# What is going on with this frequency-driven water stream?

In this video, a constantly flowing water source is attached to a speaker attached to a sine wave generator, and the water is observed with a strobe light matching the Hz (water “stands still”), 1Hz below (water “moves backward”), one 1Hz above (water “moves forward”). I get that.

The water stream also moves forward in a curve radius lengthening vertically and horizontally. (My mathematical description of the geometry is surely wrong, but you get the idea.) Gravity and additive waves?

Also, how do you account for the stream short length portion of each cycle?Between peaks of the sine wave, ie the trough of the wave? And it’s length is determined by the generating force of the speaker relative to the water’s mass?

I would love to hear this described correctly as fluid dynamics.

I don’t believe it’s going backwards. I can’t think of the name of the phenomenon, but I think it’s the same as when wheels, on camera, appear to be going backwards when they’re at just the right speed. Probably something to do with the camera being at 24 frames per second and the speaker being at 23,000 cycles per second.

You can see the pipe is vibrating with the frequency too, so it’s shaking and causing the water to go in what would, to the naked eye, be a snaking stream. In this case turning the vibration off (returning it to naked eye status) also steadies the pipe.

Sampling below the Nyquist sample rate?

I’ve edited the title at the OPs request.

samclem

It’s the rhythmic pounding of the speaker driven by the electromagnets which vibrates the hose. The taping and positing of the hose allows movement in two directions, so, instead of simply going back and forth, it is going in circles. This creates a spiral stream. The stream has breaks in it because of the vibration of the speakers. Those breaks occur at the rate of 24 times per second. Thus, a camera taking pictures at the same rate of 24 Hz will capture those breaks at the same spot giving the appearance that the water is standing still.

When the rate of the speakers is changed the spinning of the tube and the rate of those breaks change giving the appearance of what was once frozen in time either going forward or backwards.

Exactly. It is the movement of the pipe that is the key - the entire thing is no different to waving a hosepipe back and forth to make snaking water patterns in the air, except that the frequency of wiggling the hose is 24Hz, rather than about the 1 Hz you could achieve with your arm. If you saw it without the stroboscope effect it would look like a fast random spray of water, more like the spray from a sprinkler.

The next part of the trick is normal stroboscopic analysis technique. This depends upon the actual phenomenon being observed to being very regular in its motion. Just waggling about at 24 Hz would not be enough, it needs to be so stable in its operation that every time the pipe sweeps across its motion (and that is 24 times a second) the pattern of water in the air is exactly the same - right down to the individual droplet and rivulet in the air. This is the bot that is counter-intuitive. We are not used to the idea that the seemingly random spray of water you see falling is actually highly deterministic, and in one second the pattern has repeated so perfectly, 24 times, that it appears to be the same set of droplets and patterns of swirling water in the air slowed down or frozen in space. But it isn’t. In every frame of the movie every apparently frozen droplet iat a given location s actually a new droplet of water, the previous one having fallen, and actually is occupying the location of the droplet below.

The fluid flow phenomenon you want is “laminar flow”. If the flow in the pipe was turbulent you would not get this effect, but the pipe is nice and long and the flow rate in the pipe low enough that the water is flowing evenly, and is not turbulent. In this state it will tend to behave very deterministically, and as described above, every wiggle of the pipe will yield a close to perfect pattern of motion of the water as it falls. Obviously it isn’t totally perfect, and the pattern of falling water has some slight imperfections that yield an illusion that the “frozen” falling water is a little bit alive.

After that - you get to play with the frequency of the stroboscope - and that gets you the ability to create slow or reversed apparent motion - and indeed is identical to the wagon wheel effect. One term for the apparent result of mixing the frequencies of the physical motion and the stroboscope’s frequency is to heterodyne. You get artefacts from the missing that are the sum and difference of the two frequencies. So with 24 Hz as the frame rate a 25 Hz wiggle of the water yields an apparent frequency of observed motion of 25+24 = 49Hz - which we can’t observe, and and 25-24=1Hz, which is forward motion with a period of one second. When the speaker is set to 23Hz you get 23-24= -1Hz, which is an apparent period of one second going backwards.

That should, obviously, be 23 cycles per second. I was half awake and thought it said MHz. Even if that didn’t seem right for a speaker that big.

I believe this is the stroboscopic effect.

Oh, guys, I’m sorry. The phenomenon of the apparent motion I understand (when I said “I get that” I meant I understood the reason, not the just the setup).*
It’s the second and third grafs of my post where I get fouled up. If this has already been answered above, let me know.

