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  = 1; : : : ; N – 1.

a. How can we use our formula for B(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.)