1D incompressible fluid mechanics

Imagine an incompressible fluid surface in 1D - if the line is completely flat and let’s say the default density at a point is 5. So essentially our surface can be defined by f(x) = 5. Now if I ‘push’ down on the line at x = 2 (ie add some positive and negative velocity at x = 2), the surface would rise at x < 2 and x > 2 due to advection of density using the velocities just introduced. I’m trying to model this with the ability to add force like I just described. Can anyone explain how I would implement this?
I’ve looked into 1d heat diffusion and I’m pretty sure that’s what I’m looking for - I’ve implemented this and it solves diffusion just fine but I’m a bit confused on how to add forces to the model. I don’t know why I’m having issues wrapping my head around adding velocities and density advection w the velocities in 1d even though I’ve done it in 2d… I was going to reimplement navier stokes in 1d but I’m certain there’s a more simple way to model what I’m describing.

Doesn’t necessarily have to be physically accurate, just visually pleasing and maintaining a constant sum of densities.