Standing Waves Simulator

What Is the Standing Waves Simulator?

How to use the standing waves simulator

1. Read the four cards before you touch anything. They are already answering: Wavelength of this mode 1.60000 m, Frequency of this mode 250.000 Hz, Fundamental frequency 250.000 Hz and Nodes and antinodes 2 nodes, 1 antinode. The first three carry the relation they come from on their own label, and that label changes when the boundary does.
2. Count the marks on the drawing against that last card. A node is a dot on the axis with its position in metres printed beneath it; an antinode is a gold tick at the peak of the faint envelope. At the opening state there are two dots, one at each end, and one tick in the middle, which is what 2 nodes, 1 antinode means in the picture.
3. Now move Mode number n alone and watch which card refuses. Climb it to 3 and the drawing gains loops, the wavelength card falls to 0.533333 m and the frequency card rises to 750.000 Hz — while Fundamental frequency and The first four modes keep the strings they already had. That one slider is the whole argument of the page.
4. The mode number steps in whole numbers, and it has to. There is no mode 2.7 on the slider because a fraction of a half wave does not fit between two ends; the read-back prints a bare integer, and The mode condition row writes out the whole number it used. The standing waves and harmonics guide is where that condition is set out in prose, with the node positions and the worked problems.
5. Now move Wave speed v alone and watch the picture ignore you. Every frequency on the panel changes and not one shape does: same wavelength, same dots, same ticks, same bracketed span. The medium decides how fast the pattern runs through its cycle, and nothing else about it.
6. Switch Boundary at the two ends and watch all six readings follow. Free at both ends is the surprising one: against Fixed at both ends it gives the same wavelength, the same frequency, the same fundamental and the same spacing, and only the counts differ. One end fixed, one free is the one that moves the numbers, halving the fundamental and deleting every even rung.
7. Watch what the bracketed span claims in each boundary. Wherever two or more nodes are drawn it brackets the first two and labels them with Node spacing. Free at both ends in mode 1 draws only one node, so it brackets the two antinodes instead and the label says antinode spacing over the same figure. The half-open case in mode 1 has no such pair, so the span is dropped.
8. Read the compact strip as a decomposition, not as more evidence. The mode condition shows the geometry with its whole number in it, Node spacing (half a wavelength) restates half of the first card on purpose and its label says so, The first four modes is the bottom of the ladder in one unit, and Verdict is one line about the state you are in.
9. Use Pause when you want to look rather than watch. It stops the single animation loop and leaves a clean static frame, which is the right state for counting dots and reading the span. The button then reads Play; press it again to restart. Nothing on the panel is moving while the pattern swings, so no reading changes either way.
10. Reset is the way back when too many controls have moved to remember. It returns all three sliders and the boundary to their opening values, zeroes the pattern phase and leaves the loop playing. Touching any slider by hand drops the preset highlight, so no button can claim a state the cards do not show.

Dragging is the right way to see which card answers to which control, and a poor way to hit an exact figure. When you already have two of the three quantities and want the third typed out, the standing waves calculator solves the same relation for whichever one is missing, with unit menus on every box and the stopped-pipe instruction written out.

Worked example: change one thing at a time

Nothing in the tables below was worked out here. Each row is a state the lab was put into on this page and then read back from, cell by cell, so every string in them is the panel’s own. The first table prints the four cards for five published states: the first three isolate one control each, and the last two are each one half of a pair that the pairs table below completes.

Rows one and two share every control but the mode number, and that is the argument. Same length, same medium, same ends — and the fundamental column holds 250.000 Hz on both rows while the frequency beside it goes from 250.000 Hz to 750.000 Hz. Climbing the ladder does not move the ladder.

Rows one and three are the pair worth pressing yourself. Only the boundary changed, and three of the four reading columns are byte-identical down the two rows: the same wavelength, the same frequency, the same fundamental. Only the counts swap, from 2 nodes, 1 antinode to 1 node, 2 antinodes. Two pictures that look nothing alike, one set of numbers.

Row four is the one that moves the numbers rather than the picture. With one end stopped the pitch is 171.500 Hz and the wavelength has doubled to 2.00000 m; the 343.000 Hz the same column gives open at both ends is a row in the table below rather than this one. Compare row five with the water column in that second table and only the medium changes, which the wavelength column does not notice: 2.00000 m in water and 2.00000 m in steel, while the pitch crosses the kilohertz seam to 2.98000 kHz.

