Letters of Continued Interest

Hello everyone.

I was recently waitlisted at NYU Stern and Vanderbilt. I am curious how a letter of continued interest works and how I can write mine in an effective manner. What should I submit with my letter? To whom should I send it? How long should I wait before sending it? Should I submit my transcript and an additional letter of recommendation? I appreciate any input.

If you are able, please cater your responses to the schools I mentioned.