8. Extension to Higher Dimensions and Nonlinear Systems

• The 1-D wave equation
• The 2-D wave equation

• Script timing: need for numpy acceleration

• Exercise 8.1: More simple physics applications

  1. Wave tank.  Create a virtual "wave tank" by modifying the 2-D wave solver so that the left-hand side (x = -10, all y) oscillates in time with

     	u = sin(np.pi t), rx = 0, ry = 0, s = du/dt.

     You should see a nearly plane wave propagating across the domain. What is its wavelength? Is it consistent with expectations?

  2. Boundary conditions.  You should also see some boundary effects in the previous problem. You can reduce them by forcing the y derivatives to be zero at the top and bottom edges:

     	u[:,0] = u[:,1]
     	u[:,J-1] = u[:,J-2]

     and similarly for the other variables.

  3. Diffraction.  What happens if you put an absorbing barrier in the tank at x = 0 by setting

     	u = rx = ry = s = 0

     at  x = 0,  -2 < y < 2 ?

  4. 2-D interference.  Now instead of a plane wave, let's modify the left-hand boundary condition to represent two "point" sources of amplitude 5, each of height 0.25 units, at y = 2.5 and y = -2.5. Can you explain what you see? How does the resultant interference pattern change as you vary (try doubling and halving) the distance between the sources?

• Nonlinear effects