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The acoustic wave equation governs the propagation of pressure waves. Wave 
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types that obey this law tend to travel in liquids or gases where shear waves 
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or longitudinal style wave motion is not possible. An obvious example is sound 
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waves. 
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The acoustic wave equation is defined as; 
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\begin{equation} 
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\nabla ^2 p  \frac{1}{c^2} \frac{\partial ^2 p}{\partial t^2} = 0 
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\label{eqn:acswave} 
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\end{equation} 
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where $p$ is the pressure, $t$ is the time and $c$ is the wave velocity. In this 
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chapter the acoustic wave equation is demonstrated. Important steps include the 
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translation of the Laplacian $\nabla^2$ to the \esc general form, the stiff 
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equation stability criterion and solving for the displacement or acceleration solution. 
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\section{The Laplacian in \esc} 
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The Laplacian operator which can be written as $\Delta$ or $\nabla^2$, is 
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calculated via the divergence of the gradient of the object, which in this 
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example is the scalar $p$. Thus we can write; 
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\begin{equation} 
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\nabla^2 p = \nabla \cdot \nabla p = 
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\sum_{i}^n 
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\frac{\partial^2 p}{\partial x^2_{i}} 
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\label{eqn:laplacian} 
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\end{equation} 
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For the two dimensional case in Cartesian coordinates \autoref{eqn:laplacian} 
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becomes; 
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\begin{equation} 
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\nabla^2 p = \frac{\partial^2 p}{\partial x^2} 
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+ \frac{\partial^2 p}{\partial y^2} 
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\end{equation} 
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In \esc the Laplacian is calculated using the divergence representation and the 
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intrinsic functions \textit{grad()} and \textit{trace()}. The function 
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\textit{grad{}} will return the spatial gradients of an object. 
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For a rank 0 solution, this is of the form; 
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\begin{equation} 
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\nabla p = \left[ 
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\frac{\partial p}{\partial x _{0}}, 
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\frac{\partial p}{\partial x _{1}} 
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\right] 
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\label{eqn:grad} 
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\end{equation} 
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Larger ranked solution objects will return gradient tensors. For example, a 
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pressure field which acts in the directions $p _{0}$ and $p 
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_{1}$ would return; 
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\begin{equation} 
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\nabla p = \begin{bmatrix} 
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\frac{\partial p _{0}}{\partial x _{0}} & 
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\frac{\partial p _{1}}{\partial x _{0}} \\ 
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\frac{\partial p _{0}}{\partial x _{1}} & 
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\frac{\partial p _{1}}{\partial x _{1}} 
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\end{bmatrix} 
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\label{eqn:gradrank1} 
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\end{equation} 
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\autoref{eqn:grad} corresponds to the Linear PDE general form value 
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$X$. Notice however, that the general form contains the term $X 
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_{i,j}$\footnote{This is the first derivative in the $j^{th}$ 
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direction for the $i^{th}$ component of the solution.}, 
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hence for a rank 0 object there is no need to do more then calculate the 
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gradient and submit it to the solver. In the case of the rank 1 or greater 
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object, it is also necessary to calculate the trace. This is the sum of the 
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diagonal in \autoref{eqn:gradrank1}. 
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Thus when solving for equations containing the Laplacian one of two things must 
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be completed. If the object \verb!p! is less than rank 1 the gradient is 
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calculated via; 
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\begin{python} 
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gradient=grad(p) 
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\end{python} 
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and if the object is greater then or equal to a rank 1 tensor, the trace of 
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the gradient is calculated. 
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\begin{python} 
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gradient=trace(grad(p)) 
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\end{python} 
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These values can then be submitted to the PDE solver via the general form term 
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$X$. The Laplacian is then computed in the solution process by taking the 
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divergence of $X$. 
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Note, if you are unsure about the rank of your tensor, the \textit{getRank} 
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command will return the rank of the PDE object. 
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\begin{python} 
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rank = p.getRank() 
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\end{python} 
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\section{Numerical Solution Stability} \label{sec:nsstab} 
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Unfortunately, the wave equation belongs to a class of equations called 
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\textbf{stiff} PDEs. These types of equations can be difficult to solve 
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numerically as they tend to oscillate about the exact solution, which can 
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eventually lead to a catastrophic failure. To counter this problem, explicitly 
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stable schemes like the backwards Euler method, and correct parameterisation of 
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the problem are required. 
