Water Waves Simulator

What Is the Water Waves Simulator?

How to Use the Simulator

1. Read the four cards before you touch anything. They are already answering for 10 m of water under a 100 m wave: Wave speed 9.32 m/s, Which water is this? in between, sqrt(g*d) gives 9.90 m/s, and Period 10.72 s.
2. Read the sub-line under each card, because that is where it shows its working. They read from the full relation, at 10 m deep, depth is 0.1000 of a wavelength, +6.22% against the full relation and one crest passes a fixed point every 10.72 s. The third of those is the reading this lab is built around.
3. Drag Wavelength all the way left and watch the second card, not the first. At 1 m the regime has gone to deep water, the speed has fallen to 1.25 m/s, and the third card still reads 9.90 m/s — now +692.67% against the full relation. The depth never moved, so that card never moved either.
4. Now drag Wavelength all the way right. It passes deep water, then in between, then at 200 m it reads shallow water with the ratio exactly 0.0500 and the shortcut +1.62% high. By 200 000 m the speed is 9.90 m/s and the shortcut is +0.00%.
5. Hold the wavelength and move Water depth d instead. Every card changes this time, including the third, because the depth is the only thing that shortcut contains. This is the asymmetry the whole lab turns on: one rule ignores the wavelength and the other ignores the depth, and the ratio decides which you are allowed.
6. Press Run and watch what it does and does not move. The button flips its own label to Pause, animates the time slider to 2.00 periods and stops there by itself. Every physics reading on the panel is untouched: a run advances the picture and nothing else.
7. Drag Time by hand to 0.50 periods and then to 1.00 periods. At a half period the marked crest has moved half a wavelength along; at a full period the surface is back to the shape it started with. That is the definition of the speed being drawn, not a measurement of it.
8. Press Tsunami crossing the deep ocean, then The same tsunami on the shelf, and watch only the Period card. The speed falls from 197.57 m/s to 9.90 m/s and the wavelength from 200 000 m to 10 000 m, while the period moves from 1012.29 s to 1009.64 s. That is shoaling in two clicks.
9. Press Reset when you have finished. It puts the three sliders back to 10 m, 100 m and 0.00 periods and replays the opening settle, so the scene is drawn in rather than cut to.

Dragging is the right way to find out which readings move together and a poor way to land on a particular figure, because both ladders step by whole rungs rather than by metres. The five preset buttons above exist for that reason, and the written guide to water waves is where the two regimes are defined from scratch and the problems are worked with every step shown.

One thing this panel cannot do is take a depth or a wavelength that is not on a rung, and the inverse questions are beyond it altogether. The water wave speed calculator takes typed figures, and it will also run the problem backwards: a wavelength from a timed period, or a depth from a measured speed, neither of which can be read off a slider.

Worked Example: The Same 4 000 Metres, Two Different Answers

Nothing in the three tables below was worked out here. Each row is a setting the lab was put into through its own sliders and then read back from, cell by cell, so every string in them is the panel’s own. The first table is the pair the whole lab was built around.

One: one depth, one shortcut, two answers it cannot both be right about

Press Tsunami crossing the deep ocean and then Wind waves far out at sea. Only the wavelength ladder moves between them; the depth stays on the same rung, so column six is the same string twice.

Read column six first and then stop. It is 198.09 m/s in both rows, because it is worked from the depth alone and the depth did not change. The water is identical. If that shortcut were the rule for water-wave speed, these two waves would travel at the same speed, and column three says they do not.

Column seven is the size of the mistake. One wave is answered to +0.26% and the other to +1485.33%, a factor of 15.85 out, by the same arithmetic in the same water. Column five is the only thing that separates them: 0.0200 against 40.0000.

Note which way the error runs. Both shortcuts are upper bounds on the true speed, so no percentage on this panel can ever be negative and no shortcut can ever make a wave out to be slower than it is. Over the whole 600-state grid the sub-line runs from +0.00% to +17624.54% and never once carries a minus sign.

