t} \omega^2 \sin^2(\omega t + \varphi) \] -Nell'energia, e non nella fase (in quanto l'errore si accumulerebbe ciclo per ciclo), possiamo approssimare ancora e osservare che se $\gamma << \omega \imp \omega \simeq \omega_p = \frac \alpha m$. +Nell'energia, e non nella fase (in quanto l'errore si accumulerebbe ciclo per ciclo), possiamo approssimare ancora e osservare che se $\gamma << \omega \imp \omega \simeq \omega_p = \frac \alpha m$. \[ K = \frac 1 2 \alpha A^2 e^{-2\gamma t} \sin^2(\omega t + \varphi) \] @@ -590,7 +590,7 @@ Ovviamente questo vale in determinate approssimazioni, infatti non ha senso che \subsection{Seconda approssimazione} -Poicè abbiamo preso l'andamento oscillatorio dell'energia meccanica vogliamo un approssimazione migliore e più precisa. +Poiché abbiamo preso l'andamento oscillatorio dell'energia meccanica vogliamo un approssimazione migliore e più precisa. Ripartiamo dall'energia: \[ E_m = K + U = \frac 1 2 m \8 \dert{x} \9^2 + \frac 1 2 \alpha x^2 @@ -619,7 +619,7 @@ Dimostriamolo: dW = F_v dx \imp P_{visc} = \dert{W} = F_v * \dert{x} = - 2 \gamma m \8 \dert{x} \9^2 \end{align*} -Scrivamo la formula in modo chiaro: +Scriviamo la formula in modo chiaro: \[ \dert{E_m} = -2 \gamma m \8 \dert{x} \9^2 = - 2 \gamma m A^2 \omega_p^2 e^{-2\gamma t} \sin^2 (\omega t + \varphi) \] @@ -664,7 +664,7 @@ Tutto quello che resta è soltanto l'onda data dalla forzante. \begin{dimJ}{Dimostrazione della soluzione, parte 1}{dim:sol} Introduciamo una funzione complessa \(z(t)\) tale che \(x(t)=\Re[z(t)]\). - Poiché \(\cos\alpha=\Re(e^{i\alpha})\), l’equazione reale equivale alla parte reale di + Poiché \(\cos\alpha=\Re(e^{i\alpha})\), l'equazione reale equivale alla parte reale di \[ \dertt{z} + 2\gamma \dert{z} + \omega_p^2 z = \frac{F_0}{m} e^{i\omega_f t}. \] @@ -694,7 +694,7 @@ Tutto quello che resta è soltanto l'onda data dalla forzante. \dert{z_p} = i\omega_f K e^{i\omega_f t}, \qquad \dertt{z_p} = -\omega_f^2 K e^{i\omega_f t} \] - Sostituendo nell’equazione complessa: + Sostituendo nell'equazione complessa: \[ (-\omega_f^2 K + 2 i\gamma\omega_f K + \omega_p^2 K)e^{i\omega_f t} = \frac{F_0}{m} e^{i\omega_f t} \] @@ -729,13 +729,13 @@ Tutto quello che resta è soltanto l'onda data dalla forzante. \[ x(t)=A e^{-\gamma t}\cos(\omega t+\varphi_0) + B\cos(\omega_f t+\varphi) \] - che soddisfa l’equazione originale. CVD. + che soddisfa l'equazione originale. CVD. \end{dimJ} \subsection{Condizioni Iniziali} -Sfruttiamo i risultati della dimostrazione. +Sfruttiamo i risultati della dimostrazione. Poniamo ora \(D = \omega_p^2-\omega_f^2 + 2 i \gamma \omega_f\) Il modulo del denominatore è @@ -748,22 +748,22 @@ Quindi il modulo di \(K\) è B = \abs{K} = \frac{F_0/m}{\sqrt{(\omega_p^2 - \omega_f^2)^2 + (2\gamma\omega_f)^2}} \] -Scriviamo ora \(K = B e^{i\varphi}\), proprio come nella dimostrazione. -Per determinare l’angolo \(\varphi\) osserviamo che \(K\) è il reciproco (a meno di un fattore reale positivo \(F_0/m\)) del numero complesso +Scriviamo ora \(K = B e^{i\varphi}\), proprio come nella dimostrazione. +Per determinare l'angolo \(\varphi\) osserviamo che \(K\) è il reciproco (a meno di un fattore reale positivo \(F_0/m\)) del numero complesso \[ D = a + ib \qquad\text{con}\quad a = \omega_p^2 - \omega_f^2,\quad b = 2\gamma\omega_f \] -Il motivo per cui compare il reciproco risulta dalla soluzione particolare: si impone la forma \(z_p(t)=K e^{i\omega_f t}\), la si inserisce nell’equazione differenziale e i termini derivati producono un fattore complesso moltiplicativo che è proprio \(D\). -Per soddisfare l’equazione, questo fattore deve essere compensato dividendo per esso, dunque \(K\) è proporzionale a \(1/D\). -In altre parole, il numero complesso \(D\) “pesa” la risposta del sistema, e \(K\) deve contenerne il reciproco per annullarlo. +Il motivo per cui compare il reciproco risulta dalla soluzione particolare: si impone la forma \(z_p(t)=K e^{i\omega_f t}\), la si inserisce nell'equazione differenziale e i termini derivati producono un fattore complesso moltiplicativo che è proprio \(D\). +Per soddisfare l'equazione, questo fattore deve essere compensato dividendo per esso, dunque \(K\) è proporzionale a \(1/D\). +In altre parole, il numero complesso \(D\) "pesa" la risposta del sistema, e \(K\) deve contenerne il reciproco per annullarlo. Il numero \(D = a + ib\) può essere rappresentato come vettore nel piano complesso con una certa lunghezza \(\abs{D}\) e un certo angolo \(\theta\). Questo angolo soddisfa: \[ \tan\theta = \frac{b}{a} \] -perché il rapporto tra parte immaginaria e parte reale è la tangente dell’angolo del vettore. +perché il rapporto tra parte immaginaria e parte reale è la tangente dell'angolo del vettore. Dunque possiamo scrivere \[ @@ -775,7 +775,7 @@ Il reciproco ha la forma \frac{1}{D} = \frac{1}{\abs{D}} e^{-i\theta} \] -Qui si vede immediatamente che il reciproco inverte l’angolo del numero complesso. +Qui si vede immediatamente che il reciproco inverte l'angolo del numero complesso. Otteniamo quindi \[ \varphi = -\theta = -\arctan\!\left(\frac{2\gamma\omega_f}{\omega_p^2-\omega_f^2}\right) @@ -787,15 +787,15 @@ Otteniamo quindi Compresa la natura delle condizioni iniziali possiamo fare una minima discussione di quello che succede per semplici limiti notevoli. \subsubsection*{Caso \(\gamma \sim 0\)} -Quando lo smorzamento è molto piccolo, il termine transitorio decade molto lentamente e il sistema si comporta come un oscillatore quasi conservativo. +Quando lo smorzamento è molto piccolo, il termine transitorio decade molto lentamente e il sistema si comporta come un oscillatore quasi conservativo. In particolare, nel coefficiente di risposta stazionaria \[ B = \frac{F_0/m}{\sqrt{(\omega_p^2 - \omega_f^2)^2 + (2\gamma\omega_f)^2}} \] il termine \(2\gamma\omega_f\) al denominatore diventa trascurabile tranne che in prossimità della risonanza. -Ne segue che, lontano dalla risonanza, la risposta del sistema è molto piccola, mentre in prossimità di \(\omega_f \sim \omega_p\) il denominatore tende a valori molto bassi e l’ampiezza cresce sensibilmente. -Nel limite \(\gamma \to 0\) la risonanza tende a diventare infinitamente “stretta” e il sistema oscilla con ampiezza molto grande quando la frequenza di forzamento coincide con quella naturale. +Ne segue che, lontano dalla risonanza, la risposta del sistema è molto piccola, mentre in prossimità di \(\omega_f \sim \omega_p\) il denominatore tende a valori molto bassi e l'ampiezza cresce sensibilmente. +Nel limite \(\gamma \to 0\) la risonanza tende a diventare infinitamente "stretta" e il sistema oscilla con ampiezza molto grande quando la frequenza di forzamento coincide con quella naturale. \subsubsection*{Caso \(\omega_f \sim \omega_p\)} Quando la frequenza di forzamento si avvicina alla frequenza naturale, il termine @@ -806,8 +806,8 @@ nel denominatore della risposta si riduce drasticamente. Il valore di \(B\) dive \[ (2\gamma\omega_f)^2 \] -e ciò porta a un picco di ampiezza attorno alla risonanza. -Per \(\gamma\) piccolo, questo picco è particolarmente accentuato e la curva della risposta assume la tipica forma risonante molto appuntita. +e ciò porta a un picco di ampiezza attorno alla risonanza. +Per \(\gamma\) piccolo, questo picco è particolarmente accentuato e la curva della risposta assume la tipica forma risonante molto appuntita. In sintesi, per \(\omega_f \sim \omega_p\) la risposta del sistema è grande e dominata quasi completamente dal regime forzato, mentre il transitorio diventa trascurabile dopo tempi brevi. \subsubsection*{Comportamento di \(\varphi\) per \(\omega_f \sim \omega_p\)} @@ -816,15 +816,15 @@ La fase si ottiene da \varphi = - \arctan\!