Okay, some physics maven please explain this to me, and also why a few of the patterns formed by the sand are asymmetrical. (I haven’t looked up the Chladni plate experiment.)
From io9, where the notes say this:
Stop what you’re doing and watch this. It’s a video of sand. Sand skittering around on a vibrating plate, to be exact. But what happens when that sand skitters is amazing. Trust us – this is something you want to see.
What you’re watching is the Chladni plate experiment, as performed by YouTube science-and-illusion wizard Brusspup (he can also coax water into a zig-zagging stream, and make Rubik’s Cubes that aren’t Rubik’s Cubes).
Oh, and as a special treat—because you’re good enough, you’re smart enough, and doggonit, people like you—here’s the Rubik’s cube illusion and other anamorphic illusions mentioned above. Be sure to watch for the felid at the end:
h/t: Chris
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Interesting article about what might be medieval Chladni plate designs decorating a church. http://www.slate.com/articles/life/the_rosslyn_code/2011/05/the_rosslyn_code_3.html
At a given frequency, the plate vibrates at resonance, with different regions vibrating in opposite directions. The bounds between these regions are not vibrating, so the sand collects there (and if it moves slightly away, it bounces back to the nodal lines). Different frequencies produce different bounded regions, and the number of regions increases with higher frequencies.
Yes. If you think of a hose tied to a wall being oscillated at the other end, at resonant frequency there will be parts of the hose that do not move up and down at all (where sand will collect) in between the parts that move up and down the most (where sand will be propelled away with the greatest velocity). At different frequencies the length of the wave changes, and the motionless spots do as well. Think of the hose as a single cross section of the metal plate.
The plate is square, so the distance from the vibrating center to the edge of the plate is not totally uniform. The length of the “hose” changes depending on which cross section you look at. That is why you do not get spherical patterns. Also, a single cross section is influenced laterally (the plate does not fly apart), so it can be tricky to calculate exactly what the pattern will be if the distance from the source of vibration is not uniform. There may also be variations in the thickness of the plate to cause irregularities in the resonant pattern.
The only striking asymmetry I noticed was during adding more sand, which did not spread out uniformly.
Bah, I said “spherical” patterns, I should have said “circular”.
Oh the frequencies and the wave length — this makes sense now – thanks.
I would assume the asymmetry is due to imperfections in the plate causing slightly different resonant frequency boundaries across it.
Though I would await confirmation of that, I could be wrong.
I have a hunch it might more likely be due to the plate not being mounted in its exact center. That forces an anti-nodal point right there, and you’ll notice that one is always present. That is, no sand ever gathers on the mount point.
b&
Oh, this one’s easy.
Imagine a jump rope. Swing it fast enough and you get two loops going on either half. Swing it faster and you get more and more loops.
The same things’s happening to the steel plate; it’s getting vibrated the same way as the jump rope is. But the plate isn’t a rope, so the vibration patterns are different.
The sand is gathering in the spots with the least amount of vibration, analogous to the midpoint of the rope when you’ve got just two loops going.
And, just as there’s a series of speeds where the rope is stable and it’s not stable between those, there’s a series of frequencies where the plate is producing stable harmonics, which is the frequencies the woman was calling out. You should be able to plot those and find a very definite pattern.
Last…this is the exact same principle by which all musical instruments work when producing sounds using the overtone series. It’s something possible with any instrument, but only the brass rely upon it. Listen to any bugle call, and the pitches you hear are very directly analogous to both the patterns in the jump rope and on this metal plate.
Cheers,
b&
Yeah, that’s totally easy. Must be embarrassing for our host to have admitted in public that he didn’t understand it. /sarcasm/
Erm…sorry if I came across as snarky, but that wasn’t at all my intent.
As a (semi-)professional trumpet player, this is within my personal area of expertise. Explaining harmonic resonances is as easy for me as explaining DNA encoding and transcription would be for Jerry, or as explaining quantum virtual particles would be for Larry Krauss, and so on.
We can’t all be experts at everything, but of course there’re plenty of things that we can be experts at, and lots of thing in the field are easy for experts even when they’re mysterious to non-experts. Naturally — they wouldn’t be experts if that wasn’t the case!
Cheers,
b&
This is basic “standing wave” phenomena. A simple one-dimensional example is a vibrating string (such as a guitar string). Those who have studied basic physics will recall that there are specific characteristic points on a wave–crests, troughs and nodes. The nodes correspond to points where there is no vibrational motion for the wave. For an open guitar string, one of the nodes rests at the twelfth fret. The nodes in the case of the standing vibrational waves that are formed on the plate are actually called “nodal lines” in this case. The sand rests on these lines because they are not pressed by the vibrational motion of the plate, as they would be at all other points.