The strip, for the same five states

The four cards answer what; the strip underneath them shows the geometry those answers came out of. Read the mode condition column down the first two rows and the length on the right-hand side never changes — only the number of half waves packed into it does. The ladder column is where the stopped column gives itself away.

The fourth row is the only one whose ladder skips. Its rungs are 171.500, 514.500, 857.500, 1200.50 Hz, and the 343.000 Hz that the same column sounds with both ends open is not among them. The even-numbered rungs of a stopped column are not faint; the boundary never admits them at all. Note the / 4 in that row’s condition where the others carry / 2.

The whole ladder is printed in one unit, and that is a decision rather than an oversight. The opening row reads 250.000, 500.000, 750.000, 1000.00 Hz rather than switching its last member to kilohertz, because the row exists to show the 1 : 2 : 3 : 4 and a unit change half way along would hide it. The steel row is in kilohertz throughout for the same reason.

The two pairs the table above leaves half-finished

Two of the five rows are each one half of a pair, so here are both halves. The first two rows are the same half-metre air column, open and then stopped; the last two are the same metre of length carrying sound in water and then in steel. The node spacing column is the one to watch in the second pair, with one caveat: every row here is mode 1 with at least one end free, so not one of them draws two nodes — the column is printing half a wavelength, which the drawing brackets between the two antinodes in the open rows and drops altogether in the stopped one.

The octave is exact on the panel, not approximately exact. Press the two air-column buttons in turn and the pitch goes from 343.000 Hz to 171.500 Hz, which is a halving with nothing left over. Stopping one end turns it into a node, which forces a quarter of a wavelength into the length where half of one fitted before.

The medium pair is the one that proves the shape is none of its business. Water and steel differ by about a factor of four in speed, and the wavelength card, the node spacing and both counts are byte-identical across the two rows while the pitch goes from 741.000 Hz to 2.98000 kHz. Half a wavelength is the quickest wavelength measurement available, but only where two still points are drawn to measure between — and none of these four rows draws two: the open rows bracket the two antinodes instead, and the stopped row brackets nothing. The wavelength formula and units guide is where such a measured gap becomes a number you can use.

Formula and symbol reference

The lab prints its relations on itself, in a fixed line under the strip. With both ends alike the wavelength of mode n is lam = 2L / n; with one end fixed and one free it is lam = 4L / (2n-1). In both cases the frequency is f = v / lam, and the node spacing is lam / 2.

One route computes every frequency on this panel, and that is worth knowing before you read the cards. The wavelength is worked out first and the frequency is the speed divided by it, so the half-open case has no second formula to disagree with the both-ends case about. The n in f_n = n v / (2L) therefore lives inside the wavelength step, which is also why the speed cannot reach the wavelength at all.

Three buttons, several objects each. Fixed at both ends is a clamped string, held still at each end; Free at both ends is an open pipe, or — taken longitudinally — a bar with both of its ends free; One end fixed, one free is a pipe closed at one end. The buttons name the ends rather than the object because one boundary genuinely covers all of those at once. The symbol you may have met for a wavelength is written lam everywhere on the panel, so that no uppercased label can mangle it.

Where the wave speed itself comes from is a different question and a different page. That is a property of the material and not of the mode, this panel takes it as given, and for a string it is set by quantities that appear nowhere here — the guide to wave speed in different media is where that belongs. No period is printed here, so anyone arriving with one needs a single division before this panel can take it: the frequency formula guide is where that conversion is set out.

Ten rows follow and only three of them are things you drag. Four are cards, three are rows of the compact strip, and none of the seven has a control of its own. For the sliders the right-hand column gives the end stops and the step; for everything else it gives the strings the lab prints and the span it prints them over.

If you read one row twice, make it Node spacing. It is the only entry that is openly half of another reading, and its label says which half, so the panel is not quietly offering the wavelength twice and calling it two measurements.

The physics: why only certain frequencies fit

Only the wavelengths that fit a whole number of half waves between the two ends survive, so only a whole-number ladder of frequencies sounds and nothing in between does. That is the condition the first strip row writes out, and the right-hand side of it is always the length you set. Everything else on the panel is a consequence of it.