Two: the five sea states, and what each one is for

These are the buttons above with the readings they produce at 0.00 periods. The last two columns are the ones to read across: the error column swings by four orders of magnitude while the period column stays legible.

Rows one and three are the same kind of wave, and nothing about their size says so. A 50 m swell in 2 m of water and a 200 km wave under 4 km of ocean both read shallow water, both are answered by the shortcut to about one per cent, and the ratio axis in the drawing puts their markers within a third of a decade of each other.

Row five is the one with no rule of thumb at all. Harbour chop at 2 m and 10 m reads in between, and the shortcut is +21.58% out — too wrong to use and not wrong enough to be obviously wrong. Across the whole grid 102 of the 600 states are in that position.

Three: the same wave in two depths, and the reading that survives

Press Tsunami crossing the deep ocean and then The same tsunami on the shelf. This table is transposed so that the two presets are the columns, because the interesting comparison is down a column rather than across a row.

The last row is the one that makes the table worth reading. The speed falls by a factor of 19.95 and the wavelength by exactly 20.00, while the period goes from 1012.29 s to 1009.64 s — a quarter of one per cent. That 19.95 is the ratio of the two speeds before the panel rounded them for display, so dividing the two figures as the table prints them lands slightly high. Crests arrive at the rate they were generated at, so the period is what a change of depth leaves alone.

That is also why the shelf preset carries a 10 km wavelength rather than the 200 km it started with. Pairing 10 m of water with a 200 km wave would describe something of period 20192.75 s, which is five and a half hours between crests and is not that tsunami or anything else. The wavelength shortens in step with the speed; saying it is unchanged is the easiest false sentence on this topic.

If the period is the reading that survives, it is worth knowing what else it fixes. The guide to the frequency formula is where its reciprocal is set out, and that is the quantity a buoy or a stopwatch actually records offshore.

Formula and Symbol Reference: Every Slider and Every Readout

The band at the head of this page is the full relation, and both of the famous shortcuts are approximations to it rather than laws of their own. Each one is quoted with the ratio condition that licenses it, because without that condition neither of them is a true sentence.

The symbol column below uses the panel’s own names for its readings. Where a reading is a quantity with a life outside this lab, the SI unit column says so; where it is a word or a count, it says that instead. The guide to the wavelength formula and its units is where the wavelength is treated as a quantity in its own right, which this page deliberately does not do.

If you read one row twice, make it the fourth. Everything else on the panel — the regime word, both shortcut cards, the size of the gap between them — is a function of that one ratio and of nothing else about the wave.

Reading the Diagram: Three Bands and Two Scales

The drawing is laid out as three bands and the bands are reserved before anything is placed in them, so no part of it is ever painted across another. The top band carries the classification, the middle band carries the surface, and the bottom band carries the ratio that decided the top one.

The middle band’s vertical is not to scale, and that is a statement about the model rather than about the drawing. Linear theory has no amplitude in it anywhere, so there is no height for the picture to be in proportion to. The caption keeps those words at every width the band is drawn at, and where even the bare clause will not fit, the band is not drawn.

The bottom band is a ratio on a logarithmic axis, which is a different kind of quantity entirely. It spans a millionth to ten thousand, which is wide enough that every reachable state sits inside it with room over, and the two boundary ticks are exactly one decade apart at every width. Nothing on it is a length in metres, so nothing on it may be measured against the band above.

Where the height of a wave belongs as a quantity, with a definition and a unit, is the guide to amplitude. It is absent here because the model cannot produce it, not because the drawing ran out of room.

The Physics: Why One Formula Has Two Faces

One function does all of this. The full relation contains a hyperbolic tangent whose argument is the depth divided by the wavelength, scaled by two pi, and that function behaves completely differently at its two ends. Everything the panel prints is a consequence of that single curve.

At the shallow end the tangent is close to its own argument and the wavelength cancels out of the arithmetic, so the speed is fixed by the depth. Hold the depth at 2 m and run the wavelength from 2 000 m to 50 000 m, a factor of twenty-five, and the first card reads 4.43 m/s at every single rung. Shallow water is non-dispersive, and the panel shows it as a column of identical strings.

