How will the modes change across the eddy whose SSP is shown in (Sketch).

1. Consider the following eigenproblem:

The exact eigenvalues are If we solve this problem using finite differences with the standard formula, we will get approximations to these eigenvalues where N is the number of points in the mesh and k D 1; : : : ;N _ 1:

a. How can we use our formula for ˇk.N / to obtain a similar result for the approximate eigenvalues of an isovelocity acoustic problem?

b. Calculate

c. Use Richardson extrapolation to estimate from these numbers in the limit

d. Roughly, how large would N have to be to obtain this value by simple mesh

refinement?

e. How much slower would the mesh refinement be? (Solving the finite difference equations for requires roughly 20N operations.)