The mechanism is a reflection: the wave that runs out comes back, and the outgoing and returning waves add all along the length. That is one clause and no more on this page, because the general case is a subject of its own. Each particle of the medium, meanwhile, runs simple harmonic motion about the place it sits at rest, while the pattern as a whole goes nowhere.

Four readings on this panel answer to four different things, and the controls are arranged so you can see it one change at a time. From the test runs rather than from a card: the dependency grid was checked at a tolerance of zero on every non-mover, over 720 speed pairs, 756 mode pairs, 720 length pairs and 288 fixed-against-free pairs, so these are byte-identical strings rather than figures that agree to six digits.

The medium moves the three frequency readings and nothing else. The mode number moves everything except the fundamental and the ladder. The length moves everything except the counts. And the boundary moves all six, which is why switching it is the one action on the panel that can surprise you twice in a row.

The relation tying a speed to a frequency and a wavelength is the travelling-wave guide’s subject, not this page’s: here the speed is typed in and the wavelength is read off, and no travelling wave is drawn. The wave generator lab is the contrast worth pressing — there a crest really does move along the medium, and there are no fixed nodes anywhere on it.

Where standing waves break down

Every position these three sliders can reach is answered, and answered to the full precision of a double. The arithmetic is not where this page can let you down; the physics at the edges of it is — an end that does not really reflect, a medium that changes along the way, and a question about loudness or ringing that no mode condition can take.

Three boundary buttons, and not one of them is an endless medium

A wave running along a medium with no end never settles into a pattern; what builds one is the return off a boundary. One boundary gets you a pattern but not a ladder, because the condition the first strip row writes out needs a length between two ends. Both ends of the drawing are always marked for that reason, and the boundary buttons never offer a state with only one.

The panel has no card for how loud, how hard or how long

The mode condition says which wavelengths fit. What starts a mode off, how hard it answers and how long it rings on once you stop are three questions it does not touch, and no card here answers any of them. Nor does it reach the size of the swing, which is the amplitude guide’s subject and stays there.

A medium that varies along its length breaks the one-speed assumption

One slider sets one speed, and the drawing applies it from end to end. Where the medium itself varies — a string that thickens, a column warmer at one end than the other — there is no single wavelength for the panel to draw and no clean ladder for it to print, and nothing on screen would warn you. The length you give it has to be one the medium is uniform over.

The first four modes row is an idealisation, and stiffness is what it leaves out

Anything that resists being bent, as a real string does, carries its short-wavelength modes a little above the exact multiples the ladder row prints, so what you are reading is the idealisation. How far above is a question this panel cannot be asked: no figure for it on any particular instrument is published on this site, and none is invented here — verify before use.

Where the drawing puts the antinode tick, a real pipe does not

Look at where the gold tick sits in the last figure on this page: hard on the mouth of the pipe. In a real open pipe it sits a little outside, which makes the pipe behave as though it were slightly longer and drops its pitch below what this panel returns for the length you measured. Same rule as the row above — the size of that shift is not published on this site, so no number for it appears here; verify before use.

The relation is arithmetic, so some reachable states are exact and absurd

Nothing in the relation asks whether the mode is audible, or whether a real medium would carry it at that speed. Pushed to its corner the slider product reaches 240.000 kHz — a test fixture rather than a physical case, and quoted here only as the arithmetic ceiling of the three sliders. The verdict refuses to leave that unremarked: past the printed limits it stops naming the boundary and says the mode is ultrasound, or infrasound at the other end, using the roughly 20 Hz to 20,000 Hz band the guide to how sound waves travel sets out.

The pattern’s height on screen is a drawing convenience and means nothing

The envelope is drawn tall enough to read and no taller, and the rate the live curve swings at is slowed so that it can be watched. Neither carries units, neither has a readout, and the fixed line under the panel says so rather than leaving it as a silent gap. Read the dots, the ticks and the cards; do not read the height.

The bracketed span is not always measuring between two nodes

It is wherever the drawing has two or more node dots, which is most states. Free at both ends in mode 1 puts a single node in the middle, so the span is taken between the two antinodes instead and its label says antinode spacing over the same figure. One end fixed and one free in mode 1 has no half-wavelength pair on the drawing at all, so the span is dropped rather than drawn as a measurement the picture cannot support.

