Sprinkler shadows on the ground

Suppose you have an idealized American-style oscillating lawn sprinkler. It has a rotating cylindrical sprayer bar that emits a fan of water jets with a half-angle \(\theta\): that is, if \(\theta = 0°\), there is a single jet in the middle of the sprinkler, and if \(\theta = 90°\) it’s essentially spraying water in a semicircle.

Suppose that there are lots and lots of jets in the sprayer bar, which is of length \(L\), and that the water pressure emanating from all of these holes is identical. Ignoring the effects of drag and the sprayer’s rotational acceleration and jerk on the water jets, and also ignoring runoff and assuming a perfectly flat surface, what is the shape of the wet spot the sprinkler creates on the ground?

You’re going to want to solve a simple equality, right? What’s that resulting shape look like? Any given jet will be forming a parabolic trajectory; will its intersection with the ground be a familiar mathematical function? What about all of them taken as a whole? Concave edges? Convex edges? Straight edges?

Can you visualize it before doing the math?

OK. Now assuming the sprayer bar oscillates sinusoidally over time, what is the shape taken by the watering density on the ground’s surface over a long period? In other words, does more water land at the edges, the middle, or is it evenly distributed? If you render the water density as a surface (wetter being higher) what is the shape of the resulting function? Is this one a familiar shape?