\left(\frac{2\gamma\omega_f}{\omega_p^2 - \omega_f^2}\right) \] -Quando \(\omega_f\) si avvicina a \(\omega_p\), il termine \(\omega_p^2 - \omega_f^2\) tende a zero e il rapporto nella tangente diventa molto grande. -Ciò implica che l’angolo tende a +Quando \(\omega_f\) si avvicina a \(\omega_p\), il termine \(\omega_p^2 - \omega_f^2\) tende a zero e il rapporto nella tangente diventa molto grande. +Ciò implica che l'angolo tende a \[ \varphi \to -\frac{\pi}{2} \] -poiché la tangente cresce senza limite e l’arctangente tende a \(\pm \pi/2\) a seconda del segno. -Nel caso tipico con \(\gamma > 0\) e \(\omega_f\) crescente verso \(\omega_p\), l’argomento è positivo e dunque la fase risulta negativa e vicina a \(-\pi/2\). +poiché la tangente cresce senza limite e l'arcotangente tende a \(\pm \pi/2\) a seconda del segno. +Nel caso tipico con \(\gamma > 0\) e \(\omega_f\) crescente verso \(\omega_p\), l'argomento è positivo e dunque la fase risulta negativa e vicina a \(-\pi/2\). -Fisicamente questo significa che, in prossimità della risonanza, l’oscillazione imposta dal forzamento è in ritardo di un quarto di periodo rispetto alla forza esterna, comportamento caratteristico degli oscillatori debolmente smorzati. +Fisicamente questo significa che, in prossimità della risonanza, l'oscillazione imposta dal forzamento è in ritardo di un quarto di periodo rispetto alla forza esterna, comportamento caratteristico degli oscillatori debolmente smorzati. Si dice che oscillazione e forzamento sono in quadratura di fase. @@ -840,7 +840,7 @@ Come sempre sappiamo: \[ E_m = K + U \qquad K = \frac 1 2 m \8\dert{x} \9^2 \quad - U = \frac 1 2 \alpha x^2 + U = \frac 1 2 \alpha x^2 \] Facciamo la derivata della soluzione: @@ -850,8 +850,8 @@ Facciamo la derivata della soluzione: Sostituendo otteniamo: \[ - K = \frac 1 2 m B^2 \omega^2 \sin^2(\omega t + \varphi) \quad - U = \frac 1 2 \alpha B^2 \cos^2(\omega t + \varphi) + K = \frac 1 2 m B^2 \omega^2 \sin^2(\omega t + \varphi) \quad + U = \frac 1 2 \alpha B^2 \cos^2(\omega t + \varphi) \] Ricordiamo che \(\omega_p^2 = \frac \alpha m \imp \alpha = \omega_p^2 m\). @@ -885,7 +885,7 @@ L'oscillatore è quindi un moto perpetuo, ma questo non va contro le leggi della \section{Sistemi a più gradi di libertà} Partiamo da un modello molto semplice: 2 pendoli legati da una molla. I 2 pendoli sarebbero totalmente indipendenti se non fosse per la molla. -Facciamo un po' di nomencatura: +Facciamo un po' di nomenclatura: \begin{itemize} \item $m$: massa delle molle \item $K$: costante elastica della molla @@ -897,16 +897,16 @@ Facciamo un po' di nomencatura: Possiamo scrivere: \[ F_1 = - \frac{mg}{l} x_1 - K(x_1 - x_2) \qquad - F_2 = - \frac{mg}{l} x_2 - K(x_2 - x_1) + F_2 = - \frac{mg}{l} x_2 - K(x_2 - x_1) \] Scriviamo le equazioni del moto per le 2 masse: \[ m \dertt{x_1} = - \frac{mg}{l} x_1 - K(x_1 - x_2) \qquad - m \dertt{x_2} = - \frac{mg}{l} x_2 - K(x_2 - x_1) + m \dertt{x_2} = - \frac{mg}{l} x_2 - K(x_2 - x_1) \] -Sommiamo e sotraiamo le 2 equazioni: +Sommiamo e sottraiamo le 2 equazioni: \begin{align*} m \8 \dertt{x_1} + \dertt{x_2} \9 = - \frac{mg}{l} (x_1 + x_2)\\ m \8 \dertt{x_1} - \dertt{x_2} \9 = - \frac{mg}{l} (x_1 - x_2) - 2 K (x_1 - x_2) @@ -928,7 +928,7 @@ Queste sono equazioni sono indipendenti e sappiamo come risolverle. Le soluzioni sono quindi: \[ u_a = A_a \cos(\omega_a t + \varphi_a) \qquad - u_b = A_b \cos(\omega_b t + \varphi_b) + u_b = A_b \cos(\omega_b t + \varphi_b) \] Ora noi dobbiamo tornare indietro a $x_1$, $x_2$. @@ -940,7 +940,7 @@ Ora noi dobbiamo tornare indietro a $x_1$, $x_2$. Abbiamo completato lo studio del sistema. Se osserviamo i moti dei pendoli questi sono la sovrapposizione di 2 moti semplici. Nel modo $u_a$ le 2 masse si muovono insieme e la molla non fa nulla, mentre nel modo $u_b$ i pendoli si contraggono e si aprono in modo simmetrico. -Il nostro moto complesso diventa facile da studiare se pensato come la sovrapposizione di 2 moti disitinti e indipendenti. +Il nostro moto complesso diventa facile da studiare se pensato come la sovrapposizione di 2 moti distinti e indipendenti. Questi sono chiamati MODI NORMALI, nel caso specifico informalmente sono chiamati modo a pendolo e modo a respiro. @@ -952,8 +952,8 @@ Pendiamo l'esempio precedente e caliamoci nel caso specifico dove il pendolo par x_2(0) = 0 \] -Se noi vogliamo studiare quello che stà succdedendo dobbiamo scomporre la nostra condizione iniziale nei 2 modi semplici. -Facciamo una prova ragionata: prendendo il modo a pendolo e mettendolo a $-A$ avremmo che +Se noi vogliamo studiare quello che sta succedendo dobbiamo scomporre la nostra condizione iniziale nei 2 modi semplici. +Facciamo una prova ragionata: prendendo il modo a pendolo e mettendolo a $-A$ avremmo che \[ x_1(0) = -A \qquad x_2(0) = -A @@ -977,11 +977,11 @@ Abbiamo quindi scomposto il caso iniziale nei suoi 2 modi normali. \omega_b = \sqrt{\frac g l + 2 \frac k m} \] -Se osserviamo il moto della massa 1 $x_1$ ci accorigamo che è la somma di 2 oscillazioni simili ma non uguali. +Se osserviamo il moto della massa 1 $x_1$ ci accorgiamo che è la somma di 2 oscillazioni simili ma non uguali. L'effetto prodotto è quello di un oscillazione che ogni tanto interferisce in modo costruttivo ed ogni tanto in modo distruttivo. Questo fenomeno si chiamano BATTIMENTI. \[ - x_1(t) = C_a \cos(\omega_a t + \varphi_a) + C_b \cos(\omega_b t + \varphi_b) + x_1(t) = C_a \cos(\omega_a t + \varphi_a) + C_b \cos(\omega_b t + \varphi_b) \] Nel caso specifico: $\varphi_a = \varphi_b = 0$, $C_a=C_b$: @@ -992,10 +992,11 @@ Nel caso specifico: $\varphi_a = \varphi_b = 0$, $C_a=C_b$: Usiamo le formule di Prostaferesi: \[ \cos\alpha + \cos\beta = 2 \cos\!\left(\frac{\alpha + \beta}{2}\right) \cos\!\left(\frac{\alpha - \beta}{2}\right) \qquad - \alpha = \omega_a t \quad \beta = \omega_b t + \alpha = \omega_a t \quad + \beta = \omega_b t \] -Definiamo 2 variabili con nomi che sembrano casuali, ma diventerranno chiari: +Definiamo 2 variabili con nomi che sembrano casuali, ma diventeranno chiari: \[ \frac{\alpha + \beta}{2} = \frac{\omega_a + \omega_b}{2} t = \omega_M t \qquad \frac{\alpha - \beta}{2} = \frac{\omega_a - \omega_b}{2} t = \omega_{inv} t @@ -1006,8 +1007,1709 @@ Ne risulta che: x_1(t) = 2 \cos(\omega_M t) \cos(\omega_{inv} t) \] -Risuta chiaro che $\omega_{inv} << \omega_M$. +Risulta chiaro che $\omega_{inv} << \omega_M$. Se noi grafichiamo questo andamento ci accorgeremmo che $\omega_{inv}$ rappresenta l'inviluppo della curva, mentre $\omega_M$ le oscillazioni più piccole che avvengono nel "range" definito dall'inviluppo. + + +\section{Due masse collegate da tre molle} +Consideriamo ora un sistema più semplice rispetto ai pendoli sempre con 2 gradi di libertà. +Abbiamo due masse identiche collegate da tre molle tutte uguali tra loro. +Il moto avviene nel piano e assumiamo la condizione di piccoli spostamenti verticali rispetto alla distanza di riposo: +\[ + \frac{y_i}{a} \ll 1 +\] + +Definiamo: +\begin{itemize} + \item $m$: massa delle due masse + \item $K$: costante elastica delle tre molle + \item $a$: distanza di riposo tra gli attacchi delle molle + \item $y_1$: spostamento verticale della massa 1 + \item $y_2$: spostamento verticale della massa 2 +\end{itemize} + +Nel limite degli angoli piccoli valgono le approssimazioni: +\[ + \sin\theta \simeq \theta \simeq \tan\theta +\] + +Tutte le componenti orizzontali delle forze risultano di ordine superiore e quindi trascurabili. +La tensione nelle molle in equilibrio vale $T_0 = Ka$. +Quando il sistema viene deformato, le componenti verticali delle tensioni generano le forze risultanti. + +\subsection{Forze sulle masse} +Per la massa 1 le due molle (una verso la parete e una verso la massa 2) danno contributi: +\[ + F_{1y} = - T_0 \frac{y_1}{a} - T_0 \frac{y_1 - y_2}{a} +\] + +Per la massa 2 otteniamo: +\[ + F_{2y} = - T_0 \frac{y_2}{a} - T_0 \frac{y_2 - y_1}{a} +\] + +Scriviamo le equazioni del moto: +\begin{align*} + m \dertt{y_1} &= - 2 \frac{T_0}{a} y_1 + \frac{T_0}{a} y_2 \\ + m \dertt{y_2} &= - 2 \frac{T_0}{a} y_2 + \frac{T_0}{a} y_1 +\end{align*} + +Queste equazioni sono accoppiate come nel caso dei pendoli. + +\subsection{Combinazioni simmetrica e antisimmetrica} +Sommiamo e sottraiamo le due equazioni: +\[ + \dertt{(y_1 + y_2)} = - \frac{T_0}{ma} (y_1 + y_2) +\] +\[ + \dertt{(y_1 - y_2)} = - 3 \frac{T_0}{ma} (y_1 - y_2) +\] + +Introduciamo le variabili: +\[ + u_a = y_1 + y_2 \qquad + u_b = y_1 - y_2 +\] + +Otteniamo due oscillatori disaccoppiati: +\begin{align*} + \dertt{u_a} + \omega_a^2 u_a &= 0 \qquad \omega_a^2 = \frac{T_0}{ma} \\ + \dertt{u_b} + \omega_b^2 u_b &= 0 \qquad \omega_b^2 = 3 \frac{T_0}{ma} +\end{align*} + +Le soluzioni generali sono: +\[ + u_a = A_a \cos(\omega_a t + \varphi_a) \qquad + u_b = A_b \cos(\omega_b t + \varphi_b) +\] + +Riscriviamo $y_1$ e $y_2$: +\begin{align*} + y_1(t) &= \frac{u_a + u_b}{2} \\ + y_2(t) &= \frac{u_a - u_b}{2} +\end{align*} + +\subsection{Interpretazione fisica dei modi} +Come nel caso dei pendoli, i modi normali hanno una chiara interpretazione: + +\paragraph{Modo simmetrico $u_a$} +\[ + y_1 = y_2 +\] +Le masse oscillano insieme. +La molla centrale non si deforma e non modifica la tensione. + +\paragraph{Modo antisimmetrico $u_b$} +\[ + y_1 = - y_2 +\] +Le masse oscillano in opposizione di fase e la molla centrale si allunga e si accorcia durante il moto. +La frequenza è maggiore perché intervengono tre molle invece