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102 lines
3.1 KiB
ReStructuredText
102 lines
3.1 KiB
ReStructuredText
.. _api_Acous:
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.. math::
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\renewcommand{\div}{\nabla\cdot\,}
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\newcommand{\grad}{\vec \nabla}
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\newcommand{\curl}{{\vec \nabla}\times\,}
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\newcommand{\Acf}{{\mathbf A_c^f}}
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\newcommand{\Ace}{{\mathbf A_c^e}}
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\newcommand{\dcurl}{{\mathbf C}}
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\newcommand{\dgrad}{{\mathbf G}}
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\newcommand{\u}{\vec{u}}
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\renewcommand{\dudt}{\frac{\partial\u}{\partial t}}
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\renewcommand{\dphidt}{\frac{\partial\phi}{\partial t}}
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\renewcommand{\dsdt}{\frac{\partial s}{\partial t}}
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Acoustic wave equation
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**********************
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Backgrounds
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-----------
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Time domain acoustic wave equation in 1st order form:
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.. math::
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\rho\frac{\partial \phi}{\partial t} = \nabla \cdot \vec{u}+\frac{\partial s}{\partial t}\delta(\vec{r}-\vec{r}_s)
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\mu^{-1}\frac{\partial \vec{u}}{\partial t} = \nabla \phi
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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.
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By combining two equations we have 2nd order form:
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.. math::
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\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)
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By applying Fourier transform to frequency domain we have:
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.. math::
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-\rho\omega^2\phi - \div\mu\grad\phi = -\omega ^2 s \delta(\vec{r}-\vec{r}_s)
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where \\(\\omega = 2\\pi f\\) is angular frequency. Assuming contant \\(\\mu\\) and \\(\\rho\\) we have Helmholtz equation:
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.. math::
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(\grad^2 + k^2)\phi = \mu^{-1}\omega^2 s \delta(\vec{r}-\vec{r}_s)
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where \\(\ k = \\omega\\sqrt{\\frac{\\rho}{\\mu}}=\\frac{\\omega}{v} \\) is the wave propagation constant.
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Discretization of problem
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-------------------------
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To compute the solution of time or frequency dependent partial differential equation (PDE) as discussed above, we need to discretize those equations in both time and space. Our domain in real situation (i.e., earth) can be considered as infinite in space. However, in discrete space where we compute the solution of those PDE should be finite. Therefore, we need to implement artificial boundary conditions to get rid of out going wave, which propagates beyond our domain of interest. We first implement sponge boundary condition then perfectly matched boundary condition.
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Sponge boundary
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===============
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A fundamental idea of sponge boundary condition is adding damping term to acoustic wave equation:
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.. math::
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\rho\frac{\partial^2 \phi}{\partial t^2} + c\frac{\partial \phi}{\partial t}- \div\mu\grad\phi = \frac{\partial^2 s}{\partial t^2}\delta(\vec{r}-\vec{r}_s)
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Similarly in frequency domain we have:
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.. math ::
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-\rho\omega^2\phi +\imath\omega c\phi- \div\mu\grad\phi = -\omega ^2 s \delta(\vec{r}-\vec{r}_s)
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The first order form of time domain equation can be written as:
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.. math ::
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\rho\frac{\partial \phi}{\partial t} +c\phi- \nabla \cdot \vec{u}= +\frac{\partial s}{\partial t}\delta(\vec{r}-\vec{r}_s)
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\mu^{-1}\frac{\partial \vec{u}}{\partial t} = \nabla \phi
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Using staggered grid, we discretize acoustic wave equation in time (CITE):
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PML boundary
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============
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.. raw:: html
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:file: examples\refraction.html
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Notebooks
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=========
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