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There are two variables which must be considered for 
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stability when numerically trying to solve the wave equation. For linear media, 
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the two variables are related via; 
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\begin{equation} \label{eqn:freqvel} 
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f=\frac{v}{\lambda} 
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\end{equation} 
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The velocity $v$ that a wave travels in a medium is an important variable. For 
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stability the analytical wave must not propagate faster then the numerical wave 
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is able to, and in general, needs to be much slower then the numerical wave. 
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For example, a line 100m long is discretised into 1m intervals or 101 nodes. If 
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a wave enters with a propagation velocity of 100m/s then the travel time for 
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the wave between each node will be 0.01 seconds. The time step, must therefore 
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be significantly less then this. Of the order $10E4$ would be appropriate. 
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This stability criterion is known as the Courant\textendash 
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Friedrichs\textendash Lewy condition given by 
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\begin{equation} 
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dt=f\cdot \frac{dx}{v} 
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\end{equation} 
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where $dx$ is the mesh size and $f$ is a safety factor. To obtain a time step of 
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$10E4$, a safety factor of $f=0.1$ was used. 
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The wave frequency content also plays a part in numerical stability. The 
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Nyquistsampling theorem states that a signals bandwidth content will be 
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accurately represented when an equispaced sampling rate $f _{n}$ is 
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equal to or greater then twice the maximum frequency of the signal 
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$f_{s}$, or; 
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\begin{equation} \label{eqn:samptheorem} 
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f_{n} \geqslant f_{s} 
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\end{equation} 
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For example, a 50Hz signal will require a sampling rate greater then 100Hz or 
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one sample every 0.01 seconds. The wave equation relies on a spatial frequency, 
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thus the sampling theorem in this case applies to the solution mesh spacing. 
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This relationship confirms that the frequency content of the input signal 
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directly affects the time discretisation of the problem. 
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To accurately model the wave equation with high resolutions and velocities 
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means that very fine spatial and time discretisation is necessary for most 
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problems. This requirement makes the wave equation arduous to 
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solve numerically due to the large number of time iterations required in each 
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solution. Models with very high velocities and frequencies will be the worst 
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affected by this problem. 
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\section{Displacement Solution} 
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\sslist{example07a.py} 
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We begin the solution to this PDE with the centred difference formula for the 
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second derivative; 
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\begin{equation} 
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f''(x) \approx \frac{f(x+h  2f(x) + f(xh)}{h^2} 
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\label{eqn:centdiff} 
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\end{equation} 
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substituting \autoref{eqn:centdiff} for $\frac{\partial ^2 p }{\partial t ^2}$ 
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in \autoref{eqn:acswave}; 
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\begin{equation} 
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\nabla ^2 p  \frac{1}{c^2h^2} \left[p_{(t+1)}  2p_{(t)} + 
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p_{(t1)} \right] 
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= 0 
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\label{eqn:waveu} 
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\end{equation} 
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Rearranging for $p_{(t+1)}$; 
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\begin{equation} 
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p_{(t+1)} = c^2 h^2 \nabla ^2 p_{(t)} +2p_{(t)}  
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p_{(t1)} 
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\end{equation} 
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this can be compared with the general form of the \modLPDE module and it 
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becomes clear that $D=1$, $X_{i,j}=c^2 h^2 \nabla ^2 p_{(t)}$ and 
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$Y=2p_{(t)}  p_{(t1)}$. 
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The solution script is similar to others that we have created in previous 
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chapters. The general steps are; 
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\begin{enumerate} 
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\item The necessary libraries must be imported. 
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\item The domain needs to be defined. 
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\item The time iteration and control parameters need to be defined. 
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\item The PDE is initialised with source and boundary conditions. 
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\item The time loop is started and the PDE is solved at consecutive time steps. 
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\item All or select solutions are saved to file for visualisation later on. 
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\end{enumerate} 
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Parts of the script which warrant more attention are the definition of the 
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source, visualising the source, the solution time loop and the VTK data export. 