*I thank samclem for correcting the title, and also apologize once again to him, this time for resending that request just now. My screen hadn’t refreshed. What a morning.

The curve is a simple parabola - same as any object that has horizontal motion that is then dropped. Ignoring losses in the air you will have a constant horizontal component (given by the direction of the hose as driven by the speaker) and downward acceleration by gravity, which is where the squared term of the parabola comes from. The wiggling of the hose simply provides a range of horizontal components, and so a set of parabolas, bounded by the pair dictated by the extremes of the hose’s motion. Since the water is accelerating downward the wavelength in the air of each wave increases, and because the angles from the pipe diverge the amplitude of the waves increases.

I can’t actually parse the second question.

Thanks. Let’s see if I can draw the stream for my second question.

Imagine in downward parabola. I believe this a graphic of what I see:

## –

What creates the middle curve “the short one”, the more horizontal (wrong word again) spit of water? Is it “the wiggle of the hose” when the sine wave approaches (and? Or?) goes toward its trough, where I’m thinking (incorrectly?) that no signal is being generated?

I don’t see what you draw, but agree that what we see in the video is not a symmetric sine wave. I think this comes about for a couple of reasons.

One, the hose isn’t just vibrating in and out, it seems to be describing a more circular movement - or at least elliptical, and the water is thus describing more of a helix than a simple sine wave as it falls. The camera angle sometimes captures this, and yet from some other angles it ends up with one side of the helix looking flat, and thus the wave looks more like a cycloid than a sine.

Two, I don’t think the speaker ends up vibrating the hose in anything like a symmetric fashion. It is all pretty rough, and it seems pretty likely that the actual wave motion is asymmetric anyway. The speaker is probably being run quite hard, the whole mess of duct tape and hose is probably pushing the speaker cone well off centre, so that it is way past its linear range in one direction, and so on. So part of the problem will be that we never actually got a sine wave in the first place. But perspective from the camera clearly plays a big part in what we see as the motion isn’t just in one plane.

I’m also not quite sure which aspect of the image is under scrutiny, but…

The water leaving the hose has velocities in the two “sideways” directions that varies sinusoidally and a velocity in the “out the snout” direction due to the water flow. These velocities alone would make a widening helix pattern, but on top of this is superimposed the parabolic shape from falling. The resulting bent, widening helix is then viewed at a particular angle. This is enough to explain the bulk shape. A few minor details like bulges every half cycle can be seen forming from the water sloshing in the snout in the close-up 23 Hz or 25 Hz sections, but I don’t think that’s what you’re asking about.

I don’t understand what you’re not clear about.

There’s no fluid dynamics going on here – at most, just some surface tension keeping droplets together.

Each droplet falls according to (a) the impetus from the flow plus the speaker movement at the moment it leaves the tube and (b) gravity. So, as mentioned above, each droplet falls in a parabolic arc.

If we were to watch this without a strobe (eyeball it), we’d see the water splatter like all heck, in sort of a narrow cone (a 2D cone, if you get my drift.) Take a still photograph, and we’d see a picture much like the 24Hz case, only … well, a still photo. Watching a 24Hz video, we see a flip-book sequence of such still shots, which shows some structure caused by surface tension.

It’s just harmonic motion to begin with, and parabolic descent from there.

This is to me, because I obviously am not a Great Communicator, even if it is more :smack: for you, because I’m fumfa-ing around and not clear. (Is "fumfa-ing Yiddish only? I don’t even know. Meaning clear I think from context).

I’m just checking in. I thank all of you for your time. When I have a little more time myself I’ll take another whack at what I’m thinking about.

You know when you’re dealing with a child just learning to speak who is frustrated about not getting his thoughts across? I will try to use my words…

I don’t know if it helps, but for the fun of it, I threw together a quick parametric graph of the principle behaviors mentioned.

See this image. I just played with the few free quantities, with some guidance from the video, until it looked in the ballpark, so it’s not a perfect match, but it shows the sort of shape you can get using only:

• sinusoidal velocity of the nozzle in each of two directions
• “muzzle velocity” of the water
• gravity
• camera angle

The water leaving the hose is given velocity in the horizontal plane in a constantly changing direction, but this repeats, so you can divide your illustration into three parabolas.

## Third line

Now all of these are linear instead of parabolic, but that’s because you drew them that way and because these dashes don’t really given enough resolution to do a parabola anyway.