What circuit can use a falling edge to trigger this damped oscillating impulse waveform?
Since you say up to 500 &micro;s delay from the trigger edge to your exponentially decaying ringing signal is acceptable, here is a solution:



When the input trigger comes along, you have something close the switch for 500 &micro;s.  This could be a microcontroller or a dedicated timing circuit like a one-shot.  During that time, current builds up in the inductor.

When the switch opens, the inductor current will get dumped onto C1, causing its voltage to rise.  L1 and C1 are a resonant circuit that continually slosh the energy originally in the inductor back and forth between each other, generating a sine voltage on C1.  R1 dampens these oscillations.  Instead of a endless sine, which would be the result of theoretical ideal L1 and C1, you get a sine with a decaying amplitude envelope as shown in by your scope trace.  The product of L1 and C1 set the frequency.  Once you have that you can select R1 to get the desired damping.

In practise, the switch is probably best implemented by a N channel FET, but could also be a NPN transistor.