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\subsection{Pressure Sources} 
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As the pressure is a scalar, one need only define the pressure for two 
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time steps prior to the start of the solution loop. Two known solutions are 
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required because the wave equation contains a double partial derivative with 
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respect to time. This is often a good opportunity to introduce a source to the 
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solution. This model has the source located at it's centre. The source should 
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be smooth and cover a number of samples to satisfy the frequency stability 
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criterion. Small sources will generate high frequency signals. Here, when using 
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a rectangular domain, the source is defined by a cosine function. 
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\begin{python} 
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U0=0.01 # amplitude of point source 
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xc=[500,500] #location of point source 
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# define small radius around point xc 
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src_radius = 30 
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# for first two time steps 
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u=U0*(cos(length(xxc)*3.1415/src_radius)+1)*\ 
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whereNegative(length(xxc)src_radius) 
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u_m1=u 
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\end{python} 
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\subsection{Visualising the Source} 
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There are two options for visualising the source. The first is to export the 
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initial conditions of the model to VTK, which can be interpreted as a scalar 
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surface in Mayavi2. The second is to take a cross section of the model which 
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will require the \textit{Locator} function. 
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First \verb!Locator! must be imported; 
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\begin{python} 
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from esys.escript.pdetools import Locator 
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\end{python} 
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The function can then be used on the domain to locate the nearest domain node 
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to the point or points of interest. 
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It is now necessary to build a list of $(x,y)$ locations that specify where are 
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model slice will go. This is easily implemented with a loop; 
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\begin{python} 
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cut_loc=[] 
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src_cut=[] 
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for i in range(ndx/2ndx/10,ndx/2+ndx/10): 
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cut_loc.append(xstep*i) 
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src_cut.append([xstep*i,xc[1]]) 
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\end{python} 
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We then submit the output to \verb!Locator! and finally return the appropriate 
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values using the \verb!getValue! function. 
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\begin{python} 
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src=Locator(mydomain,src_cut) 
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src_cut=src.getValue(u) 
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\end{python} 
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It is then a trivial task to plot and save the output using \mpl 
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(\autoref{fig:cxsource}). 
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\begin{python} 
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pl.plot(cut_loc,src_cut) 
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pl.axis([xc[0]src_radius*3,xc[0]+src_radius*3,0.,2*U0]) 
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pl.savefig(os.path.join(savepath,"source_line.png")) 
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\end{python} 
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\begin{figure}[h] 
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\centering 
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\includegraphics[width=6in]{figures/sourceline.png} 
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\caption{Cross section of the source function.} 
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\label{fig:cxsource} 
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\end{figure} 
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\subsection{Point Monitoring} 
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In the more general case where the solution mesh is irregular or specific 
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locations need to be monitored, it is simple enough to use the \textit{Locator} 
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function. 
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\begin{python} 
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rec=Locator(mydomain,[250.,250.]) 
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\end{python} 
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When the solution \verb u is updated we can extract the value at that point 
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via; 
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\begin{python} 
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u_rec=rec.getValue(u) 
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\end{python} 
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For consecutive time steps one can record the values from \verb!u_rec! in an 
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array initialised as \verb!u_rec0=[]! with; 
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\begin{python} 
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u_rec0.append(rec.getValue(u)) 
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\end{python} 
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It can be useful to monitor the value at a single or multiple individual points 
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in the model during the modelling process. This is done using 
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the \verb!Locator! function. 
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\section{Acceleration Solution} 
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\sslist{example07b.py} 
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An alternative method to the displacement solution, is to solve for the 
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acceleration $\frac{\partial ^2 p}{\partial t^2}$ directly. The displacement can 
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then be derived from the acceleration after a solution has been calculated 
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The acceleration is given by a modified form of \autoref{eqn:waveu}; 
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\begin{equation} 
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\nabla ^2 p  \frac{1}{c^2} a = 0 
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\label{eqn:wavea} 
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\end{equation} 
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and can be solved directly with $Y=0$ and $X=c^2 \nabla ^2 p_{(t)}$. 