The mathematics necessary to explain this is of the type you would encounter in a course on partial differential equations. In mathematical terms, there is a second order partial differential equation (called the wave equation). There are specific waveforms that can exist at specific frequencies (called eigenfrequencies), and the shape of these waveforms is determined by two things: (1) where the vibration is emminating from–in this case, the center of the plate–and, (2) the conditions imposed on the wave at the boundary of the plate (called “boundary conditions” in math-y terms).
The rest is a bit more esoteric. If you could see my white board right now, I could explain it a LOT better.
JW
Yep; it is a nice application of Fourier analysis.
Reblogged this on Enlightening, i'n't it? and commented:
interesting. very very interesting
I wonder if octaves produce the same sort of patterns.
The nodes only appear at every multiple of the fundamental frequency (which would be a fraction of 345Hz), the other frequencies just make a mess. They are not exactly octaves because there will be some irregularities in the structure of the plate, but close enough. I’m guessing they did not go below 345Hz because the plate would be vibrating quite vigorously, the sand would bounce up and down too much to fall into a pattern.
Standing/stationary waves! The sound waves are reflecting back from the edges of the plate, interfering with themselves in a regular pattern. Where the sand gathers is an node, the plate isn’t vibrating. Two waves moving towards each other at certain points one wave will be going up and the other down, the resultant amplitude at that point on the wave is 0, so no vibrations to knock sand away. Where there is no sand the plate is vibrating as you would expect. I hope this made sense.
You can do a similar experiment in most rooms, if you clear the walls from books first, set a speaker placed at ear level to play a pure tone (best effect between 20-200hz) and walk around the room. Your ear will find a node where the tone will be very quiet. The sound waves will be doing the same thing to the air in your room as the sand on the plate.
Presumably a circular plate would give quite different (simpler?) patterns.
Offhand I think a circular plate with the sound-generator moving vertically at the centre, like that square one, would give only concentric rings of sand. Struck from the side like a bell, though, it would give interesting patterns.
As several commenters have noted, the experiment is forming standing waves in the near square plate, with the forcing coming from the center. Salt is transported to the standing nodes.
The nodes are found by solving the wave equation for a solid (here a square thin membrane) and the boundary condition of a loose edge. I have never been forced (heh) to solve a 3D problem with a point forcing, but I’m fairly certain that you need to incorporate that too. Typically a radial equation as here results in some form of Bessel function.
Asymmetries forms because of imperfections, because the amount of salt will vary over the plate both stochastically and because of the imperfections, and because the change rate in forcing will vary between choice of nodes and how it affects the different wave regions.
An extensive account of Chladni’s life and work, deep on scientific and cultural context (although heavily Germanic in style), is found at:
http://monoskop.org/Ernst_Chladni
Of particular interest are the links to the original illustrations of Chladni’s work on acoustics, as well as its aftermath, including the fundamental 1909 paper by Walter Ritz, who derived equations for Chladni figures from a least action principle, which I always found an admirable physical insight.
Of anecdotal interest: the link to Hans Jenny’s work, a pioneer in the field of cymatics. As my first exposure to Chladni figures dates back to a demonstration by Jenny shortly before his death, I recall my triple tiers of puzzlement: the wonders of acoustics made visible through Chladni figures, the rigorous experimental setup by Jenny, and the mind-boggling anthroposophical fluff with which he garnished his explanations.
Thank you for that link, I loved the section on Rejection of the monochord. In Newton’s Principia, Book 8 Acoustics, where Newton wrote his analysis “follows the law of the pendulum,” today we would say “a time-harmonic factor is implied and suppressed” in our notation. But a time-harmonic analysis doesn’t explain physics I care about, like how a musical instrument starts a sound, and much of our perception of a sound depends on how the sound starts. Your link says, “This dismissal of the Pythagorean monochord and calculation is Chladni’s revolution, his modernity.”
Also that page shows bowing a Chladni plate as some science before we had instruments to specify frequency in cycles per second. For Chladni to bow a given spatial pattern, he must have been watching the plate, listening for the musical pitch of a given response, and giving the bow the right speed and pressure to get a visual result. That required some eye-ear-hand coordination!
Demonstrations of Chladni that I saw in the 70’s used a violin bow (I remember one with a metal plate in the shape of a violin) before introducing electronics.
Reblogged this on Mark Solock Blog.