At the deep end the tangent is close to one and the depth drops out instead, so the speed is fixed by the wavelength and longer waves are faster. Sit on 5 000 m of water: a 500 m wave reads 27.94 m/s and a 2 000 m wave reads 55.88 m/s. Four times the wavelength is exactly twice the speed, at a ratio of 10.0000 and 2.5000 respectively.

Between those two ends there is no shortcut, and the lab is blunt about how much room that occupies. Of the 600 reachable states, 202 read shallow water, 296 read deep water and 102 read in between. More than one state in six is a case where you have to use the full relation or accept an error you have not measured.

Even at the boundaries the shortcuts are approximations, and the panel prints the size of it. Set 5 m under a 100 m wave for a ratio of exactly 0.0500 and the third card reads +1.62% against the full relation; set 50 m for exactly 0.5000 and the deep-water row reads 12.50 m/s against a real 12.47 m/s. A boundary is where an approximation becomes tolerable, not where it becomes exact.

Why the depth should enter the arithmetic at all comes down to one mechanism, and this is the only line on the page that mentions it: under every crest the water itself runs round closed loops whose reach extends some way downwards, and a seabed shallow enough to cut across those loops is a seabed the wave notices. Anything further about how the water moves belongs to the guide to transverse and longitudinal waves, which is also where what type of wave this is gets settled.

Be blunt about what the simulator is doing, because a panel that keeps agreeing with itself can look like a panel that has found something out. It has not. The simulator measures nothing: it is told the depth and the wavelength and works out the rest, so the crest crossing one wavelength per period is the definition of the speed built into it rather than discovered by it, and the honesty line under the drawing says so in its own words.

What the two ladders are genuinely for is the other direction. They let you hold one quantity still and move the other, then watch the answer walk away from a shortcut you were about to trust — and that is a thing no still picture of a sea can show you. It is also why the time slider is a slider and not only a Run button: every instant is a state you can set and read.

Where It Breaks Down

Every state the two ladders can reach is answered, and the arithmetic is not where this lab can let you down. What it cannot do is model a real sea, and what it chooses not to do is treat its own idealisations as facts. Six limits sit under every reading on this page.

There is no wave height anywhere in the model, and that is the big one

Linear theory is a small-amplitude theory, so the height of the wave is not a quantity this lab contains. It follows that nothing here can tell you how tall a wave gets, how much energy it carries or how far up a beach it runs, and no figure for any of those appears on this page. A shoaling wave does rear up — the speed and the wavelength both fall and the energy has to go somewhere — but the direction is all that is supported, and the drawn surface in the middle band is a fixed pixel height for exactly this reason.

Nothing here predicts breaking

A real wave breaks when it becomes too steep, and that is precisely the point at which a small-amplitude theory stops applying. So the moment you most want a prediction is the moment every figure on this panel stops being valid. There is no steepness criterion printed anywhere in the lab, because the model cannot produce one, and a lab that printed one would be inventing it.

The depth is one number and a real seabed slopes

One constant depth under the whole wave is what lets a single speed exist at all. A real beach shelves continuously, so the speed changes continuously with it, and the two-click shoaling comparison in the third table above is a snapshot of a gradual process rather than two states any wave is ever in at once. The lab has no way to represent a slope, and the companion calculator has no way either.

The wavelength ladder stops at 1 m on purpose

Gravity is the only restoring force in this model, and for very short waves on water it is not the force in charge: surface tension is, below roughly 1.7 cm of wavelength. Rather than answer in a region where the physics is the wrong physics, the ladder’s lowest rung is 1 m and the lab simply cannot be taken there. The guide to surface tension is the force that takes over at the bottom of that range.

Four things the open sea has that this model does not

Wind, flow, friction within the water and the turning of the Earth are all outside the relation, so the first card gives a speed measured against still water and against nothing else. Put a tidal stream under the same wave and the crests reach you at a different rate; this lab has no box for that and will not do the addition. Whether the medium rather than the depth is what changes is a different question entirely, and the guide to wave speed in other media is where it belongs.

