Wave equation with decay band

Nick Trefethen, November 2010

Here are eigenmodes 1, 2, 10, 20 of the wave equation on [-pi/2,pi/2]:

LW = 'linewidth'; lw = 1.6; FS = 'fontsize'; fs = 8;
L = chebop(-pi/2,pi/2);
L.op = @(u) diff(u,2);
L.bc = 'dirichlet';
nn = [1 2 10 20]; nmax = max(nn);
[V,D] = eigs(L,nmax);
for j = 1:4
  n = nn(j);
  v = V(:,n);                 % pick out nth eigenvector
  v = v/norm(v,inf);          % normalize to have amplitude 1
  lam = D(n,n);               % nth eigenvalue
  subplot(4,1,j)
  plot(v,LW,lw)
  axis([-pi/2 pi/2 -1.6 1.6])
  if j < 4, set(gca,'xtick',[]), end
  text(.3,1.1,sprintf('mode %d         lam = %6.3f',n,lam),FS,fs)
end

Here are the same, but for the wave equation with a decay band:

figure
a = 0.2;
x = chebfun('x',[-pi/2 pi/2]);
middle = (abs(x)<=a);
L.op = @(x,u) diff(u,2) + (2/a)*middle.*diff(u);   % decay band
nn = [1 2 10 40]; nmax = max(nn);
[V,D] = eigs(L,nmax);
for j = 1:4
  n = nn(j);
  v = V(:,n);                 % pick out nth eigenvector
  v = v/norm(v,inf);          % normalize to have amplitude 1
  lam = D(n,n);               % nth eigenvalue
  subplot(4,1,j)
  hold off, fill(a*[-1 1 1 -1],1.6*[-1 -1 1 1],[1 .8 .8])
  hold on, plot(v,LW,lw)
  axis([-pi/2 pi/2 -1.6 1.6])
  if j < 4, set(gca,'xtick',[]), end
  text(.3,1,sprintf('mode %d         lam = %6.3f',n,lam),FS,fs)
end