Stationary States

Homework Statement

dJ(x,t)/dx = -dY^2 / dt , where y is the wave equation, and the d's represent partial derivatives. I want to make an assumption that I can describe the wave equation as a stationary state, so my question is the following:

What is the definition of a stationary state and can it be used to describe a free particle incident on a finite potential step from the left? This is not a specific question for the problem, but I need to know in order to make an assumption to solve it.

I'm not that clued up on this myself, but stationary states appear to be similar to solving for standing waves on a string.

The assumption is made that the wavefunction y(x,t) = Bsin(kx +/- wt) (B arbitrary, k the wave number, w the angular frequency, x position and t time) and the wave is bouncing back and forth within a potential well "box". The stationary states correspond to the particular circumstances that produce standing wave patterns.