CS 493/693 Lecture, Dr. Lawlor
I claim simple wave dynamics requires three variables:
X variation in height drives X velocity: steep cliff facing right will resolve itself by fluid flowing to the right.
- X velocity
- Y velocity
Y variation in height drives Y velocity the same way.
Velocity "divergence" drives height: if all velocities are converging on one point, that point gets higher.
These three things should make waves. Let's try this shader:
This is exactly the discrete form of the partial differential equations at Wikipedia for the shallow water wave equations.
Sadly, this is unstable--it explodes into weird zebra patterns.
There's a simple fix: blur things out, by averaging the neighbor values.
vec4 blur=0.25*(L+R+T+B); /* neighborhood average */
Instead, Wikipedia recommends pushing back against velocities: big
velocities will tend to damp out this way, but you need to add so much
viscosity, nothing interesting happens. Piles of tar are not
known for their cool ripples.
Curiously, the unstable zebra patterns are almost prettier than the waves we're trying to simulate!
Here's the full 2D version, with X and Y change:
Initial conditions make a big difference: if you have nonzero X and Y
initial velocity, then your waves will move at that velocity.
float HtChange=L.x-R.x+B.y-T.y; /* = divergence of velocity */
The "ht" and "vel" constants are related: the ratio is the scale factor between initial height and velocity.
In a long simulation, everything slowly gets darker, I think due to
roundoff. We can pull the average closer to any value we like by
blending it in, similar to blur, although this destroys waves in
progress. It's better to add in a "stimulus" based on the
difference between the current average and your target (think Keynes).