WP Protected Page behavior

[HandsOn]

I checked WP Codex for support documentation to see if I had setup a protected page properly, it sure seemed simple enough, and apparently I had.

But I am a little surprised by the behavior of the password protection...

The visitor enters the correct password once and thereafter is never prompted again for the password unless cookies are deleted. I guess in some ways this could be considered a feature , however if several people are sharing an office computer and only one is allowed access well then this is a problem, unless cookies are routinely deleted.

The way I would like it to behave would see the visitor being prompted for the password with each new browser session.
(I would instruct the secure users to make sure they end their browser session once they are finished working otherwise others could get into the protected page).

Anyone have thoughts on this or recommendations?

This is my first website and I am not sure how to code the deletion of cookies once the session is terminated.

Note that the data that I am securing is not super sensitive but I thought that WP would provide better security than what I described above.

Thanks as always for the great support.
Donna