  Two different methods are used to study the existence and stability of the
(1+1)-dimensional $\Phi^4$ oscillon. The variational technique approximates it
by a periodic function with a set of adiabatically changing parameters. An
alternative approach treats oscillons as standing waves in a finite-size box;
these are sought as solutions of a boundary-value problem on a two-dimensional
domain. The numerical analysis reveals that the standing wave's
energy-frequency diagram is fragmented into disjoint segments with
$\omega_{n-1} < \omega < \omega_{n-2}$, where $\omega_n=\frac{2}{n+1}$. In the
interval $(\omega_{n-1}, \omega_{n-2})$, the structure's small-amplitude wings
are formed by the $n$-th harmonic radiation ($n=2,3, ...$). All standing waves
are practically stable: perturbations may result in the deformation of the
wave's radiation wings but do not affect its core. The variational
approximation involving the first, zeroth and second harmonic components
provides an accurate description of the oscillon with the frequency in
$(\omega_1, \omega_0)$, but breaks down as $\omega$ falls out of that interval.
