Finally, if that quantity under the radical sign is negative,

then we have the underdamped case and this is our solution.

Again we have the decaying exponential e to the -alpha t,

and it's multiplying a sinusoidal term which

oscillates at a frequency omega d, which is called the damped resonant frequency.

Now it's interesting to compare the quantity under this radical sign

with the quantity that we were using to determine

which is the natural response form.

This is the negative of that.

So in the underdamped case, alpha squared minus omega naught squared is negative,

but what we do is reverse the order, making a positive value

under the radical sign to define our damped resonance frequency.