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Transport Phenomena (Wave equation \( \dfrac{\partial^2 \phi}{\partial…
Transport Phenomena
Wave equation \( \dfrac{\partial^2 \phi}{\partial t^2} = c_0 \dfrac{\partial^2 \phi}{\partial x^2} \)
Initial condition:
\( \phi(x,0) = \alpha(x)\)
\(\frac{\partial \phi}{\partial t}(x,0) = \beta(x) \)
D’Alembert’s solution:
\(
\phi(x,t) = \dfrac{1}{2}\left( \alpha(x-c_0t)+\alpha(x+c_0t) \right) + \dfrac{1}{2c_0}\int_{x-c_0t}^{x+c_0t} \beta(s)ds
\)
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wave-like solution:
\( \phi(x,t) = \hat{\phi}e^{i(kx-\omega t)} \)
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for finding dispersion relation
(an equation with \(\omega\) and \(k\)),
eg \(\omega(k) = k(c_0 - k^2)\)
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Kinematic Wave Equation
\( \dfrac{\partial \phi}{\partial t} + c(\phi)\dfrac{\partial \phi}{\partial x} = 0 \)
slowly varying wavetrain
\( \eta(x,t) = A(x,t)e^{i\theta(x,t)} \)
\( A(x,t) \) : amplitude
\( \theta(x,t) \) : phase
local wave number
\( K(x,t) = \dfrac{\partial \theta}{\partial x}(x,t)\)
local frequency
\( \omega(x,t) = -\dfrac{\partial \theta}{\partial t}(x,t)\)
plane wave solution:
\( \phi(x,t) = \frac{1}{2\pi} \int_{-\infty}^\infty A(k)e^{i(kx-\omega t)}dk \)