Soft matter / 03
Why jelly wobbles and steel does not
Steel rings. Jelly wobbles. The difference is not that one vibrates and the other does not, it is that one vibrates far too fast to see.
Two ways to deform something
You can squeeze a material, changing its volume, or you can shear it, sliding one layer past another without changing its volume at all. Materials resist these very differently, and gels resist them about as differently as it is possible to.
A gel is mostly water, and water is nearly incompressible. Squeezing a jelly means forcing water out through pores so fine that it will not go. So the resistance to compression, the bulk modulus, is essentially that of water: a couple of gigapascals, comparable to many hard materials.
Shearing is a completely different story. Sliding one layer past another only requires stretching the loose chains between junction zones, and those are easy to stretch. The shear modulus of a soft gel is measured in kilopascals. Published values vary widely with formulation, but a soft dessert gel of order a few kilopascals is a reasonable picture.
A jelly is roughly a million times easier to shear than to squeeze. That single ratio is the whole story.
Shear waves at walking pace
A wobble is a shear wave travelling through the gel, reflecting off the surfaces and coming back. The speed of a shear wave depends on the shear modulus and the density:
c = √(G / ρ)Put in numbers for a soft gel. Take a shear modulus of a few kilopascals and a density close to water at a thousand kilograms per cubic metre. The square root of a few thousand over a thousand is somewhere between one and two, so the wave travels at a metre or two per second, about walking pace.
Now do the same for steel, with a shear modulus near eighty gigapascals and a density around 7,800. The square root of that ratio is a bit over three thousand metres per second.
So a shear wave crosses a ten-centimetre jelly in something like a tenth of a second, and a ten-centimetre steel bar in thirty microseconds. The jelly oscillates a few times a second, which is squarely inside the range your eye resolves. The steel oscillates tens of thousands of times a second, which your eye cannot follow and your ear registers as a ring.
The jelly is not doing something exotic. It is doing the same thing, slowly enough to watch.
Why a big jelly wobbles more slowly
The frequency of the fundamental mode depends on how long the wave takes to cross the object and return, so it scales as the wave speed divided by the size:
f ∝ c / L = √(G / ρ) / LThe consequence is worth stating plainly. Double the size of a jelly, made from the same recipe, and the wobble frequency roughly halves. A large banquet jelly visibly heaves. A teaspoon of the same mixture quivers quickly. Nothing about the material changed; only the distance the wave has to travel did.
The same scaling explains why a firmer recipe wobbles faster. More gelatin means a higher shear modulus, a faster wave, and a higher frequency for the same size.
Why the wobble stops
A purely elastic object would oscillate forever. Jelly does not, because it is viscoelastic: partly a spring, partly a damper.
Rheologists split the response into two parts. The storage modulus, written G′, is the elastic part, the energy stored and given back on each cycle. The loss modulus, G″, is the viscous part, the energy converted to heat and never returned. Anything with G′ larger than G″ behaves as a solid; the crossover point where they are equal is used to define the moment a cooling solution becomes a gel.
The ratio G″/G′ is called tan delta, and it is a direct measure of how quickly oscillations die. A gel with a small tan delta rings on for several seconds. One with a large tan delta stops almost immediately and feels more like a paste.
On the interactive jelly, the internal damping control is exactly this knob. Turn it down and the specimen keeps wobbling for a long time. Turn it up and it settles almost at once. The firmness control changes the other term, the stiffness, which changes the frequency rather than the decay.
Why it holds its volume while it shakes
Watch the wobble closely and the jelly changes shape constantly while its volume stays put. That is the incompressibility again, and it is the reason the motion is a shear wave rather than a pressure wave. Materials that cannot change volume have only one way to deform, and every wobble you see is that one way, repeated.
This is also the hardest property to fake in a simulation. A model that lets volume drift quietly collapses into a bag, however good the surface looks. The interactive specimen enforces volume explicitly for exactly this reason.