Strengthening a half wall

First post here and looking for some advice:

Last year I constructed a half wall to better define a room in my home. The wall itself is free standing and not tied into any other walls and is 8 feet long x 3 feet tall. I constructed it by nailing and lag bolting a bottom 2x4 plate with studs every 16" and a top plate nailed to them. I have a double 2x4 on both ends of the wall. Wall is complete with drywall and an oak shelf on top.

So, the problem...as you probably could already guess...it wobbles. I plan to install hardwood floors in the upcoming weeks and would like to strengthen the wall some how.

My idea is...cut open both ends of the wall... add a double 2x4 to each end that will extend below the floor into the basement ceiling. (The wall runs parallel to the floor joists and falls in between 2 joists) I plan to sandwich the new 2x4's between two 2x6's in between the floor joists below and nail/lag bolt through all.

Will this work or am I wasting my time? Any other alternatives? The structure of the wall itself is secure...I believe my problem is the methold i used to secure it to the floor.

Thanks

 

Thats what I was going to suggest. seems like the only option.