$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
A simple vibration problem
$$
u^{\prime\prime}(t) + \omega^2u = 0,\quad u(0)=I,\ u^{\prime}(0)=0,\ t\in (0,T]
$$
Exact solution:
$$
u(t) = I\cos (\omega t)
$$
\( u(t) \) oscillates with constant amplitude \( I \) and
(angular) frequency \( \omega \).
Period: \( P=2\pi/\omega \).