di una sola. + +\subsection{Generalizzazione e limite continuo} +Un sistema con $N$ gradi di libertà possiede esattamente $N$ modi normali. +Ogni configurazione del moto può essere scritta come combinazione lineare dei modi. + +Aumentando il numero delle masse: +\begin{itemize} + \item il primo modo è sempre quello a frequenza più bassa e con tutte le masse in fase + \item gli altri modi presentano un numero crescente di inversioni di fase +\end{itemize} + +Nel limite $N \to \infty$ otteniamo una corda continua con infiniti modi normali: +\[ + \text{modo fondamentale}, \quad \text{secondo modo}, \quad \text{terzo modo}, \dots +\] +Ogni oscillazione possibile della corda può essere espressa come combinazione lineare dei suoi modi normali. +Questo principio è alla base della decomposizione di Fourier e della descrizione ondulatoria dei sistemi con infiniti gradi di libertà. +\begin{comment} + +\section{Dai sistemi discreti al continuo} +Vogliamo ora fare un passo ulteriore: passare dal caso discreto ad un sistema continuo. +Per farlo consideriamo una catena di masse uguali collegate da molle identiche, una ''catena'' di masse e molle, che può essere pensata come un modello di una corda elastica. + +Riprendiamo la configurazione di riferimento. +La massa generica la indichiamo con indice $i$, mentre le due masse adiacenti si trovano nelle posizioni $i-1$ e $i+1$. +Indichiamo con $y_i(t)$ lo spostamento della massa $i$ lungo la direzione verticale. + +Le forze elastiche che agiscono sulla massa $i$ dipendono solo dagli allungamenti delle due molle adiacenti. +Possiamo quindi scrivere la forza risultante: +\[ + F_i = - T_0 \8 y_i - y_{i-1} \9 - T_0 \8 y_i - y_{i+1} \9 +\] + +$T_0$ rappresenta la costante elastica equivalente delle molle. +Raccogliendo i termini: +\[ + F_i = - T_0 \8 - y_{i-1} + 2 y_i - y_{i+1} \9 +\] + +Applicando la seconda legge di Newton otteniamo l'equazione del moto per la massa $i$: +\[ + m \dertt{y_i} = - T_0 \8 - y_{i-1} + 2 y_i - y_{i+1} \9 +\] + +Questa è una relazione ricorsiva che vale per qualsiasi massa della catena. + +\subsection{Passaggio al continuo} +Ora immaginiamo che le masse siano sempre più vicine tra loro, separate da una distanza $a$. +Definiamo la variabile spaziale continua $x$ tale che: +\[ + y_i(t) \to y(x,t) \qquad + y_{i-1}(t) \to y(x - a, t) \qquad + y_{i+1}(t) \to y(x + a, t) +\] + +L'equazione del moto diventa quindi: +\[ + \dertt{y(x,t)} = - \frac{T_0}{m} \8 - y(x - a,t) + 2 y(x,t) - y(x + a,t) \9 +\] + +A questo punto sfruttiamo che $a$ è molto piccolo rispetto alla lunghezza d'onda delle oscillazioni che ci interessano. +Possiamo quindi espandere $y(x \pm a, t)$ in serie di Taylor: + +\begin{align*} + y(x + a,t) &= y(x,t) + \derx{y}(x,t)\, a + \frac12 \derxx{y}(x,t)\, a^2\\ + y(x - a,t) &= y(x,t) - \derx{y}(x,t)\, a + \frac12 \derxx{y}(x,t)\, a^2 +\end{align*} + +Sostituendo nell'equazione del moto e osservando che i termini lineari in $a$ si cancellano: +\[ + - y(x-a,t) + 2y(x,t) - y(x+a,t) = - \derxx{y}(x,t)\, a^2 +\] + +L'equazione del moto diventa: +\[ + \dertt{y} = \frac{T_0}{m} a^2 \derxx{y} +\] + +Introduciamo ora la densità lineare di massa: +\[ + \rho_l = \frac{m}{a} +\] + +Da cui: +\[ + \frac{T_0}{m} a^2 = \frac{T_0}{\rho_l} +\] + +Otteniamo così la forma finale dell'equazione: +\[ + \dertt{y(x,t)} = \frac{T_0}{\rho_l} \derxx{y(x,t)} +\] + +Questa è l' \textbf{equazione d'onda unidimensionale}, chiamata anche equazione di d'Alembert. + +\[ + \dertt{y} = v^2 \derxx{y} \qquad + v = \sqrt{\frac{T_0}{\rho_l}} +\] + +La quantità $v$ rappresenta la velocità di propagazione delle onde lungo la corda. +Questa equazione è fondamentale perché descrive: + +\begin{itemize} + \item le onde meccaniche sulle corde + \item le onde sonore in una dimensione + \item la propagazione delle onde elettromagnetiche nel vuoto (in forma vettoriale) + \item le onde gravitazionali nel limite di campo debole +\end{itemize} + +La soluzione generale di questa equazione è una funzione che si propaga senza deformarsi: +\[ + y(x,t) = f(x - vt) + g(x + vt) +\] + +Questo rappresenta 2 onde che si muovono in direzioni opposte con velocità $v$. +\end{comment} + + +\section{Derivazione continua dell'equazione d'onda} +Nelle lezioni precedenti abbiamo visto come, aumentando il numero di gradi di libertà, aumenti anche il numero dei modi normali. +Nel caso di due pendoli collegati da una molla avevamo due modi: il modo a pendolo e il modo a respiro. +Ora vogliamo capire che cosa accade quando portiamo il numero di gradi di libertà a valori molto grandi fino ad avvicinarci ad un sistema continuo. + +Per motivarci partiamo dal modello già affrontato: masse collegate da molle. +Generalizziamo la configurazione precedente considerando non soltanto due masse collegate, ma una sequenza molto lunga di masse tutte uguali collegate fra loro da molle identiche. +Pensiamo a una "collana di perle" infilate su un elastico. + +\subsection{Ipotesi fisiche fondamentali} +\begin{itemize} + \item \textbf{Corda perfettamente flessibile}: non oppone resistenza alla curvatura, non esiste rigidità intrinseca. + \item \textbf{Tensione uniforme}: la tensione interna è ovunque uguale a $T_0$ e non cambia durante l'oscillazione. + \item \textbf{Oscillazioni piccole}: + \[ + \abs{\derx{\psi}} \ll 1 + \] + Questa condizione implica che gli angoli $\alpha$ e $\beta$ formati dall'elemento di corda con l'orizzontale siano piccoli. + \item \textbf{Inestensibilità al primo ordine}: la lunghezza reale di un elemento $d\ell$ coincide con $dx$ al primo ordine. + Infatti: + \[ + d\ell = dx \sqrt{1 + \8 \derx{\psi} \9^2} \simeq dx + \] + Il termine di ordine secondo $\8\derx{\psi}\9^2$ è trascurabile. + \item \textbf{Gravità trascurabile}: la tensione è molto maggiore del peso per unità di lunghezza, così la componente verticale della tensione compensa l'eventuale contributo gravitazionale. +\end{itemize} + +Queste ipotesi definiscono il regime di validità dell'equazione d'onda per una corda. + +\subsection*{Forza sulla massa generica} +Indichiamo con $y_i(t)$ lo spostamento verticale della massa $i$. +Le masse adiacenti esercitano forze elastiche pari alla costante elastica $T_0$ moltiplicata per l'allungamento relativo. +Nel caso iniziale la massa 1 sentiva: +\[ + F_1 = - T_0 \8 y_1 - y_0 \9 - T_0 \8 y_1 - y_2 \9 +\] + +$y_0$ è lo spostamento della massa alla sua sinistra e $y_2$ quello della massa a destra. + +Generalizzando alla massa $i$: +\[ + F_i = - T_0 \8 y_i - y_{i-1} \9 - T_0 \8 y_i - y_{i+1} \9 +\] + +Raccogliendo: +\[ + F_i = - T_0 \8 - y_{i-1} + 2 y_i - y_{i+1} \9 +\] + +Applicando Newton: +\[ + m \dertt{y_i} = - T_0 \8 - y_{i-1} + 2 y_i - y_{i+1} \9 +\] + +Questa equazione è valida per quasi qualsiasi massa della catena ed è la forma discreta dell'equazione del moto. + +\subsection{Passaggio alla descrizione spaziale continua} +Introduciamo la distanza costante $a$ tra una massa e la successiva. +Se la catena è molto fitta possiamo pensare che la coordinata della massa $i$ sia: +\[ + x = i a +\] + +In questo modo: +\[ + y_i(t) \to y(x,t) \qquad + y_{i-1}(t) \to y(x - a, t) \qquad + y_{i+1}(t) \to y(x + a, t) +\] + +Sostituendo: +\[ + \dertt{y(x,t)} = - \frac{T_0}{m} \8 - y(x - a, t) + 2 y(x,t) - y(x + a, t) \9 +\] + +Fino a qui non è stata fatta alcuna approssimazione: questa scrittura è ancora esatta per qualsiasi valore di $a$. + +\subsection{Espansione di Taylor per a piccolo} +Ora usiamo l'ipotesi fondamentale del passaggio al continuo: la separazione $a$ tra le masse è molto più piccola della lunghezza d'onda delle oscillazioni che vogliamo descrivere. + +Espandiamo attorno a $x$: +\[ + y(x + a,t) = y(x,t) + \derx{y}(x,t)\, a + \frac12 \derxx{y}(x,t)\, a^2 +\] + +\[ + y(x - a,t) = y(x,t) - \derx{y}(x,t)\, a + \frac12 \derxx{y}(x,t)\, a^2 +\] + +Sostituiamo nella combinazione dell'equazione discreta: +\[ + - y(x-a,t) + 2 y(x,t) - y(x+a,t) +\] + +Sostituendo esplicitamente: +\begin{align*} + - y(x-a,t) &= - y(x,t) + \derx{y}(x,t) a - \frac12 \derxx{y}(x,t) a^2 \\ + 2 y(x,t) &= 2 y(x,t) \\ + - y(x+a,t) &= - y(x,t) - \derx{y}(x,t) a - \frac12 \derxx{y}(x,t) a^2 +\end{align*} + +Sommiamo tutto con estrema attenzione: +\[ + - y(x,t) + \derx{y}\, a - \frac12 \derxx{y}\, a^2 + 2 y(x,t) - y(x,t) - \derx{y} a - \frac12 \derxx{y}\, a^2 +\] + +Osserviamo: +- i termini $y(x,t)$ si cancellano completamente +- i termini lineari $\derx{y} a$ si cancellano +- rimangono solo i termini quadratici: +\[ + - \derxx{y}(x,t) a^2 +\] + +Dunque: +\[ + - y(x-a,t) + 2 y(x,t) - y(x+a,t) = - a^2 \derxx{y}(x,t) +\] + +Sostituendo nell'equazione