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After each iteration the displacement is reevaluated via; 
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\begin{equation} 
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p_{(t+1)}=2p_{(t)}  p_{(t1)} + h^2a 
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\end{equation} 
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\subsection{Lumping} 
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For \esc, the acceleration solution is prefered as it allows the use of matrix 
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lumping. Lumping or mass lumping as it is sometimes known, is the process of 
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aggressively approximating the density elements of a mass matrix into the main 
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diagonal. The use of Lumping is motivaed by the simplicity of diagonal matrix 
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inversion. As a result, Lumping can significantly reduce the computational 
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requirements of a problem. Care should be taken however, as this 
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function can only be used when the $A$, $B$ and $C$ coefficients of the 
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general form are zero. 
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More information about the lumping implementation used in \esc and its accuracy 
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can be found in the user guide. 
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To turn lumping on in \esc one can use the command; 
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\begin{python} 
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mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().HRZ_LUMPING) 
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\end{python} 
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It is also possible to check if lumping is set using; 
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\begin{python} 
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print mypde.isUsingLumping() 
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\end{python} 
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\section{Stability Investigation} 
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It is now prudent to investigate the stability limitations of this problem. 
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First, we let the frequency content of the source be very small. If we define 
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the source as a cosine input, then the wavlength of the input is equal to the 
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radius of the source. Let this value be 5 meters. Now, if the maximum velocity 
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of the model is $c=380.0ms^{1}$, then the source 
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frequency is $f_{r} = \frac{380.0}{5} = 76.0 Hz$. This is a worst case 
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scenario with a small source and the models maximum velocity. 
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Furthermore, we know from \autoref{sec:nsstab}, that the spatial sampling 
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frequency must be at least twice this value to ensure stability. If we assume 
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the model mesh is a square equispaced grid, 
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then the sampling interval is the side length divided by the number of samples, 
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given by $\Delta x = \frac{1000.0m}{400} = 2.5m$ and the maximum sampling 
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frequency capable at this interval is 
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$f_{s}=\frac{380.0ms^{1}}{2.5m}=152Hz$ this is just equal to the 
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required rate satisfying \autoref{eqn:samptheorem}. 
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\autoref{fig:ex07sampth} depicts three examples where the grid has been 
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undersampled, sampled correctly, and over sampled. The grids used had 
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200, 400 and 800 nodes per side respectively. Obviously, the oversampled grid 
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retains the best resolution of the modelled wave. 
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The time step required for each of these examples is simply calculated from 
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the propagation requirement. For a maximum velocity of $380.0ms^{1}$, 
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\begin{subequations} 
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\begin{equation} 
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\Delta t \leq \frac{1000.0m}{200} \frac{1}{380.0} = 0.013s 
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\end{equation} 
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\begin{equation} 
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\Delta t \leq \frac{1000.0m}{400} \frac{1}{380.0} = 0.0065s 
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\end{equation} 
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\begin{equation} 
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\Delta t \leq \frac{1000.0m}{800} \frac{1}{380.0} = 0.0032s 
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\end{equation} 
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\end{subequations} 
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Observe that for each doubling of the number of nodes in the mesh, we halve 
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the time step. To illustrate the impact this has, consider our model. If the 
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source is placed at the center, it is $500m$ from the nearest boundary. With a 
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velocity of $380.0ms^{1}$ it will take $\approx1.3s$ for the wavefront to 
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reach that boundary. In each case, this equates to $100$, $200$ and $400$ time 
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steps. This is again, only a best case scenario, for true stability these time 
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values may need to be halved and possibly halved again. 
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\begin{figure}[ht] 
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\centering 
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\subfigure[Undersampled Example]{ 
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\includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm,clip]{figures/ex07usamp.png} 
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\label{fig:ex07usamp} 
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} 
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\subfigure[Just sampled Example]{ 
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\includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm,clip]{figures/ex07jsamp.png} 
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\label{fig:ex07jsamp} 
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} 
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\subfigure[Over sampled Example]{ 
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\includegraphics[width=0.45\textwidth,trim=0cm 6cm 5cm 6cm,clip]{figures/ex07nsamp.png} 
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\label{fig:ex07nsamp} 
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} 
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\caption{Sampling Theorem example for stability investigation} 
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\label{fig:ex07sampth} 
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\end{figure} 
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