To the left of the barrier, a single plane wave moving from left to right and incident on the barrier with angular frequency and wave number given be the de Broglie relations. To the right of the barrier, the superposition of two circular waves centered at each of the two slits radiating outward in concentric circles with the same frequency and wave number as the incident wave.
Use complex exponentials to represent all three waves.
For the plane wave:
u(x,t) = e i (kx - wt)