del moto: +\[ + \dertt{y(x,t)} = \frac{T_0}{m} a^2 \derxx{y(x,t)} +\] + +\subsection{Introduzione della densità lineare} +Siccome il sistema si sta avvicinando a un continuo, introduciamo la densità lineare di massa: +\[ + \rho_l = \frac{m}{a} +\] + +Da cui: +\[ + \frac{T_0}{m} a^2 = \frac{T_0}{\rho_l} +\] + +L'equazione diventa: +\[ + \dertt{y(x,t)} = \frac{T_0}{\rho_l} \derxx{y(x,t)} +\] + +Questa è la celeberrima \textbf{equazione d'onda unidimensionale}, o equazione di d'Alembert: +\[ + \dertt{y} = v^2 \derxx{y} \qquad + v = \sqrt{\frac{T_0}{\rho_l}} +\] + +\subsection{Significato fisico} +L'equazione mostra che la forza che agisce su un punto della corda è proporzionale alla sua \textbf{curvatura} $\derxx{\psi}$: + +- maggiore curvatura $\Rightarrow$ maggiore forza di richiamo + +- il moto risultante è una perturbazione che si propaga con velocità costante $v$ + +La soluzione generale è la somma di due onde che viaggiano in direzioni opposte: +\[ + \psi(x,t) = f(x - vt) + g(x + vt) +\] + +dove $f$ e $g$ sono determinate dalle condizioni iniziali. + + + +\section{Struttura dell'equazione d'onda} +Ora che abbiamo ottenuto l'equazione di d'Alembert per una corda vibrante, vogliamo capire in modo più profondo che cosa essa stia descrivendo. +Ricordiamo la forma generale: +\[ + \dertt{\psi(x,t)} = \frac{T_0}{\rho_l} \derxx{\psi(x,t)} +\] + +Questa è una \textbf{equazione differenziale alle derivate parziali} del secondo ordine. +La nostra incognita non è più una funzione di una sola variabile, come nello studio dell'oscillatore armonico, ma una funzione di due variabili indipendenti: lo spazio $x$ e il tempo $t$. + +\subsection{Significato delle derivate parziali} +In una funzione $\psi(x,t)$ le derivate parziali vengono effettuate tenendo costante una delle due variabili. +Per comprenderne il significato: +\begin{itemize} + \item $\derx{\psi}$ è la pendenza del profilo della corda fissato un certo istante $t$ + \item $\dert{\psi}$ è la velocità di uno specifico punto della corda, fissata una certa posizione $x$ + \item $\derxx{\psi}$ misura la \textbf{curvatura} della corda nel punto $x$ + \item $\dertt{\psi}$ è l'accelerazione del punto $x$ della corda +\end{itemize} + +È dunque un'equazione che collega accelerazione e curvatura. +Dal punto di vista fisico questo è esattamente ciò che ci si aspetta: quando il profilo della corda è incurvato, la tensione interna produce una forza che riporta il sistema verso la forma rettilinea. + +\subsection{Proprietà dell'equazione} +Osserviamo alcune caratteristiche fondamentali: +\begin{itemize} + \item \textbf{Equazione lineare}: compaiono solo derivate lineari della funzione incognita. + Ne segue immediatamente il \textbf{principio di sovrapposizione}: se $\psi_1$ e $\psi_2$ + sono soluzioni, qualunque combinazione lineare $a \psi_1 + b \psi_2$ è ancora una soluzione. + Questo sarà essenziale quando introdurremo i modi normali della corda. + \item \textbf{Equazione omogenea}: non è presente alcun termine dipendente solo da $x$ e $t$. + Questo significa che la corda non è soggetta a forzanti esterne durante l'evoluzione. + \item \textbf{Equazione del secondo ordine}: compaiono derivate seconde sia rispetto allo spazio + sia rispetto al tempo. Le condizioni iniziali e al contorno dovranno essere fornite di + conseguenza. +\end{itemize} + +\subsection{Interpretazione fisica: il ruolo della curvatura} +Riscriviamo l'equazione in forma più compatta introducendo il parametro: +\[ + v = \sqrt{\frac{T_0}{\rho_l}} +\] + +Abbiamo ricavato come velocità caratteristica del mezzo. +Allora: +\[ + \dertt{\psi} = v^2 \derxx{\psi} +\] + +La parte sinistra rappresenta l'accelerazione verticale del punto di corda in $x$. +La parte destra rappresenta la curvatura locale del profilo, moltiplicata per la quantità $v^2$ che ha le dimensioni di una velocità al quadrato. + +Una curvatura positiva implica un'accelerazione che riporta la corda verso la posizione di equilibrio, proprio come accadrebbe per un oscillatore armonico. +Questo non è sorprendente: abbiamo visto esplicitamente che la corda continua può essere pensata come il limite di un sistema di infiniti oscillatori armonici accoppiati. +L'equazione d'onda è quindi la loro sintesi. + +\subsection{Analisi dimensionale} +Per comprendere meglio il significato del parametro $v$, osserviamo le sue dimensioni: +\[ + [T_0] = \text{forza} = \text{kg m s}^{-2} \qquad + [\rho_l] = \text{kg m}^{-1} +\] + +quindi: +\[ + \frac{T_0}{\rho_l} = \frac{\text{kg m s}^{-2}}{\text{kg m}^{-1}} = \text{m}^2 \text{s}^{-2} +\] + +ovvero: +\[ + \sqrt{\frac{T_0}{\rho_l}} = \text{velocità} +\] + +Questo conferma che $v$ rappresenta la velocità con cui una perturbazione si propaga lungo la corda. +Si tratta dunque della \textbf{velocità di propagazione dell'onda meccanica} in quel mezzo. + +\subsection{Spazio e tempo non sono indipendenti nella soluzione} +L'equazione d'onda impone un vincolo molto forte sulla forma delle soluzioni: non è possibile combinare $x$ e $t$ in modo arbitrario. +Affinché la relazione +\[ + \dertt{\psi} = v^2 \derxx{\psi} +\] + +sia soddisfatta, la funzione $\psi$ deve dipendere da $x$ e $t$ attraverso combinazioni del tipo: +\[ + x - v t \qquad x + v t +\] + +Lo vedremo in modo rigoroso quando cercheremo la soluzione generale. +Per ora è importante intuire che la soluzione ha la forma di una perturbazione che si \textbf{sposta rigidamente} lungo l'asse $x$, con velocità costante $v$. + +\subsection{Condizioni iniziali e condizioni al contorno} +Poiché l'equazione è del secondo ordine in entrambe le variabili, occorrono: +\begin{itemize} + \item \textbf{due condizioni iniziali} per ogni punto dello spazio: + \[ + \psi(x,0) \qquad \dot\psi(x,0) + \] + che specificano la forma iniziale della corda e la velocità iniziale di ogni suo punto + \item \textbf{due condizioni al contorno} per determinare completamente la soluzione nel caso + di una corda di lunghezza finita +\end{itemize} + +Il caso più importante, e quello che studieremo, è quello della corda con estremi fissi: +\[ + \psi(0,t) = 0 \qquad \psi(L,t) = 0 +\] + +che darà luogo all'emergere di modi normali perfettamente analogo al caso discreto. + +\subsection{La costante v come legame con i modi normali} +Il parametro $v^2$ determina il legame tra accelerazione e curvatura. +Nel caso discreto avevamo: +\[ + m \dertt{x_i} = - k \8 x_i - x_{i-1} \9 - k \8 x_i - x_{i+1} \9 +\] + +Nel limite continuo, è divenuto proprio l'equazione d'onda. +Ci aspettiamo quindi che le soluzioni siano combinazioni di modi normali, ciascuno con la propria frequenza, e che tali modi costituiscano una base completa delle soluzioni. + + + +\section{La velocità di propagazione e il legame con i modi normali} +Riprendiamo il ragionamento interrotto: abbiamo osservato che il coefficiente che lega $\dertt{\psi}$ a $\derxx{\psi}$, cioè il rapporto $\frac{T_0}{\rho_l}$, possiede le dimensioni di una velocità al quadrato. +Definendo: +\[ + v = \sqrt{\frac{T_0}{\rho_l}} +\] + +Otteniamo l'equazione d'onda nella forma compatta: +\[ + \dertt{\psi} = v^2 \derxx{\psi} +\] + +Ora vogliamo capire che ruolo giochi questa quantità $v$. +Per farlo torniamo per un attimo al caso degli oscillatori armonici accoppiati. + +\subsection{Dalla catena discreta al continuo: interpretazione unitaria} +Nel caso discreto, per una catena di masse collegate da molle, avevamo equazioni del moto del tipo: +\[ + m \dertt{y_i} = - k \8 2 y_i - y_{i-1} - y_{i+1} \9 +\] + +Abbiamo visto che le soluzioni generali erano combinazioni di modi normali, ciascuno con una propria frequenza caratteristica, determinata dalla struttura della catena. +Ogni modo era un'oscillazione collettiva in cui tutte le masse oscillavano alla stessa frequenza, pur muovendosi con ampiezze relative diverse. + +Nel limite in cui rendiamo la distanza tra le masse sempre più piccola e il numero delle masse tende all'infinito, questo insieme di modi discreti diventa un insieme continuo di modi. +Il passaggio: +\[ + y_{i \pm 1} \to y(x \pm a) \qquad + a \to 0 +\] + +Ci ha portati, attraverso l'espansione di Taylor, all'equazione d'onda. +L'operatore discreto +\[ + y_{i+1} - 2 y_i + y_{i-1} +\] + +si è trasformato nella curvatura $\derxx{\psi}$. + +Questo ci dice che la struttura profonda dell'equazione d'onda è la stessa del sistema di oscillatori accoppiati: la corda continua non è altro che un numero infinito di oscillatori armonici accoppiati infinitamente vicini. + +\subsection{Trama fisica dell'equazione d'onda} +Adesso possiamo interpretare in modo intuitivo la forma