I’ve used FEM/FEA software for many years to determine the resonance frequencies (among other things) of race car chassis and frames. It’s called modal analysis, or frequency analysis. All structures will have natural harmonics, but the more complex the structure, the more difficult it is to find those harmonics. In this instance, the location of the mount, thickness, density, size, and flatness of the plate will determine the harmonics and node locations. As stated by others, each harmonic is a combination of competing forces cancelling each other out. Where they cancel out is where the sand will wind up. If the plate has even slight variations of any of the above, the resonance frequencies for that area will be different than other areas of the plate.
The asymmetries above 4100 Hz seem to be largely due to asymmetrical distribution of sand; there seems to be quite a bit more on the half-plate nearest the camera. A generous salting around 4600 Hz seems to restore much of the missing symmetry.
I wonder to what extent the mass of the sand acts as a secondary forcing function that favors patterns with nodal lines where the sand is already piled up. So the side with less sand can mutate faster and change patterns more often than the side with more sand.
If the plate was circular and really really uniform, you would be looking at Bessel Functions.
I’m not sure if there is a math name for square plate waves, two-dimentionial Fourier Functions perhaps?
I call them vibration isolation nightmare functions.
I presume that researchers have set up a numerical model that can predict the Chladni plate experiment patterns versus given frequency inputs.
I looked around on Google for a nifty graphing applet that could graphically draw patterns versus a given frequency, but could not find any.
Maybe some gentle readers could point me in the proper direction for an online applet. 🙂
Ha, I found one after all!
http://www.phy.davidson.edu/stuhome/jimn/java/modes.html
I liked the Stuart Smalley reference and the crazy anamorphic illusions that accompanied it.
I knew when you asked for a physic mavens help it would be good,and it was.LOved the music better then the sand or salt that was moving around.
When I teach freshmen chemistry students about atomic orbitals, I sometimes make reference to standing wave patterns on drumheads and the like, such as you can see here. If you go to from which this is linked you can learn more about acoustic waves. The salt (sand?) collects in te nodes of standing waves. As for the particular frequencies (and associated wavelengths), those are the particular frequncies at which the plate can oscillate (as in the old “is it live or is it Memorex” commercials).
…go to the page from which…
What happens when the plate and its mounting are circular? Or triangular?
I found more diagrams of Chladni figures online. They remind me of blackwork embroidery patterns.
What patterns appear in the 20-25 hz range (cat purr frequency)?
At those frequencies the wavelength will be measured in tens of meters, not centimeters. So there simply isn’t room for any standing wave patterns to form. Instead, the entire plate basically moves up and down in unison.
Yes, except that real objects (the plate) always generate higher harmonics.
“What happens when the plate and its mounting are circular? Or triangular?”
Different patterns. Probably visually interesting ones at different frequencies.
Anything you change in: Driving frequency, plate shape, stiffness, weight, or uniformity, or the supports for the plate will change the patterns and the frequencies at which the visually interesting ones occur.
Of course the anamorphic illusions concluded with a cat…
Standing wave, resonance . . . check out the Tacoma Narrows bridge collapse, 1940!
Galloping Gertie, as they called it!
The Tacoma Narrow Bridge (the original one) had insufficient torsional stiffness. There is no net applied torsional loading, from either the dead load or the live load, on the bridge deck, and it was therefore simply missed in the design. (They forgot about the torsional loading that would be imposed by wind.) It’s an example of aerodynamic flutter: Resonance in the structure combined with alternating loading from the varying angle of attack. Many airplanes have shaken themselves to pieces by the same effect.
sub
Wikipedia has an article on what is pretty much the same phenomenon, just on a circular membrane with stationary points along the full circumference, and no force driving the oscillation from the middle. They have a basic discussion of the maths and good illustrations of the effect. In the example above, sand is driven to the nodes (stationary points) by the horizontal component of the impacts with the membrane, with gravity ultimately balancing the vertical component.
http://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane
The morphing into various ‘designs’ is a treat for our pattern recognition abilities! Most of them would make excellent embellishing of tiles and wallpaper, lending an Art Deco flair.
Plate vibrations (out of plane). The areas clear of particles are the maximum amplitude zones, the particles collect in the node (minima) areas.
As the frequency driving the plate is changed, the vibration pattern changes, producing a different sand pattern. There are various patterns given the stiffness of the plate, its size, and the wave length of the driving sound. They usually drive with a signal generator and a pure-sine wave (or as pure as the generator, amplifier, and speaker can generate).
There is no requirement that the pattern be symmetric. Many objects vibrate assymmetrically. And: Perfect symmetry does not exist in the real world in middle-world sized objects. (I.e.: Don’t expect symmetry.)