The first card is the speed of the crests, and the third grid row is not the same thing

The pattern advances at the speed on the first card; the water itself goes nowhere in particular and the energy travels at the speed in the third grid row. In deep water those two differ by a factor of two — 12.50 m/s against 6.25 m/s on the sea preset — and quoting the first as the speed at which something floating would be carried is the most common way to misread this panel.

Where the Depth Rule Is Actually Used

Working out how long a wave takes to cross an ocean

Warning systems compute arrival times from a depth chart, because in the shallow-water regime the speed is the depth and nothing else. The lab gives the scale of it: at 4 000 m, with a ratio of 0.0200, a 200 km wave moves at 711.26 km/h, and at 10 m the same wave is down to 35.66 km/h. A deep basin is crossed in hours and the last few kilometres are not.

Turning a buoy’s period record into a forecast inshore

Offshore instruments record the period, which is exactly the reading a change of depth leaves alone, so a period measured far out still describes the wave when it arrives. The shoaling table above is that in two settings: 1012.29 s against 1009.64 s while the speed falls twentyfold. Anyone working the other way — wavelength from a period — needs the inverse the sliders cannot do.

Deciding whether a dredged channel will change how waves arrive

Deepening a channel makes the water faster for a shallow-water wave, so the part of a crest over the channel runs ahead of the part beside it and the crest turns. The direction is always the same: a crest swings towards the slower, shallower side. The panel gives the two speeds but not the angle; for the swell preset’s period, this cluster’s reference computation turns a 30-degree arrival offshore in 50 m of water into 7.60 degrees on a 2 m shelf, and the guide to refraction is where the law behind that belongs.

Knowing when a long wave will keep its shape across a shallow sea

In the shallow-water regime every wavelength travels at the same speed, so a long disturbance crossing a shelf sea does not spread out into its components on the way. The lab shows it as a column of repeated strings: at 2 m of depth, the first card reads 4.43 m/s for every wavelength from 2 000 m to 50 000 m. That is why surge and tide-scale waves are modelled with the depth alone.

Explaining why the long swell from a distant storm arrives first

In deep water longer waves are genuinely faster, so a storm’s output sorts itself out over the ocean and the long stuff reaches the coast days ahead of the short. The lab’s own numbers at 5 000 m: 27.94 m/s for a 500 m wave against 55.88 m/s for a 2 000 m one. Set the depth shallow instead and the sorting stops happening at all.

Working out which waves a harbour entrance actually has to stop

Two waves in the same approach depth can be in different regimes, which is why a marina screen that works against swell can fail against chop. In 2 m of water a 50 m swell reads shallow water at 4.38 m/s while a 10 m chop reads in between at 3.64 m/s, and only one of those is predictable from the depth. For waves crossing into a different medium rather than a different depth, the wave speed lab is the one to open.

Where to Go Next

A panel can put you in a state and let you move one thing at a time, and that is the end of what a panel is for. Water waves is where the subject itself is written out: both regimes defined separately with their own conditions, the decision between them, the shoaling chain worked through, and problems with every step shown.

Two guides sit either side of this one. The guide to wave speed in other media is the separate question of what fixes a wave speed when the medium rather than the depth is what changes, and the guide to transverse and longitudinal waves is where what type of wave a water wave is gets answered properly.

Four more pick up what this lab leaves out. The wavelength formula and units guide treats the wavelength as a quantity, the frequency formula guide is the period’s reciprocal, the amplitude guide is where the height of a wave belongs, and the surface tension guide is the force that takes over below the ladder’s lowest rung. The refraction guide is the law behind crests turning towards a shore.

On the tool side, the water wave speed calculator takes typed depths and wavelengths and runs the problem backwards as well as forwards, the standing waves calculator is the other wave relation with a boundary in it, and the wave speed lab and the transverse and longitudinal waves lab cover the two neighbouring pictures. Beyond them there is the simulation library, an index of calculators, and the blog for everything written out at length.