dell'equazione: +\[ + \dertt{\psi} = v^2 \derxx{\psi} +\] + +La derivata seconda spaziale misura quanto la corda è incurvata. +Se la corda è localmente curva, allora la tensione produce una forza di richiamo diretta verso la posizione rettilinea. +L'accelerazione risultante è esattamente $\dertt{\psi}$. + +Il fatto che la costante di proporzionalità sia $v^2$ indica che la risposta dinamica della corda dipende dalle sue proprietà: +\begin{itemize} + \item la tensione $T_0$ stabilisce quanto la corda ''tira'' per tornare in equilibrio + \item la densità lineare $\rho_l$ stabilisce quanta massa viene accelerata +\end{itemize} +Una tensione maggiore produce onde più veloci, una densità lineare maggiore le rallenta. + +\subsection{La natura propagativa delle soluzioni} +Una conseguenza straordinaria dell'equazione d'onda è che spazio e tempo non possono entrare nella soluzione in modo arbitrario. +La funzione $\psi$ deve dipendere da $x$ e $t$ attraverso combinazioni del tipo: +\[ + x - v t \qquad x + v t +\] + +e questo non è un caso: sono proprio questi due argomenti a rappresentare onde che si propagano senza deformarsi verso destra e verso sinistra. + +La forma più generale delle soluzioni è infatti: +\[ + \psi(x,t) = f(x - v t) + g(x + v t) +\] + +dove $f$ e $g$ sono funzioni arbitrarie determinate dalla configurazione iniziale della corda. + +Questo risultato può sembrare sorprendente, ma segue direttamente dal fatto che: +\[ + \dertt{\psi} = v^2 \derxx{\psi} +\] + +impone che ogni ''pezzo'' della perturbazione si muova con velocità costante $v$, esattamente come se l'intero profilo della corda fosse trascinato rigidamente lungo l'asse $x$. + +\subsection{Il ruolo delle condizioni iniziali e delle condizioni al contorno} +A differenza dello oscillatore armonico, qui abbiamo una funzione di due variabili. +Per determinare completamente la soluzione servono: +\begin{itemize} + \item \textbf{due condizioni iniziali} + \[ + \psi(x,0) \qquad \dert{\psi}(x,0) + \] + che specificano forma e velocità della corda all'istante iniziale + \item \textbf{due condizioni al contorno}, necessarie quando la corda ha lunghezza finita +\end{itemize} + +Il caso più importante, e fisicamente più rilevante, è quello degli estremi fissi: +\[ + \psi(0,t) = 0 \qquad \psi(L,t) = 0 +\] + +Questo vincolo modifica radicalmente l'insieme delle soluzioni possibili: solo alcune funzioni $f$ e $g$ saranno ammissibili, e vedremo che ciò conduce alla quantizzazione dei modi normali, analogamente a quella vista per due masse accoppiate, ma con un numero infinito di modi. + +\subsection{Verso la ricerca dei modi normali per la corda} +Ora che abbiamo capito la struttura matematica dell'equazione d'onda e la natura delle sue soluzioni propagative, possiamo affrontare il passo successivo: trovare le oscillazioni proprie della corda, cioè i \textbf{modi normali}. + +Come nel caso degli oscillatori accoppiati, ogni modo normale avrà una frequenza propria e rappresenterà un'oscillazione ''pura''. +La soluzione generale sarà la somma di tutti questi modi. + +Nella prossima sezione introdurremo il metodo della separazione delle variabili, che ci permetterà di identificare esplicitamente tali modi. + + + +\subsection{Condizioni iniziali e completezza dei modi normali} +Abbiamo ottenuto che i modi normali della corda lunga $l$ vincolata agli estremi sono descritti da: +\[ + \psi_n(x,t) = A_n \sin(k_n x) \cos(\omega_n t + \varphi_n) +\] + +dove: +\[ + k_n = \frac{n \pi}{l} \qquad + \omega_n = v k_n = v \frac{n \pi}{l} +\] +e $v = \sqrt{\frac{T_0}{\rho_l}}$ è la velocità delle onde sulla corda. + +A questo punto dobbiamo capire come descrivere una qualunque oscillazione della corda che non sia un singolo modo normale. +Nel caso con due gradi di libertà avevamo: +\[ + x_1(t) = c_a u_a(t) + c_b u_b(t) +\] +dove $u_a$ e $u_b$ erano i due modi normali. + +Ora, nel continuo, il discorso si generalizza a: +\[ + y(x,t) = \sum_{n=1}^{\infty} A_n \sin(k_n x) \cos(\omega_n t + \varphi_n) +\] + +Questa è la soluzione generale della corda vibrante vincolata agli estremi. +È del tutto analoga al caso discreto, ma ora la somma contiene infiniti modi normali. + +\subsection{Significato fisico delle componenti spaziale e temporale} +Per un singolo modo normale: +\[ + \psi_n(x,t) = Y_n(x) T_n(t) +\] + +dove: +\[ + Y_n(x) = \sin(k_n x) \qquad + T_n(t) = \cos(\omega_n t + \varphi_n) +\] + +Osserviamo due aspetti fondamentali: +\begin{itemize} + \item In ogni modo normale \textbf{tutti i punti della corda oscillano con la stessa frequenza} $\omega_n$. + \item Le ampiezze spaziali cambiano lungo la corda secondo la forma $Y_n(x)$, che contiene nodi e ventri. +\end{itemize} + +Questo è esattamente ciò che avevamo osservato nel caso discreto dei pendoli accoppiati: ogni modo normale è un oscillatore armonico indipendente. + +\subsection{Condizioni iniziali per la corda continua} +Quando avevamo un solo oscillatore armonico, servivano: +\[ + x(0) \qquad \dot x(0) +\] + +Nel caso con due gradi di libertà servivano: +\[ + x_1(0) \quad x_2(0) \qquad + \dot x_1(0) \quad \dot x_2(0) +\] + +Ora, nel caso continuo, abbiamo un numero infinito di punti, ognuno dei quali è un grado di libertà. +Quindi dobbiamo specificare due funzioni continue: +\[ + y(x,0) = f(x) +\] +\[ + \dot y(x,0) = g(x) +\] + +Queste due funzioni rappresentano: +\begin{itemize} + \item la forma iniziale della corda + \item la distribuzione iniziale delle velocità lungo la corda +\end{itemize} + +\subsection{Ruolo dei modi normali nella ricostruzione della soluzione} +Poiché i modi normali $\sin(k_n x)$ formano una base completa per le funzioni che soddisfano le condizioni al contorno $y(0,t)=0$ e $y(l,t)=0$, possiamo espandere la condizione iniziale come: +\[ + f(x) = \sum_{n=1}^{\infty} a_n \sin(k_n x) +\] + +e analogamente: +\[ + g(x) = \sum_{n=1}^{\infty} b_n \sin(k_n x) +\] + +I coefficienti $a_n$ e $b_n$ si determinano mediante le usuali formule di ortogonalità dei seni: +\[ + a_n = \frac{2}{l} \int_0^l f(x) \sin(k_n x) \, dx +\] +\[ + b_n = \frac{2}{l} \int_0^l g(x) \sin(k_n x) \, dx +\] + +Una volta noti i coefficienti, la soluzione generale si costruisce automaticamente: +\[ + y(x,t) = \sum_{n=1}^{\infty} \8 a_n \cos(\omega_n t) + \frac{b_n}{\omega_n} \sin(\omega_n t) \9 \sin(k_n x) +\] + +Questa è la forma completa della soluzione dell'equazione di d'Alembert per una corda vincolata agli estremi. +È la controparte continua e infinitamente estesa di ciò che avevamo già visto con due oscillatori accoppiati. + +\subsection{Interpretazione finale} +Abbiamo quindi stabilito alcuni punti centrali: +\begin{itemize} + \item una corda continua ha infiniti gradi di libertà + \item esistono infiniti modi normali, ciascuno con la propria frequenza $\omega_n$ + \item ogni modo è un oscillatore armonico perfettamente indipendente dagli altri + \item qualsiasi moto della corda è una combinazione lineare di questi modi + \item le condizioni iniziali determinano i coefficienti della combinazione lineare +\end{itemize} + +Abbiamo quindi completato il passaggio: +\[ + \text{sistemi discreti a pochi gradi di libertà}\quad + \longrightarrow \quad + \text{sistemi continui con infiniti modi} +\] + +e abbiamo visto come l'equazione di d'Alembert racchiuda al suo interno l'intera dinamica della corda vibrante. + + + +\section{Onde progressive e significato fisico della velocità v} +Consideriamo il generico modo normale della corda vincolata agli estremi: +\[ + \psi_n(x,t) = A_n \sin(k_n x) \cos(\omega_n t) +\] +con +\[ + k_n = \frac{n \pi}{l} \qquad \omega_n = v k_n \qquad v = \sqrt{\frac{T_0}{\rho_l}} +\] + +Scriviamo il prodotto seno per coseno tramite le identità ricavate con le formule di Eulero. +Usiamo: +\[ + \sin\alpha = \frac{e^{i\alpha} - e^{-i\alpha}}{2i} \qquad + \cos\beta = \frac{e^{i\beta} + e^{-i\beta}}{2} +\] + +Allora +\[ + \sin(k_n x)\cos(\omega_n t) = \frac{1}{4i} \8 e^{i(k_n x + \omega_n t)} - e^{-i(k_n x + \omega_n t)} + e^{i(k_n x - \omega_n t)} - e^{-i(k_n x - \omega_n t)} \9 +\] + +e riconvertendo in seni: +\[ + \sin(k_n x)\cos(\omega_n t) = \frac12 \8 \sin(k_n x + \omega_n t) + \sin(k_n x - \omega_n t) \9 +\] + +Otteniamo quindi la decomposizione fondamentale: +\[ + \psi_n(x,t) = \frac{A_n}{2} \sin(k_n x + \omega_n t) + \frac{A_n}{2} \sin(k_n x - \omega_n t) +\] + +Le due funzioni sono onde della forma: +\[ + f(x,t) = \sin(k_n x \pm \omega_n t) +\] +Scriviamo $\omega_n = v k_n$: +\[ + \sin(k_n x \pm v k_n t) = \sin\!