The workbench.

Notes on the method

What a sweep is. One configuration - cell, fluid, contact line - held fixed while the drive frequency walks a range step by step. At each step the tool asks the same two questions the sandbox answers live: which pattern would the surface choose, and how hard must you shake for it to appear. A study that once meant a long computer run compresses into seconds because the catalogue of patterns is solved once and only the onset question is asked per step.

Sweeping the cell instead. The swept variable need not be the drive. Hold the drive fixed and walk the contact angle θ, the depth h, the surface tension σ, or the viscosity ν, and the tool answers the same two questions at every step - which pattern, how hard a shake - for a family of cells or fluids instead of a family of drives. There is a price: change the cell and every pattern's own frequency moves, so the whole catalogue must be rebuilt at each step rather than solved once. That rebuild is exactly what made such sweeps long computer runs in the research code; here it is tens of milliseconds per step, so a full cell sweep still lands in seconds. Two readings change with the axis. The half-the-drive line lies flat, because the drive no longer varies. And in the ladder, a faint dot now marks the cell where a pattern comes exactly into tune with the drive - patterns drift in and out of tune as the cell changes, and the winner bars show which one the surface actually takes. A θ sweep always runs the robin wall condition, since the contact angle acts through nothing else; sweeping through 90° passes the free-wall limit on the way.

Where the angle bites. The wall is a smaller and smaller part of what shapes a pattern as the pattern gets finer, so the contact angle picks the winner outright only where patterns are coarse - small cells, slow drives. Sweep θ there and the winner hands over several times; sweep it for a fine pattern at a higher drive and the identity barely moves while the threshold still does - in this site's standard cell at 24.4 Hz the winning pattern holds across most of the range while the shake needed to raise it triples. One counting note: a wetting wall (θ below about 87°) holds one extra low pattern with no spokes, so the ring counts of that family read one higher there - the research code's own convention, kept here. The charts stitch bands and tune-dots by physical continuity across that seam, so a change of label alone never draws a handoff or a dot; the CSV keeps each angle's own labels.

Sweeping the fluid. Surface tension can be the swept variable too, with the density and viscosity held. It walks the same way a cell sweep does, rebuilding the catalogue at each step, but it bites where the contact angle does not: surface tension is the part of a pattern's stiffness that grows as the pattern gets finer, so raising it lifts every fine pattern's own frequency, and to stay in tune with a fixed drive the surface answers with a coarser pattern - the winner hands over as you raise it. Two honest caveats travel with the axis. In the underlying relation surface tension enters only as the ratio of tension to density, so sweeping the density at fixed tension would trace the same curves mirrored - tension is the one exposed here. And holding the three fluid properties apart is a modelling move, not something a real liquid does: warming it, or adding alcohol, shifts tension and viscosity together. The compare mode rests on the same fact: it sets whole liquids side by side, and what separates them in the model is their tension-to-density ratio and their viscosity, never density on its own.

Sweeping the viscosity. Viscosity is the odd one out, and usefully so. It is absent from the frequency relation entirely, so it never retunes a pattern: thickening the fluid does not move where the patterns sit. What it moves is the threshold - the shake needed to raise a pattern - which climbs steadily as the fluid gets more sluggish, roughly in step with the viscosity. So a viscosity sweep is read on the onset chart, not the ladder: the A_c curve rises, answering "how much harder must I drive this thicker fluid." Two consequences follow. Because the patterns never move, the catalogue is solved once, not rebuilt per step - the fast sweep on this page. And the ladder carries the winner bars but few or no tune-dots: with each pattern's frequency all but fixed in viscosity, little comes into tune as the fluid thickens (the Wilson model adds a small viscous shift that can nudge one through; Lamb leaves the frequencies exact, so none do). The winner can still hand over, though: a fine pattern's damping grows faster than a coarse one's, so past a certain thickness the surface gives up the fine pattern for a coarser one it can still raise.

Two ways to get each pattern's frequency. Every answer downstream starts from the catalogue: each pattern's own natural frequency. Rayleigh-Ritz finds it by energy balance - the research method behind this site, good to about 1% against laboratory measurement. The dispersion relation is the textbook shortcut: quicker to write down, but it misses by 2-45% depending on the pattern and the contact line, worst for the pinned waterline and the lowest patterns. The shortcut is offered here deliberately, so the two can be swept side by side and the difference seen rather than asserted.

Why wrong frequencies wreck thresholds. This is the unintuitive one. The onset curve of each pattern is a narrow valley in the plane of drive frequency against shaking strength, and the valley floor sits at exactly twice the pattern's own frequency. Shift that frequency by a few percent and the whole valley moves sideways - so at a fixed drive you may no longer be standing in the valley you thought, but on the wall of another. The winning pattern changes; the predicted threshold can be wrong by a factor of ten; and the error runs in both directions, because a misplaced valley is sometimes accidentally closer to the drive than the true one. Choosing the dispersion shortcut together with the Mathieu onset solver is therefore a controlled experiment: the onset machinery is identical in every respect, so whatever separates the two curves is frequency error alone, made visible. A few percent in frequency does not stay a few percent - valleys are sharp, and it becomes wrong winners and tenfold threshold errors.

Two dampings. The threshold depends on how fast the fluid drains motion. Lamb (1932) is one classical formula for every pattern; Wilson et al. (2022) computes each pattern's own friction from its shape - which is why it needs the Rayleigh-Ritz shapes and is unavailable under the shortcut. Both count friction in the body of the fluid only: real dishes rub at the walls and the waterline too, so measured thresholds sit above these predictions - they are honest lower bounds, as everywhere on this site.

Reading the charts. With the Mathieu solver on, the first chart is the roof over all the valleys at once: each dip is one pattern's valley floor, each climb ends where the next pattern undercuts it, and the short ticks along the top mark the drive frequencies where the crown changes heads. In the modes-only register it is a staircase instead: the winner's own frequency hugging the half-the-drive line, each tread the band of drives one pattern captures. Below either, the same sweep is drawn as the mode ladder - spoke count up the page, drive frequency across. The faint dots are every pattern the cell can hold, each at twice its own frequency: the rungs. The solid bars are the winners, and each capture band sits astride its rung - the surface climbing the ladder rung by rung as the drive rises, which is the shape this whole site keeps describing, here drawn by measurement. Most dots never earn a bar: at any one drive several patterns sit near resonance at once, all sharing the one surface, and the one that reaches its onset first - best in tune, least friction - takes it; the rest stay candidates. A bar ends the same way: not because its pattern stopped resonating, but because a neighbour began resonating better. The CSV download carries the full ledger for every step, with the configuration and the stated limits written into its header.