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docs/source/api_Acous.rst

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+.. _api_Acous:
+
+
+.. math::
+
+    \renewcommand{\div}{\nabla\cdot\,}
+    \newcommand{\grad}{\vec \nabla}
+    \newcommand{\curl}{{\vec \nabla}\times\,}
+    \newcommand{\Acf}{{\mathbf A_c^f}}
+    \newcommand{\Ace}{{\mathbf A_c^e}}
+    \newcommand{\dcurl}{{\mathbf C}}
+    \newcommand{\dgrad}{{\mathbf G}}
+    \newcommand{\u}{\vec{u}}
+    \renewcommand{\dudt}{\frac{\partial\u}{\partial t}}
+    \renewcommand{\dphidt}{\frac{\partial\phi}{\partial t}}
+    \renewcommand{\dsdt}{\frac{\partial s}{\partial t}}
+
+Acoustic wave equation
+**********************
+
+Time domain acoustic wave equation in 1st order form:
+
+.. math::
+
+        \rho\frac{\partial \phi}{\partial t} = \nabla \cdot \vec{u}+\frac{\partial s}{\partial t}\delta(\vec{r}-\vec{r}_s)
+
+        \mu^{-1}\frac{\partial \vec{u}}{\partial t} = \nabla \phi
+
+where \\(\\rho\\) is the density, \\(\\mu\\) is the adiabatic compression modulus, \\(\\delta(\\vec{r}-\\vec{r}_s)\\) represents point source, and \\(\\phi\\) is the pressure.
+
+By combining two equations we have 2nd order form:
+
+
+.. math::
+
+        \rho\frac{\partial^2 \phi}{\partial t^2} - \div\mu\grad\phi = \frac{\partial^2 s}{\partial t^2}\delta(\vec{r}-\vec{r}_s)
+
+By applying Fourier transform to frequency domain we have:
+
+.. math::
+
+        -\rho\omega^2\phi - \div\mu\grad\phi = -\omega ^2 s \delta(\vec{r}-\vec{r}_s)
+
+where \\(\\omega = 2\\pi f\\) is angular frequency. Assuming contant \\(\\mu\\) and \\(\\rho\\) we have Helmholtz equation:
+
+.. math::
+
+        (\grad^2 + k^2)\phi = \mu^{-1}\omega^2 s \delta(\vec{r}-\vec{r}_s)
+
+where \\(\ k = \\omega\\sqrt{\\frac{\\rho}{\\mu}}=\\frac{\\omega}{v} \\) is the wave propagation constant.
+
+
+Discretization of problem
+-------------------------
+
+Artificial Boundary condition
+-----------------------------
+
+Sponge boundary
+===============
+
+PML boundary
+============
+
+
+.. raw:: html
+    :file: examples\refraction.html
+
+
+Backgrounds
+===========
+
+
+Notebooks
+=========
+