Probe the Eigenproblem of a Wave Operator
Find the four smallest eigenvalues and eigenfunctions of a generalized wave equation over a 1D region.
Set up a generalized wave operator .
In[1]:=
![Click for copyable input](assets.en/probe-the-eigenproblem-of-a-wave-operator/In_14.png)
\[Gamma] = 1.3; c = 1.1;
op = D[u[t, x], {t, 2}] + \[Gamma] D[u[t, x], {t, 1}] -
c^2 D[u[t, x], {x, 2}] + \[Gamma] u[t, x];
Find the four smallest eigenvalues and eigenfunctions over a 1D region.
In[2]:=
![Click for copyable input](assets.en/probe-the-eigenproblem-of-a-wave-operator/In_15.png)
{vals, funs} = NDEigensystem[op == 0, u[t, x], t, {x, 0, \[Pi]}, 4];
Inspect the eigenvalues.
In[3]:=
![Click for copyable input](assets.en/probe-the-eigenproblem-of-a-wave-operator/In_16.png)
vals
Out[3]=
![](assets.en/probe-the-eigenproblem-of-a-wave-operator/O_8.png)
Visualize the real and imaginary parts of the eigenfunctions. Notice the eigenfunctions come in conjugate pairs like the eigenvalues.
In[4]:=
![Click for copyable input](assets.en/probe-the-eigenproblem-of-a-wave-operator/In_17.png)
Grid[Partition[
Plot[Evaluate[ReIm[#]], {x, 0, \[Pi]}, PlotRange -> .5,
PlotLegends -> {HoldForm@Re[f], HoldForm@Im[f]}] & /@ funs, 2]]
Out[4]=
![](assets.en/probe-the-eigenproblem-of-a-wave-operator/O_9.png)