Let's examine the underdamped case experimentally using a method
that we call a sine sweep.
Recall that 20 times the log of the amplitude is the same
as taking 20 times the log of the output amplitude
over the input amplitude of the two sine waves.
So when we do a sine sweep, we look at the input amplitude and
the corresponding output amplitude at various frequencies.
Now let's go ahead and do our sine sweep.
Let's take a look at the input-output behavior of this circuit.
The input is shown in green here, on the oscilloscope, and
the output is shown in blue.
You notice that they have virtually the same amplitude.
This is for a frequency of 1000 hertz.
They're almost on top of one another.
If I increase the frequency of the input very slowly,
in a sine sweep, see, I'm increasing the frequency, and
you see what's happening to the output amplitude, it's getting bigger.