Two engines agree readout for readout, and Safari was never run at all

From running the lab in two browsers rather than from a card: the published states were put side by side in Chrome and in Firefox, 19 assertions passed and none failed. There is a hole in that, and it is Safari: the WebKit build on this machine is installed but cannot be launched without administrator rights, so Safari was never exercised and nothing here is evidence about it.

Where standing waves are actually used

Finding out what a length will sound at before you cut it

This is the question most people arrive with, and it is two slider moves. Set the length, set the medium, leave the mode number at 1, and Fundamental frequency is the pitch that length will give: 250.000 Hz for the opening string, 100.000 Hz for the 1.200 m one. Shortening the vibrating part is the only thing a finger on a fretboard does.

Telling the harmonics of one string apart without retuning it

Press the first two buttons in turn. The pitch goes from 250.000 Hz to 750.000 Hz, three loops appear where there was one, and the fundamental card does not budge. That is what a player means by touching a harmonic: the same string, the same tuning, a different rung of the same ladder.

Two presses on the air-column buttons drop the pitch by an octave

Press A half-metre air column, open at both ends and then The same column, closed at one end: 343.000 Hz becomes 171.500 Hz and the ladder loses every even rung. That is why a stopped pipe of a given length sounds an octave below an open one, and why its tone is a different thing rather than the same tone lower. The 343 m/s those two presets use is the speed of sound in air this site publishes.

Telling the material apart from the geometry with two preset buttons

Press the water column and then the steel one. The wavelength card, the spacing and both counts hold their strings while the pitch goes from 741.000 Hz to 2.98000 kHz. Anyone who has to decide whether a measured pitch is telling them about the geometry or about the material has that separation in front of them in two presses.

Getting a wavelength out of a gap you can actually measure

The still points on a real vibrating object are the easiest thing on it to find, and the gap between two of them is half a wavelength in every boundary this panel offers. Read Node spacing against the dots on the drawing, double it, and that is the wavelength: 0.266667 m of spacing on the third harmonic of the opening string, against a 0.533333 m wavelength card.

Reading the verdict row when the arithmetic runs past hearing

Because the relation is arithmetic it reaches past hearing in both directions, and the verdict row says which side you are on using the band this site publishes. Pull the length up and the speed down and it reads that the mode is infrasound at 0.500000 Hz; push the other way and it reaches the ultrasound band that sonar and medical imaging work in. The decibel calculator is the tool for how loud rather than how high.

What else a length will hold, four rungs at a time on the strip

Anything with two ends and one speed along it has a mode condition of this shape, which is why the panel prints the bottom four rungs rather than only the one you asked for. Microwave and radio cavities are sized by exactly this arithmetic, and are named here and no further. Not every resonator is a length between two ends, though: the Helmholtz resonator calculator handles the cavity-and-neck sort, which this panel cannot draw and should not be asked to.

Where to go next

A panel can show you which readings a control is allowed to move, and that is the end of what a panel is for. Standing Waves and Harmonics is where the subject itself is written out: what a standing wave is, where the nodes sit, the harmonic ladder in full, the worked problems and the misconceptions. The calculator that takes those figures typed heads the list of related tools below.

Three things this panel leans on but never explains have pages of their own. A wave that actually goes somewhere, and how its speed, pitch and wavelength are tied together, is the transverse and longitudinal waves guide. The motion of one particle rather than the whole pattern is the simple harmonic motion guide. Why the second slider reads what it reads for a given material is the guide to wave speed in different media.

Further out, the wavelength formula and units guide takes the wavelength by itself, the amplitude guide is the height this panel refuses to print, the guide to how sound waves travel is where the audible band in the verdict comes from, and the speed of sound guide covers the 343 m/s that the two air-column buttons sit at.

One lab on this site looks startlingly like this one, and the resemblance is real. The infinite-well lab solves the same eigenvalue problem — only whole numbers of half waves fit between two walls, with the nodes marked — for a different subject: its readings are about a probability density rather than a pitch, and it prints no wavelength and no frequency. It is already published and is not changing, and that is the whole of the claim.

Beyond that there is the simulation library to work through, an index of calculators for when typing beats dragging, and the blog for everything written out at length.