In the spring, while Joab's army besieged Rabbah, where did David stay?

Answer: Jerusalem

At the time when kings go out to war, David sent Joab with the army but remained in Jerusalem, where he saw Bathsheba.

“In the spring, at the time when kings march out to war, David sent out Joab and his servants with the whole army of Israel. They destroyed the Ammonites and besieged Rabbah, but David remained in Jerusalem.”— 2 Samuel 11:1 (BSB)