Some people believe that they can improve (for instance) the sound performance of their guitars (or violins) by observing these patterns on the top plate (usually, or back plate) and adjusting the plate/bracing to acheive certain patterns at certain frequencies. Although it seems theoretically possible, I am extremely skeptical of this claimed ability.
JBlilie,
Oh I don’t know, percussionists seem to do much the same thing in tuning their kettledrums, by tweaking some sort of tensioners (I’m not a drummer) around the edge.
‘Course, they do this by ear, not by sprinkling sand on the drumheads. Presumably they are hearing overtones as well as adjusting the fundamental resonance. The former influences the “quality” of the sound.
We need a program to calculate these patterns. With different plate shapes and different placements of the vibrating point.
For example I presume a circular plate will just give concentric rings. But what if the vibration is off center?
A circular plate can haz other modes of vibration, with stationary nodes on a diameter (slicing the circle into 2, 4, 6 etc. equal segments). Timothy Hughbanks above has linked to slo-mo vids of drum heads oscillating, and mentioned analogies with atomic orbitals, which can be spherically symmetric or have 2, 3, 4 or more radial ‘lobes’ (antinodes).
Reblogged this on foolsmusings and commented:
Cool
I’m pretty sure you have them at your sports fixtures too, but anamorphic advertisements sprayed into the grass are common at our big games, and readily give the illusion of standing upright in the TV presentation, which is bizarre when a player runs “through” them.
I assume they are simply done by projecting the advertisement on to the grass from the same vantage point that the camera will have, and painting over the distorted image on the grass.
Actually, I think the fancy designs on the field are now done with some sort of oversized mobile inkjet printer. At least, it’s the way I’d do it.
b&
(A bit long-winded, but it should cover it..)
As many have already said, these are the stationary vibration (In the sense that the spatial pattern is constant and evey point simply oscillates up and down at a set frequency) modes of a square plate driven at its center, and the sand accumulates along the nodal lines (Those parts of the spatial pattern that have 0 vibration amplitude).
The answer to the question of the assymetrical patterns isn’t as simple as we would like to think – “The plate+mounting are not perfectly symmetrical”.
Though this is critical to the solution, it does not tell the whole picture.
The problem of the stationary vibration modes can be stated concisely as an eigenvalue problem (http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) relating the oscillation frequency, what is known as the wavenumber of the oscillation, and the pattern of oscillation.
In general, the higher the frequency, the higher the wavenumber correlating to it. The wavenumber is in a sense the reciprocal of the wavelength exhibited by the system, so as the frequency of excitation rises, a higher wavenumber solution is exhibited – that is to say, the effective wavelength (spatial extent) of the solution is shortened, and we see patterns with more and more small scale features.
For certain discrete frequencies, there will be a solution. For some frequencies there will be several solutions differing in form. For instance, for a certain frequency we’ll call f, there could be two solutions, A1 and A2. Since they are both solutions for the same frequency f, they are termed degenerate solutions.
A1 and A2 could be like mirror images of each other, though each of them is assymetrical with respect to the plate as a whole.
Physically speaking, if we oscillate the plate at some arbitrary frequency, we’re trying to excite a stationary mode of vibration by delivering energy to the system, until such point where the amplitude of vibration is large enough so that the rate of dissipation to heat energy exactly cancels out the energy we’re exciting the system with.
Oscillating at our degenerate frequency f, we would excite both mode A1 and mode A2. If the details of the excitation are exactly symmetrical (square plate, centered mount), they would be excited to an equal degree and we would see a symmetrical image.
If, however, the plate is not exactly square, or the mount is not exactly centered, there is some breaking of this symmetry. The modes A1 and A2 themselves would not be changed to any noticable degree, but the degeneracy, the fact they are both excited at frequency f, would be broken! Each would now be excited at slightly different frequencies, f1 and f2, close, but not equal to f.
This effect of “degeneracy lifting” can even be seen in the video!
At around the 2:00 mark, you can see the patterns at 4129 Hz and 4173 Hz are nearly identical. Seeing how the natural unit of frequency is around 345 Hz for this particular plate, the difference between the two could very well be a form of degeneracy lifiting around some intermediate frequency, ~4150 Hz.
And one final point on the issue of harmonics – the plate does have some base frequency, 345 Hz, but since it is not a simple system such as a taut string, it is not likely that it will exhibit a series of simple harmonics, just as a circular drum does not exhibit them… For me to explain this I would need to resort to a tiny bit more mathematics, so I’ll just stop here…
Reblogged this on Sarvodaya and commented:
This is the most remarkable thing I’ve seen in awhile; science as an artform.