Python Specials #8 3D WAVE SIMULATION
This code uses a finite-difference method to solve the wave equation in 3D, with the initial condition of a single pulse in the center of the domain. The solution is plotted at each time step using Matplotlib's 3D plotting capabilities. You can adjust the parameters such as nx, ny, and nt to change the size of the domain and the duration of the simulation.
Here is a detailed explanation of the code:
The code begins by importing the necessary libraries. numpy is used for numerical computations, matplotlib is used for visualization, and Axes3D is used to enable 3D plotting.
Next, some parameters are set up. nx and ny determine the number of points in the x and y directions, respectively. nt is the number of time steps. c is the wave speed. dx, dy, and dt are the spatial and temporal step sizes.
The initial conditions for the wave are set up. x and y are arrays of coordinates spanning the domain, created using np.linspace. X and Y are 2D arrays created using np.meshgrid that represent the coordinates of each point in the domain. u is a 2D array of zeros representing the wave amplitude at each point. In this case, a single pulse is initialized at the center of the domain.
A loop over time steps begins. At each time step, the wave equation is solved using finite differences. The un variable stores a copy of the previous time step's solution. The equation used to update u is:
u[1:-1, 1:-1] = (un[1:-1, 1:-1] - c * dt / dx * (un[1:-1, 1:-1] - un[1:-1, :-2])
c * dt / dy * (un[1:-1, 1:-1] - un[:-2, 1:-1]))
This equation updates the interior points of u based on the previous time step's solution (un) and the wave speed (c) and spatial and temporal step sizes (dx, dy, dt). The indices [1:-1, 1:-1] are used to exclude the boundary points, which are updated separately.
After u has been updated, the boundary conditions are enforced. In this case, the boundary values are simply set to zero.
Finally, the solution is plotted using Matplotlib's plot_surface function. A new figure is created at each time step, and the zlim argument is used to set the range of the z-axis.
That's a brief summary of what's happening in the code. Let me know if you have any further questions!
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