Transient Response Suppression
Ok, this is not an issue, though I would like to see user’s response to determine if it is useful or not. Let us solve it in another way. For author’s information:the reason that your transducer starts to ring is because you stop your drive too hard. The switch-off causes a step stimuli and it rings usually longer than you drive it inversely due to the high quality factor. Some of the signal waveform demonstrate the transducer is not driven at its central frequency, so you can see it not following the tuning. To solve your issue, all you need is a soft switch off at the end of the pulse train.

In case you don’t follow my suggestion, you may realise the drive through an analogue switching mechanism where at the end of pulse train, you switch to a capacitive drive branch to produce an damped waveform, this will make your transducer a lot quieter. I understand your work may have already finished, but you don’t need to struggle with the math, only an analogue implementation and the parametric choice of the capacitive drive.  

In fact, a lot more questions should be asked first before providing a solution, e.g. your coupling mechanism with the propagation medium which may be either solid or liquid, though that ringing is much different in either case. Looking at the central frequency of the transducer, this analogue switcher is quickest solve because it is too high for digital controlled active damper to sample and process. I believe some commercially available medical or Nondestructive test equipment have similar transient compensation if the ringing is a matter to the reflected wave, or if they want to switch the phase of the drive signal much quicker.