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Wave equation in two-dimensions

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1. Derive numerical solutions to the wave equation in two-dimensions using a second-order accurate central difference scheme for derivations in both space and time. 
2. Write a new MATLAB script that implements the finite-difference solutions to the 2D wave equations.
3. What happens when the wave reaches boundary? What assumption within the
implementation of the finite difference scheme causes this to happen? 
4. Increase the time step beyond the stability limit. What happens to the wave
field? Plot a snapshot of the field as the instability is beginning.

5. Using a time step set to the stability limit, save the value of the pressure field
at the grid position (40, 40) at each time point during the time loop. Store the values in a
new variable p_out, and plot this at the end of the simulation against time in seconds.
Label the axis of the plot.

6. Reduce the variance of the Gaussian initial pressure to dx^2, and plot the
resulting time signal at the same grid position. What has happened to the recorded
signal? What is the name of this phenomena, and why does it become more apparent
when the initial pressure distribution is steeper?

7. Plots should be included as part of the report. Include your MATLAB code as an
appendix to your report. Fully comment every line of code.