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 θ: that is, if θ=0°, there is a sin­gle jet in the mid­dle of the sprin­kler, and if θ=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?

No Eng­lish house spar­rows to speak of this sum­mer. Maybe a pair. We typ­i­cally have a dozen or more, fight­ing over the birdseed.

Chip­ping spar­rows trilling every­where, all the time, audi­ble dawn to dusk. Flit­ting around on the ground, pick­ing over the seed leav­ings, squeaky-​​wheeling all the day. We typ­i­cally see one pair.

A rose-​​breasted gross­beak cou­ple in the area, mak­ing melo­di­ous improve­ments on the robins’ nor­mal tunes. Much more professional-​​sounding. Never seen one.

Not one house or pur­ple finch to be seen. Crow­less. Notably so. We used to have stream­ing night-​​flights of them, even in sum­mer, head­ing to the roost of the day. Star­ling short­age, too.

Dou­ble the nor­mal num­ber of blue­jays, and the dis­tinc­tive sounds of squalling babies from the nest­ing tree. New devel­op­ment, there.

No flick­ers.

Car­olina wren cou­ple, bob­bing about and pulling at the weed piles I leave around. Never seen one before this April.

Only con­stant: mourn­ing doves. Same three, far as I can tell, that have been here for a decade.

What gives?