Persamaan Gelombang

Konsep Dasar

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Persamaan dasar gelombang

\(y = A \sin (\omega t - kx)\)

 

\(y\) = simpangan getar

\(x\) = jarak dari sumber getar

\(A\) = amplitudo

\(t\) = waktu getar

\(\omega\) = kelajuan angular = \(2 \: \pi \: f = \dfrac{2\pi}{T}\)

\(k\) = bilangan gelombang = \(\dfrac{2\pi}{\lambda}\)

 

 

Cepat rambat gelombang

\(v = \lambda \:.\: f\)

 

 

Fase dan sudut fase

Fase = \(\varphi = \dfrac tT - \dfrac {x}{\lambda}\)

Sudut fase = \(\theta = \omega t - kx\)

 

 

Beda fase

Beda fase 2 titik berbeda pada waktu yang sama

\begin{equation*} \begin{split} \Delta \varphi & = \varphi_2 - \varphi_1 \\\\ \Delta \varphi & = \left(\frac{t}{T} - \frac{x_2}{\lambda}\right) - \left(\frac{t}{T} - \frac{x_1}{\lambda}\right) \\\\ \Delta \varphi & = \cancel {\frac{t}{T}} - \frac{x_2}{\lambda} - \cancel {\frac{t}{T}} + \frac{x_1}{\lambda} \\\\ \Delta \varphi & = \frac{x_1 - x_2}{\lambda} \\\\ \Delta \varphi & = \frac{\Delta x}{\lambda} \end{split} \end{equation*}

Beda fase suatu titik pada waktu yang berbeda

\begin{equation*} \begin{split} \Delta \varphi & = \varphi_2 - \varphi_1 \\\\ \Delta \varphi & = \left(\frac{t_2}{T} - \frac{x}{\lambda}\right) - \left(\frac{t_1}{T} - \frac{x}{\lambda}\right) \\\\ \Delta \varphi & = \frac{t_2}{T} - \cancel {\frac{x}{\lambda}} - \frac{t_1}{T} + \cancel {\frac{x}{\lambda}} \\\\ \Delta \varphi & = \frac{t_2 - t_1}{T} \\\\ \Delta \varphi & = \frac{\Delta t}{T} \end{split} \end{equation*}

 

 

Cepat getar 

\(v_y = \dfrac{dy}{dt} = \omega A \cos (\omega t - kx) \)

\(v_y = \omega \sqrt{A^2 - y^2}\)

 

 

Percepatan getar 

\(a = \dfrac{dv}{dt} = - \omega^2 A \sin (\omega t - kx) \)

\(a = - \omega^2 y \)

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