Lecture 16, 18.376, Tue Apr 11, 2023.
Summary
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Continue with Quasi-Modes; and finish section 3
    "Wave equation in an interval: Dirichlet/Radiation BC (example 3)"
of the notes: Examples of pde with the Laplace Transform.

Point out how the poles of the Green's function that give rise to the
"spectral decomposition" obtained in these notes [i.e.: equations (3.14)
and (3.15)] are only accessible because we are looking at the problem in
0 < x < 1, with a radiation BC at x = 1. If the full 0 < x < infty interval
is considered, the Green's function ceases to make sense for re(s) < 0
[same phenomena explained in prior lecture], and the poles cannot be
accessed.

Mention other similar examples, like quasi-modes in the atmosphere, if
a radiation BC is imposed at the tropo-pause [much harder problem, but
same mechanics].

At the lecture's end a question was asked about equations (3.14) and (3.15).
Equations seem puzzling, since
the alpha_n depend only on alpha = u(x, 0)
and the beta_n only on beta = u_t(x, 0)
yet u(x, t) depends on both ... even at t = 0.
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