\left[k_n(x \pm vt)\right] +\] + +Introduciamo una variabile di fase: +\[ + \phi_{\pm}(x,t)=k_n(x \pm vt) +\] + +Una funzione $f(x \pm vt)$ è una traslazione rigida nel tempo. +Se definiamo: +\[ + g(x) = \sin(k_n x) +\] + +allora: +\[ + g(x - vt) \ \text{si sposta verso destra con velocità}\ v +\] +\[ + g(x + vt) \ \text{si sposta verso sinistra con velocità}\ v +\] + +Dunque ogni modo normale è somma di due onde progressive uguali che viaggiano in direzioni opposte: +\[ + \psi_n(x,t) = \frac{A_n}{2} g(x - vt) + \frac{A_n}{2} g(x + vt) +\] + +La velocità $v$ non è la velocità dei punti della corda, bensì la velocità con cui si trasmette la perturbazione. + +L'equazione d'onda garantisce che: +\[ + \text{se } y(x,t)=f(x - vt) \quad \text{allora} \quad \dertt y = v^2 \derxx y +\] +e lo stesso vale per $g(x + vt)$. + +Questo mostra che: +\begin{enumerate} + \item I modi normali sono onde stazionarie formate da 2 onde progressive opposte + \item Le onde progressive viaggiano con velocità $v = \sqrt{T_0 \rho_l^{-1}}$ + \item I punti della corda oscillano verticalmente, ma l'informazione si trasferisce orizzontalmente con velocità $v$ +\end{enumerate} + +Per una corda generica, la soluzione completa: +\[ + y(x,t) = \sum_{n=1}^{\infty} A_n \sin(k_n x) \cos(\omega_n t + \varphi_n) +\] + +può essere vista come somma di onde progressive: +\[ + y(x,t) = F(x - vt) + G(x + vt) +\] + +dove $F$ e $G$ derivano dalle condizioni iniziali. + + + +\section{Soluzione generale dell'equazione d'onda e metodo di d'Alembert} +Nel caso della corda vincolata agli estremi abbiamo visto che ogni modo normale può essere +riscritto come somma di due onde che si muovono in direzioni opposte. +Per il modo $n$: +\[ + \psi_n(x,t) = A_n \sin(k_n x) \cos(\omega_n t + \varphi_n) +\] +si ottiene +\[ + \psi_n(x,t) = \frac{A_n}{2} \sin(k_n x + \omega_n t + \varphi_n) + \frac{A_n}{2} \sin(k_n x - \omega_n t + \varphi_n) +\] + +Usando $\omega_n = v k_n$: +\[ + \sin(k_n x \pm \omega_n t + \varphi_n) = \sin\!\left[k_n(x \pm vt) + \varphi_n\right] +\] + +Si vede quindi che ogni modo normale è somma di due onde della forma +\[ + f(x - vt) \qquad g(x + vt) +\] + +La soluzione generale della corda vincolata è +\[ + \psi(x,t) = \sum_{n=1}^{\infty} A_n \sin(k_n x)\cos(\omega_n t + \varphi_n) +\] + +e poiché ogni termine è somma di un'onda progressiva e una regressiva si può scrivere +\[ + \psi(x,t) = F(x - vt) + G(x + vt) +\] + +dove $F$ e $G$ dipendono dalle condizioni iniziali. + +\subsection{Metodo di d'Alembert} +Consideriamo ora l'equazione d'onda nella forma +\[ + \derxx{\psi} = \frac{1}{v^2}\dertt{\psi} +\] + +Definiamo le nuove variabili: +\[ + \xi = x - vt \qquad \eta = x + vt +\] + +La funzione $\psi(x,t)$ diventa $\psi(\xi,\eta)$. + +Calcoliamo le derivate usando la regola della funzione composta. + +\paragraph{Derivata prima in $x$} +\[ + \frac{\partial\psi}{\partial x} = \frac{\partial\psi}{\partial \xi}\frac{\partial \xi}{\partial x} + \frac{\partial\psi}{\partial \eta}\frac{\partial \eta}{\partial x} = \psi_\xi + \psi_\eta +\] + +\paragraph{Derivata seconda in $x$} +Deriviamo di nuovo: +\[ + \frac{\partial^2\psi}{\partial x^2} = \psi_{\xi\xi} + 2\psi_{\xi\eta} + \psi_{\eta\eta} +\] + +\paragraph{Derivata prima in $t$} +\[ + \frac{\partial\psi}{\partial t} = \psi_\xi \frac{\partial \xi}{\partial t} + \psi_\eta \frac{\partial \eta}{\partial t} = - v \psi_\xi + v \psi_\eta +\] + +\paragraph{Derivata seconda in $t$} +\[ + \frac{\partial^2\psi}{\partial t^2} = v^2\psi_{\xi\xi} - 2v^2\psi_{\xi\eta} + v^2\psi_{\eta\eta} +\] + +Ora imponiamo l'equazione d'onda: +\[ + \psi_{\xi\xi} + 2\psi_{\xi\eta} + \psi_{\eta\eta} = \psi_{\xi\xi} - 2\psi_{\xi\eta} + \psi_{\eta\eta} +\] + +I primi e gli ultimi termini si cancellano. Rimane: +\[ + 4\psi_{\xi\eta} = 0 + \qquad\Rightarrow\qquad + \psi_{\xi\eta} = 0 +\] + +Integrare rispetto a $\xi$ implica che $\psi_\eta$ non dipende da $\xi$: +\[ + \psi_\eta(\xi,\eta) = f(\eta) +\] + +Integrare rispetto a $\eta$ dà: +\[ + \psi(\xi,\eta) = F(\eta) + G(\xi) +\] + +Tornando alle variabili originali: +\[ + \psi(x,t) = F(x + vt) + G(x - vt) +\] + +Questa è la soluzione più generale dell'equazione d'onda in una dimensione. + +\subsection{Proprietà delle soluzioni} +Una funzione del tipo $\psi(x,t)=F(x - vt)$ rappresenta un'onda che si sposta verso destra. +Una funzione del tipo $\psi(x,t)=G(x + vt)$ rappresenta un'onda che si sposta verso sinistra. + +La forma dell'onda non cambia durante la propagazione: viene traslata con velocità $v$. + +Il vincolo ottenuto nell'ipotesi di derivata spaziale piccola +\[ + \left|\frac{\partial\psi}{\partial x}\right| \ll 1 +\] +garantisce che la corda rimanga tesa e che il modello sia valido. + +\subsection{Onde armoniche} +Consideriamo ora una singola onda armonica: +\[ + \psi(x,t) = A \sin(kx - \omega t + \varphi) +\] + +I parametri soddisfano: +\[ + k = \frac{2\pi}{\lambda} \qquad + \omega = 2\pi\nu \qquad + \omega = v k +\] + +La lunghezza d'onda è la distanza tra due massimi successivi. +Il periodo è il tempo tra due oscillazioni successive di un punto fissato. +La velocità dell'onda vale: +\[ + v = \frac{\omega}{k} = \lambda \nu +\] + +A parità di ampiezza, l'energia trasportata è proporzionale ad $A^2$. + +\subsection{Interpretazione geometrica} +Per un tempo fissato $t_0$ la dipendenza spaziale è: +\[ + \psi(x,t_0) = A \sin(kx + \text{costante}) +\] +che è un'onda sinusoidale nello spazio. + +Per una posizione fissata $x_0$ la dipendenza temporale è: +\[ + \psi(x_0,t) = A \sin(\omega t + \text{costante}) +\] +che è un'oscillazione armonica nel tempo. + +Il movimento dei punti della corda è verticale. +La propagazione dell'onda è orizzontale con velocità $v$. +La perturbazione trasporta energia e quantità di moto. + + + +\section{Potenza e trasporto di energia nelle onde} +Una soluzione dell'equazione di d'Alembert che rappresenti un'onda progressiva ha la forma +\[ + \psi(x,t) = f(x - vt) +\] + +La derivata temporale è +\[ + \dert{\psi} = \frac{d f}{d(x - vt)} \dert{x - vt} = - v f' +\] + +La derivata spaziale è +\[ + \derx{\psi} = f' +\] + +Da queste due espressioni segue +\[ + \dert{\psi} = - v \derx{\psi} +\] + +che vale per un'onda progressiva. +Per un'onda regressiva si ottiene invece +\[ + \dert{\psi} = + v \derx{\psi} +\] + +Consideriamo ora il trasporto di energia lungo la corda. +La tensione ha una componente verticale pari a +\[ + T_\psi = - T_0 \frac{\partial \psi}{\partial x} +\] + +Per un piccolo spostamento verticale +\[ + d\psi = \dert{\psi} \, dt +\] + +Il lavoro elementare è +\[ + dL = T_\psi\, d\psi = - T_0 \derx{\psi} \dert{\psi}\, dt +\] + +La potenza istantanea è +\[ + P(x,t) = \frac{dL}{dt} = - T_0 \derx{\psi} \dert{\psi} +\] + +Per un'onda progressiva usiamo $\dert{\psi}=- v \derx{\psi}$ e otteniamo +\[ + P = + T_0 v \8 \derx{\psi} \9^2 +\] + +Per un'onda regressiva, usando $\dert{\psi}=+ v \derx{\psi}$ si ottiene +\[ + P = - T_0 v \8 \derx{\psi} \9^2 +\] + +Il segno distingue la direzione del trasporto. +Se $\psi(x,t)=f(x - vt)$ allora anche $P(x,t)$ è funzione di $x - vt$: +\[ + P(x,t)=P(x - vt) +\] + +Dunque anche la potenza si propaga come un'onda con velocità $v$. +Consideriamo ora un'onda armonica progressiva +\[ + \psi(x,t)=A\sin(kx - \omega t) +\] + +Allora +\[ + \derx{\psi} = A k \cos(kx - \omega t) \qquad + \dert{\psi} = - A \omega \cos(kx - \omega t) +\] + +La potenza istantanea diventa +\[ + P(x,t) = T_0 A^2 k \omega \cos^2(kx - \omega t) +\] + +La funzione $\cos^2$ ha periodo $\pi$. La potenza media su un periodo si calcola come +\[ + \overline{P} = \frac{1}{\pi} \int_0^\pi T_0 A^2 k \omega \cos^2\phi \, d\phi +\] + +Usando +\[ + \frac{1}{\pi}\int_0^\pi \cos^2\phi\, d\phi = \frac12 +\] + +Segue +\[ + \overline{P} = \frac12 T_0 A^2 k \omega +\] + +Poiché $\omega = vk$ otteniamo la forma equivalente +\[ + \overline{P} = \frac12 T_0 v k^2 A^2 +\] +che mostra in modo chiaro la dipendenza quadratica dell'energia trasportata dall'ampiezza dell'onda. + +Per interpretare la struttura energetica confrontiamo la fase $\phi = kx - \omega t$ dell'onda con l'espressione di $\cos^2\phi$ nella potenza. +Quando $\psi$ è massima o minima si ha $\cos\phi=0$ e dunque $P=0$. +Quando $\psi$ passa per lo zero si ha $\cos\phi=\pm 1$ e dunque +\[ + P_{\max} = T_0 A^2 k \omega +\] + +La potenza è massima dove l'elemento di corda è nel punto di massimo stiramento locale, cioè dove $\derx{\psi}$ è massima. +In tali punti anche $\dert{\psi}$ è massima per un'onda armonica, quindi l'elemento possiede contemporaneamente massima energia cinetica e massima energia elastica. + +La potenza propagata è quindi modulata da una funzione armonica e si muove con velocità $v$. +Graficamente