Any complete set of functions, not just eigenfunctions, can be used to construct a solution to a linear PDE if we know how these functions evolve in time. Dirac delta functions are a perfectly acceptable alternative to the trigonometric eigenfunctions that were studied previously. We can approximate a Dirac delta function using a narrow Gaussian if we are careful not to make the width, a , smaller than the grid spacing in the simulation.
Figure shows how an initial saw-tooth disturbance is decomposed into delta functions.
Select the equation type and then select contour plot. Stop the simulation after the first rendering of the contours. Continue to run the program until the disturbance reaches the walls. Place the initial Gaussian off center. You have found the Green's function for a particular PDE if you were able to answer the last question.