Consider the second order differential equation as follows $$ \frac{d^2 f(t)}{dt^2} = \alpha(t) f(t), $$
where the variable $t$ is time. A result very often used in Physics is that, if $\alpha(t)$ varies in time much faster than $f(t)$, one can replace it by its average in time
$$ \frac{d^2 f(t)}{dt^2} \approx \overline{\alpha(t)} f(t), $$
where
$$ \overline{\alpha(t)} = \frac{1}{\pi} \int_0^\pi \alpha(t) dt.$$
This result sounds very intuitive to me, but I can't seem to prove it. Can I have a hint or a reference for this?