3.3. Exercises for time-harmonic waves#
Exercise 1#
Solve the scattering problem with a first order absorbing boundary condition on a domain \([0,1]^2\setminus \bar\Omega_0\) for a compact domain \(\Omega_0\subset[0,1]^2\) with a plane-wave incident field. Visualize the incident, scattered and total field.
Exercise 2#
Solve the Helmholtz problem on the geometry from square_inf.ipynb for right hand side \(\hat f(x)=\exp(-30(x^2+y^2))\) with homogeneous Dirichlet boundary conditions on the boundary of the square-shaped scatterer and first-order absorbing boundary conditions on the outer boundary. Plot the \(L^2\)-norm of the solution depending on \(k^2/\pi^2\in[0,11]\). How do you interpret the results? What happens if the right hand side is a Gaussian peak which is slightly off-centered?
Exercise 3#
Solve the time-domain problem corresponding to the Helmholtz equation from Exercise 2 for time harmonic right hand sides \(f(t,x) = \cos(\omega t)\exp(-30((x-0.2)^2+y^2))\) for \(\omega=\sqrt 2\pi,\sqrt 3\pi,\sqrt {5}\pi\). Plot the \(L^2\)-norm of the solution with respect to time.
Exercise 4#
Argue that the discrete resonance problem corresponding to Exercise 2 is of the form: find \(\mathbf x\in\mathbb C^N,\omega\in\mathbb C\)
for certain matrices \(\mathbf M_j\). Rewrite (3.3) as a (twice as large) generalized linear eigenvalue problem in \(\omega\) by introducing \(\mathbf y = \omega \mathbf x\). Solve the resulting linear generalized eigenvalue problem and plot the resulting eigenvalues and eigenfunctions.