Hello folks, newbie here
It has been a long while… 
I have a 3 second piece of “siren sound” that I would like to extend.
But cutting and pasting doesn’t sound seamless, and the joining point
is clearly recognizable. What is the best method/approach to
accomplish this task?
Thank you for your help!
Goony 
Try this. Cut & paste, then hit the arrow to land exactly at the edit point. Maximize to where you can see the precise point of edit. The edit should be on the horizontal line. If it’s not, just clip the excess off. Be sure that you leave enough to make a complete sine wave cycle. There should be a full cycle containing one full positive curve and it’s counterpart - one full negative curve (above and below the horizontal line, respectively.)
You should be checking all of your edits at the microscopic level of view. All edits should be made at the greatest point of silence. Eventually, with a little practice, you will be making the edits down at that view level, choosing to point of silence first, then cutting and pasting. This should do for starters, especially with this kind of edit.
A [u]crossfade[/u] (fade-out, fade-in, and overlap) makes a smooth transition. Depending on the situation the crossfade can be a few milliseconds, or up to several seconds if you want to fade one sound into the other (such as with music if you want to slowly fade from one song to the next).
Of course a cross fade does subtract from the total length because of the overlap.
Or, to help with what Jebbers is suggesting, there is an option to [u]select/cut on zero crossings[/u]. (That’s not necessary with a crossfade.)
Thank you both for your brilliant comments!!! 
I am gonna try all of your suggestions…
Goony