la forma di $\psi(x,t)$ e quella di $P(x,t)$ hanno lo stesso periodo spaziale ma non gli stessi zeri: +\[ + \psi = 0 \quad \Rightarrow \quad P = P_{\max} +\] + +\[ + \psi = \pm A \quad \Rightarrow \quad P = 0 +\] + +Questo descrive perfettamente il trasferimento di energia associato all'onda armonica sulla corda. + + + + +\section{Dissipazione radiativa e oscillatore accoppiato alla corda} +Consideriamo una massa $m$ collegata ad una molla di costante $k$ e, nello stesso punto, ad una corda tesa di tensione $T_0$. +Indichiamo con $y(t)$ lo spostamento verticale della massa e con $\psi(x,t)$ lo spostamento della corda, con $\psi(0,t)=y(t)$. + +L'equazione dell'oscillatore isolato sarebbe +\[ + m \dertt y = - k y +\] + +La presenza della corda introduce nuove forze dovute alla tensione $T_0$. +Nel punto $x=0$ la corda esercita due contributi orizzontali proiettati verticalmente: +\[ + F_1 = T_0 \frac{\partial\psi}{\partial x}\Big|_{x=0^+} \qquad + F_2 = - T_0 \frac{\partial\psi}{\partial x}\Big|_{x=0^-} +\] + +Per un'onda che si propaga verso $+x$ (onda progressiva) la relazione d'Alembert semplificata è +\[ + \dert\psi = - v \derx\psi +\] + +quindi +\[ + \derx\psi = - \frac{1}{v} \dert\psi +\] + +Per un'onda regressiva vale invece +\[ + \dert\psi = + v \derx\psi +\] + +Poiché il moto dell'oscillatore genera a destra onde progressive e a sinistra onde regressive, si può usare +\[ + \derx\psi\Big|_{x=0^\pm} = - \frac{1}{v}\dert\psi(0,t) +\] + +Poiché $\psi(0,t)=y(t)$: +\[ + \dert\psi(0,t) = \dert y(t) +\] + +La somma delle due forze della corda diventa +\[ + F_1 + F_2 = - T_0 \derx\psi(0^+,t) - T_0 \derx\psi(0^-,t) = - 2 T_0 \Big( - \frac{1}{v} \dert y \Big) +\] +cioè +\[ + F_1 + F_2 = - \frac{2T_0}{v} \dert y +\] + +L'equazione del moto è quindi +\[ + m \dertt y = - k y - \frac{2T_0}{v} \dert y +\] + +che riscritta nella forma usuale diventa +\[ + \dertt y + 2\gamma \dert y + \omega_0^2 y = 0 +\] + +con +\[ + \gamma = \frac{T_0}{m v} \qquad + \omega_0^2 = \frac{k}{m} +\] + +Questa è l'equazione di un oscillatore armonico smorzato. +Lo smorzamento non è dovuto ad attrito ma alla perdita di energia trasportata via dalla corda sotto forma di onde progressive e regressive. +È una dissipazione radiativa. + +La soluzione è +\[ + y(t) = A e^{-\gamma t}\cos(\omega t + \varphi) \qquad + \omega = \sqrt{\omega_0^2 - \gamma^2} +\] + +La massa quindi oscilla con ampiezza che decade esponenzialmente. +Ogni oscillazione genera sulla corda un'onda che si allontana con velocità $v$. +Dopo un tempo $t$ il fronte dell'onda si trova a +\[ + x = vt +\] + +La forma dell'onda sulla corda è data da +\[ + \psi(x,t) = y(t - x/v) +\] + +per $x>0$ e da +\[ + \psi(x,t) = y(t + x/v) +\] +per $x<0$. + +L'inviluppo della perturbazione lungo la corda è +\[ + \abs{\psi(x,t)} = A e^{-\gamma (t - x/v)} +\] +per $t > x/v$. + +La parte iniziale dell'onda (generata quando l'ampiezza era maggiore) si trova più lontano, mentre la parte più vicina alla massa è legata all'ampiezza attuale $A e^{-\gamma t}$. +La corda mostra quindi una forma modulata da un inviluppo esponenziale che si propaga rigidamente con velocità $v$. +Se si applica una forzante al punto $x=0$, ad esempio +\[ + y_{\text{forz}}(t) = B\cos(\Omega t) +\] + +L'oscillatore produce onde sinusoidali sulla corda di frequenza $\Omega$ e ampiezza determinata dal bilancio tra energia fornita dalla forzante ed energia radiata lungo la corda. +Dopo il transiente, la corda trasporta energia costante in entrambe le direzioni con potenza media +\[ + \overline{P} = \frac12 T_0 A^2 k \Omega +\] +per una componente armonica di numero d'onda $k$. + +Questo modello meccanico mostra che la radiazione ondosa agisce come uno smorzamento viscoso e permette il trasferimento di energia lungo la corda. +Il fenomeno è analogo alla dissipazione di energia acustica per un altoparlante o alla radiazione elettromagnetica emessa da un dipolo oscillante. + + + +\section{Onde acustiche nei fluidi} +Passiamo da onde trasversali in un mezzo unidimensionale alle onde longitudinali in un fluido tridimensionale. +Un fluido non ha forma propria ed è caratterizzato dall'incapacità di sostenere sforzi di taglio. +La grandezza fondamentale per descriverlo è la densità +\[ + \rho = \frac{M}{V} +\] +dove $M$ è la massa contenuta nel volume $V$. + +Un'altra grandezza centrale è la pressione. +Per una superficie $S$, se su di essa agisce una forza totale $F_\perp$ perpendicolare alla superficie, la pressione è +\[ + P = \frac{F_\perp}{S} +\] + +Microscopicamente $F_\perp$ deriva dagli urti delle molecole del fluido sulle pareti. +Il contributo della $i$-esima particella è +\[ + F_{i\perp} = \vec F_i \cdot \hat n +\] +e la pressione totale è +\[ + P = \frac{1}{S}\sum_i F_{i\perp} +\] + +L'unità di misura è il Pascal, che vale +\[ + \si{\Pa} = \si{\N\per\m\squared} +\] + +Consideriamo ora la pressione atmosferica. +La colonna d'aria sopra una superficie $S$ produce una forza peso totale +\[ + F = \int_0^H g(z)\rho(z) S \, dz +\] +dove $\rho(z)$ diminuisce con l'altezza. + +La pressione atmosferica è quindi +\[ + P = \int_0^H g(z)\rho(z)\, dz +\] +e al livello del mare vale circa + +\[ + P_{\text{atm}} \approx 10^5 \ \si{\Pa} +\] + +Per una superficie di area $A$, la forza esercitata dall'atmosfera è +\[ + F = P_{\text{atm}} A +\] +e per una mano con $A = 10^{-2} \ \text{m}^2$ si ottiene +\[ + F = 10^3 \ \si{\N} +\] +ossia l'equivalente del peso di circa $100$ kg. +Non percepiamo questa forza perché il nostro corpo contiene fluido interno alla stessa pressione: il principio di Pascal richiede +\[ + P_{\text{int}} = P_{\text{est}} +\] + +Il principio di Pascal afferma che in un fluido incomprimibile e in equilibrio statico la pressione è uniforme. +In un sistema idraulico, applicando una forza $F'$ su un pistone di area $S'$ e ottenendo una forza $F$ su un pistone di area $S$, si ha +\[ + \frac{F'}{S'} = \frac{F}{S} +\] +da cui +\[ + F = F' \frac{S}{S'} +\] +che permette la moltiplicazione delle forze. + +Passiamo ora alla generazione delle onde acustiche. +Un mezzo fluido può essere compresso: agendo su uno strato del fluido, si produce una variazione locale di densità $\rho$ e di pressione $P$. +Lo strato successivo viene spinto da quello precedente e la perturbazione si propaga. +Se lo spostamento delle particelle è lungo la stessa direzione di propagazione dell'onda, la perturbazione è longitudinale. + +Se consideriamo un pistone che comprime un gas, il primo strato viene spostato e compresso: +\[ + \rho \to \rho + \Delta\rho \qquad + P \to P + \Delta P +\] + +Lo strato successivo viene messo in moto dalla forza trasmessa dal primo e la perturbazione procede. +Il moto locale delle particelle ha accelerazione dovuta alla differenza di pressione tra strati adiacenti: +\[ + \rho \, \dert{u} = - \derx{P} +\] +dove $u(x,t)$ è la velocità locale del fluido. + +La continuità di massa impone che, se il fluido si comprime in una regione, la densità varia secondo +\[ + \dert{\rho} = - \rho_0 \derx{u} +\] +dove $\rho_0$ è la densità di equilibrio. + +Le variazioni di pressione e densità sono legate dalla risposta elastica del fluido. +Per piccole variazioni si assume una relazione lineare +\[ + \Delta P = B \frac{\Delta\rho}{\rho_0} +\] +dove $B$ è il modulo di comprimibilità del mezzo. + +Combinando le tre relazioni: +\[ + \rho_0 \dert u = - \derx{P} \qquad + \dert{\rho} = - \rho_0 \derx{u} \qquad + \Delta P = B \frac{\Delta\rho}{\rho_0} +\] + +si ottiene l'equazione d'onda per la pressione (o per la densità): +\[ + \derxx{P} = \frac{1}{v^2}\dertt{P} \qquad + v = \sqrt{\frac{B}{\rho_0}} +\] + +Questa è l'equazione che descrive le onde sonore. +La perturbazione longitudinale di densità e pressione si propaga con velocità +\[ + v = \sqrt{\frac{B}{\rho_0}} +\] +che dipende dal mezzo: i liquidi hanno grande $B$ e quindi velocità acustica maggiore dei gas. + +Le onde acustiche sono quindi oscillazioni longitudinali di pressione e densità, governate da una equazione d'onda formalmente identica a quella ottenuta per la corda vibrante, ma con grandezze fisiche diverse. + + + +\section{Onde acustiche: densità, pressione e spostamento} +Consideriamo un fluido contenuto in un tubo di sezione costante $S$. +Ogni grandezza dipenderà da $x$ e $t$. +Lo spostamento delle particelle rispetto alla posizione di equilibrio è $\psi(x,t)$ ed è parallelo alla direzione di propagazione. +Assumiamo +\[ + \left| \derx{\psi} \right| \ll 1 +\] +in modo che le deformazioni locali siano piccole. + +Un volumetto iniziale del fluido compreso tra $x$ e $x + dx$ ha volume +\[ + dV_0 = S\, dx +\] + +Dopo una perturbazione, la sua faccia sinistra si sposta di $\psi(x,t)$ e la faccia destra di $\psi(x+dx,t)$. +Il nuovo volume è +\[ + dV = S \8 \psi(x+dx,t) + dx - \psi(x,t) \9 +\] + +La variazione di volume è +\[ + \Delta V = dV - dV_0 = S \8 \psi(x+dx,t) - \psi(x,t) \9 +\] + +Dividendo e moltiplicando per $dx$: +\[ + \Delta V = S\, dx \frac{\psi(x+dx,t) - \psi(x,t)}{dx} +\] +che nel limite $dx\to 0$ diventa +\[ + \Delta V = dV_0 \derx{\psi} +\] + +Dunque il volume finale è +\[ + dV = dV_0 \8 1 + \derx{\psi} \9 +\] + +La massa contenuta nel volumetto resta costante. +La densità iniziale è +\[ + \rho_0 = \frac{dm}{dV_0} +\] + +la densità dopo la perturbazione è +\[ + \rho = \frac{dm}{dV} = \frac{dm}{dV_0 \8 1 + \derx{\psi} \9} +\] + +Sviluppando al primo ordine: +\[ + \rho \simeq \rho_0 \8 1 - \derx{\psi} \9 +\] + +La variazione di densità è quindi +\[ + \Delta\rho = \rho - \rho_0 = - \rho_0 \derx{\psi} +\] + +Passiamo ora alle forze dovute alla pressione. +Il volumetto è soggetto alla pressione sulla faccia sinistra $P(x,t)$ e sulla faccia destra $P(x+dx,t)$. +Le forze sulle due facce sono +\[ + F_A = P(x,t) S \qquad + F_B = - P(x+dx,t) S +\] + +La forza risultante è +\[ + F = F_A + F_B = S \8 P(x,t) - P(x+dx,t) \9 +\] + +Nel limite differenziale +\[ + F = - S\, dx\, \derx{P} +\] + +La massa del volumetto è +\[ + dm = \rho_0 S\, dx +\] + +La sua accelerazione è +\[ + \dertt{\psi}(x,t) +\] + +Applicando la seconda legge di Newton: +\[ + \rho_0 S\, dx\, \dertt{\psi} = - S\, dx\, \derx{P} +\] + +Eliminando $S\, dx$ otteniamo la relazione fondamentale: +\[ + \rho_0 \dertt{\psi} = - \derx{P} +\] + +Abbiamo quindi la prima equazione che lega lo spostamento $\psi$ alla pressione $P$. +Per ottenere l'equazione d'onda sarà necessario trovare una relazione che esprima $P$ in funzione di $\psi$ o di $\rho$. +Poiché +\[ + \Delta\rho = - \rho_0 \derx{\psi} +\] + +Una relazione tra variazione di pressione e variazione di densità chiuderà il sistema. + + + +\section{Relazione pressione-densità e chiusura dell'equazione acustica} +Dalla lezione precedente avevamo ottenuto la relazione dinamica +\[ + \rho_0 \dertt{\psi} = - \derx{P} +\] + +Lega lo spostamento $\psi(x,t)$ alla variazione di pressione $P(x,t)$. +Questa equazione però contiene due funzioni incognite. +Serve una seconda relazione che colleghi $P$ e $\rho$. + +Dalla variazione di densità avevamo già trovato +\[ + \Delta \rho = - \rho_0 \derx{\psi} +\] + +Ora vogliamo ricavare come varia la pressione quando varia la densità. + +Poiché $P$ non dipende solo da $\rho$ ma anche dalla temperatura $T$, occorre considerare +\[ + P = P(\rho, T) +\] + +La variazione infinitesima di $P$ è +\[ + dP = \pder{P}{\rho}_{T} d\rho + \pder{P}{T}_{\rho} dT +\] + +Per chiudere il sistema assumiamo che la trasformazione sia isoterma, cioè +\[ + dT = 0 +\] + +Quindi +\[ + dP = \pder{P}{\rho}_{T} d\rho +\] + +Nell'equilibrio termodinamico il legame tra pressione e densità per un gas ideale può essere dedotto dalla legge dei gas perfetti. +In questa fase ci interessa solo il fatto che la derivata parziale +\[ + \pder{P}{\rho}_{T} +\] + +Costante del mezzo. +Definiamo allora il modulo di comprimibilità isoterma +\[ + \beta = \pder{P}{\rho}_{T}\Big|_{\rho=\rho_0} +\] + +Sostituendo $d\rho = \Delta\rho$ otteniamo +\[ + \Delta P = \beta\, \Delta\rho +\] + +Usando la relazione precedente +\[ + \Delta\rho = - \rho_0 \derx{\psi} +\] + +Segue +\[ + \Delta P = - \beta \rho_0 \derx{\psi} +\] + +Poniamo ora $\Delta P = P(x,t)$, essendo la variazione rispetto alla pressione di equilibrio. +Allora +\[ + P = - \beta \rho_0 \derx{\psi} +\] + +Sostituiamo questa espressione nella relazione dinamica +\[ + \rho_0 \dertt{\psi} = - \derx{P} +\] + +Cioè +\[ + \rho_0 \dertt{\psi} = - \derx\!\left( - \beta \rho_0 \derx{\psi} \right) +\] + +Otteniamo +\[ + \rho_0 \dertt{\psi} = \beta \rho_0 \derxx{\psi} +\] + +Semplificando $\rho_0$ si arriva a +\[ + \dertt{\psi} = \beta \derxx{\psi} +\] + +Questa è l'equazione d'onda di d'Alembert per le onde acustiche in condizioni isoterme: +\[ + \dertt{\psi} = v^2 \derxx{\psi} \qquad + v^2 = \beta +\] + +Poiché $\beta = \pder{P}{\rho}_T$ possiamo scrivere la velocità dell'onda come +\[ + v = \sqrt{\frac{\beta}{\rho_0}} +\] + +Questa espressione fornisce la velocità delle onde sonore in un fluido isoterma. +La struttura dell'equazione è identica a quella trovata per la corda vibrante, ma con diverse grandezze fisiche: il ruolo di $T_0$ è ora svolto da $\beta$ e il ruolo della massa per unità di lunghezza è svolto da $\rho_0$. + +La natura universale di questa equazione è notevole. +L'equazione di d'Alembert descrive onde meccaniche sulla corda, onde sonore nei fluidi, onde elettromagnetiche nel vuoto e onde gravitazionali nella relatività generale lineare. + +In ogni caso compare la stessa struttura matematica +\[ + \dertt{\Phi} = v^2 \derxx{\Phi} +\] +per una variabile $\Phi$ che rappresenta di volta in volta spostamento, pressione, campo elettrico, campo magnetico o variazione metrica dello spazio tempo. + +Nelle prossime lezioni vedremo come alla perturbazione di spostamento $\psi$ corrisponda anche una perturbazione di pressione $P(x,t)$ e di densità $\rho(x,t)$, che soddisfano a loro volta la stessa equazione d'onda. + + + +\section{Equazioni d'onda per pressione e densità} +Partiamo dalla relazione già ricavata tra densità e spostamento +\[ + \Delta\rho = - \rho_0 \derx{\psi} +\] + +e dalla sua inversa +\[ + \derx{\psi} = - \frac{\Delta\rho}{\rho_0} +\] + +Inseriamo questa relazione nell'equazione d'onda per $\psi$ +\[ + \dertt{\psi} = \frac{\beta}{\rho_0} \derxx{\psi} +\] + +Sostituendo +\[ + \derxx{\psi} = - \frac{1}{\rho_0} \derx{(\Delta\rho)} +\] + +si ottiene +\[ + \dertt{\psi} = - \frac{\beta}{\rho_0^2}\, \derx{(\Delta\rho)} +\] + +Deriviamo ora entrambi i membri rispetto a $x$: +\[ + \derx{\left( \dertt{\psi} \right)} = - \frac{\beta}{\rho_0^2}\, \derxx{(\Delta\rho)} +\] + +Poiché $\psi$ è funzione regolare, possiamo scambiare l'ordine delle derivate +\[ + \dertt{\left( \derx{\psi} \right)} = - \frac{\beta}{\rho_0^2}\, \derxx{(\Delta\rho)} +\] + +Usiamo di nuovo la relazione tra densità e spostamento +\[ + \derx{\psi} = - \frac{\Delta\rho}{\rho_0} +\] + +Allora +\[ + \dertt{\left( - \frac{\Delta\rho}{\rho_0} \right)} = - \frac{\beta}{\rho_0^2} \derxx{(\Delta\rho)} +\] + +Moltiplichiamo per $-\rho_0$: +\[ + \dertt{(\Delta\rho)} = \frac{\beta}{\rho_0} \derxx{(\Delta\rho)} +\] + +Questa è una equazione di d'Alembert per la densità: +\[ + \dertt{(\Delta\rho)} = v^2 \derxx{(\Delta\rho)} \qquad + v^2 = \frac{\beta}{\rho_0} +\] + +La stessa procedura vale per la pressione. +Poiché abbiamo mostrato che +\[ + \Delta P = \beta\, \frac{\Delta\rho}{\rho_0} +\] + +Le variazioni di pressione soddisfano la medesima equazione +\[ + \dertt{(\Delta P)} = v^2 \derxx{(\Delta P)} +\] + +Esistono quindi tre onde distinte ma simultanee: +\begin{itemize} + \item Onda di spostamento $\psi(x,t)$ + \item Onda di densità $\Delta\rho(x,t)$ + \item Onda di pressione $\Delta P(x,t)$ +\end{itemize} + +Tutte si propagano con la stessa velocità +\[ + v = \sqrt{\frac{\beta}{\rho_0}} +\] + +Consideriamo ora una sorgente che impone uno spostamento armonico alla superficie $x=0$: +\[ + \psi(0,t) = A \cos(\omega t) +\] + +Questa condizione produce nel fluido una soluzione ondosa progressiva +\[ + \psi(x,t) = A \cos(kx - \omega t) \qquad + k = \frac{\omega}{c} +\] + +La densità segue da +\[ + \Delta\rho = - \rho_0 \derx{\psi} +\] + +Quindi +\[ + \Delta\rho(x,t) = A \rho_0 k \sin(kx - \omega t) +\] + +La pressione segue da +\[ + \Delta P = \beta \frac{\Delta\rho}{\rho_0} +\] +e dunque +\[ + \Delta P(x,t) = A \beta k \sin(kx - \omega t) +\] + +Le onde di pressione e densità sono quindi in fase tra loro, mentre risultano sfasate di $\pi/2$ rispetto all'onda di spostamento. +In particolare: +\[ + \psi \propto \cos(kx - \omega t) \qquad + \Delta\rho \propto \sin(kx - \omega t) \qquad + \Delta P \propto \sin(kx - \omega t) +\] + +L'onda fisicamente osservabile è l'onda di pressione, mentre l'onda di spostamento rappresenta l'oscillazione locale delle particelle del fluido. +Ogni particella del mezzo oscilla attorno alla posizione di equilibrio, mentre l'onda si propaga. + +Le animazioni mostrano esattamente questo: le particelle oscillano avanti e indietro, ma le zone di compressione e rarefazione si muovono lungo l'asse $x$ con velocità $v$. +Le creste dell'onda di densità sono le regioni più compresse, quelle di pressione sono identiche, e si muovono con la stessa legge. \end{document}