Standing Waves Calculator

What Is the Standing Waves Calculator?

How to use the standing waves calculator

1. Choose the unknown. The Solve for menu opens on Frequency of this mode, and the other two choices are Wave speed in the medium and Length of the string or column. Any two of the three, plus the mode number, fix the third.
2. Type the length. It takes m, cm, mm or in and opens on 0.800 m. Measure the vibrating part between the two ends, not the whole string on the reel.
3. Type the wave speed. It takes m/s, cm/s, km/h or ft/s and opens on 400 m/s. This page takes the speed as given rather than deriving it from anything.
4. Type the mode number. A whole number from 1 to 8, an input in every mode, so it never leaves the form. A typed 2.7 is rounded to 3 and anything above 8 is held at 8, and the working names the number it used.
5. Read the answer. The headline always carries four significant figures, with trailing zeros dropped, in the unit that box was showing — switching to scientific notation outside the range 0.0001 to 10,000,000. Set the unit while the quantity is still an input and then point Solve for back at it.
6. Read the wavelength chip first. It is 2L/n, and at the defaults it reads 1.60000 m. Change only the wave speed and it does not budge, because the ends set the shape and the medium only sets the pitch.
7. Read the fundamental chip. At the defaults it reads 250.000 Hz, and at n = 3 it still reads 250.000 Hz while the headline climbs to 750 Hz. Climbing the ladder does not change the ladder.
8. Read the counts chip. At the defaults it reads 2 nodes, 1 antinode; at n = 3 it reads 4 nodes, 3 antinodes. Those are the counts for a string fixed at both ends or a pipe open at both ends.
9. Read the ladder chip. It prints the first four modes in one unit so the pattern stays readable: 250.000, 500.000, 750.000, 1000.00 Hz at the defaults. Change the mode number and this chip does not move either.
10. Open Show working. The steps form the wavelength, rearrange the relation, and then give the ladder, the fundamental, the node spacing, the instruction for a pipe closed at one end and your figures in every unit the boxes offer.

This page exists for the one step that turns a length into a pitch: the mode condition. The standing waves and harmonics guide is the prose half of it, with the node positions, the worked problems and the misconceptions set out in full. Where the wave speed in the box comes from is a different question and a different page: the guide to wave speed in different media is what sets it for a string or a column, and nothing on this calculator derives it.

Three mistakes account for most wrong answers here. The first is using the mode number of the harmonic you can hear while measuring the length of the whole string; the second is entering a closed pipe's own length instead of twice it, which comes out an octave too high. The third is reading the fundamental chip as the answer when the headline is the mode you asked for — at n = 1 they agree, and at every other mode number they do not.

Worked example: change one thing at a time

The table starts at the defaults, moves the unknown through all three quantities, then changes only the mode number, only the wave speed and only the length, walks every unit menu, runs the closed-pipe trick, and finally reaches the entries the calculator declines. Rows 1 to 3 are one string solved three ways. Every answer and chip cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 to 3 are one string with the Solve for menu in each of its three positions, and all four chips are identical down all three because the chips describe the state rather than the mode. That is not free: they are worked out from the answer substituted back into the state, so the mode in which the length is the unknown still has a length to take a wavelength of. The residual is exactly zero in all three directions, because no rounding enters this relation anywhere.

Rows 4 to 7 climb the mode number and nothing else. The headline and the wavelength move, the counts move, and the fundamental and ladder chips do not budge from 250.000 Hz and 250.000, 500.000, 750.000, 1000.00 Hz. Rows 6 and 7 are the integer clamp: a typed 2.7 returns the same 750 Hz as mode 3, and a typed 12 the same 2000 Hz as mode 8.

Rows 8 and 9 move the length alone. Double it and the pitch halves to 125 Hz; halve it and the pitch doubles to 500 Hz, with the whole ladder moving with it. This is the one control that moves every figure on the page.

Rows 10 to 12 all print the fixed-at-both-ends counts: in rows 11 and 12 the Nodes and antinodes column counts the doubled length rather than the pipe, and in row 10 it stands in for an open pipe, whose counts are 1 node, 2 antinodes. The counts chip is the both-ends case, below, translates both. A 0.500 m column open at both ends sounds 343 Hz; the same column closed at one end goes in as 1.000 m and sounds 171.5 Hz, exactly an octave lower. Row 12 takes its next real mode with n = 3 and gets 514.5 Hz — and the 343.000 Hz sitting in that ladder chip is not a mode of the closed pipe.

Rows 13 to 15 change only the medium, or only the mode on a longer string. Water and steel on the same 1.000 m column return the identical 2.00000 m wavelength and the identical counts, while the pitch goes from 741 Hz to 2980 Hz — and the fundamental chip crosses the Hz-to-kHz seam to read 2.98000 kHz. Row 15 is a 1.200 m string in its fifth mode, sounding 500 Hz on a 100.000 Hz fundamental.

Rows 16 to 23 change nothing but the units, and every one of them returns the same string. One frequency reads 250 Hz, 0.25 kHz and 0.00025 MHz; the length typed as 80 cm, 800 mm or 31.4960629921 in is the same 0.800 m; and the speed typed as 40000 cm/s, 1440 km/h or 1312.33595801 ft/s is the same 400 m/s. Those are the round trips through the multipliers people most often invert.

Rows 24 and 25 are the two back-solves worth doing. A string sounding concert A at 440 Hz in a medium carrying 320 m/s must be 0.3636 m long; and a 0.900 m length whose second harmonic is 300 Hz is carrying waves at 270 m/s on a 150.000 Hz fundamental. Note how the fundamental chip in row 25 is half the frequency you typed, because you typed a second harmonic.

Rows 26 to 31 sit outside the domain and are declined there rather than answered. A zero or negative length has no mode behind it, a zero speed carries no wave, a zero frequency in the length mode would need an infinitely long string, a negative frequency a negative length, and a mode number below 1 is not a mode. In each case the widget says so in words instead of printing an infinity.

Formula and symbol reference

One condition does all the work here. A wave trapped between two ends is reflected back on itself, and the returning wave only reinforces the outgoing one when a whole number of half wavelengths fits the length: L = n lam / 2. Rearranged, that gives the wavelength of mode n as lam = 2L/n, and nothing else is allowed.

The frequency follows at once, because a wave of that wavelength running at the medium's speed completes v / lam cycles a second: f = n v / (2L). Rearranged the other two ways that is v = 2Lf/n and L = n v / (2f), and those are the three positions of the Solve for menu. Setting n to 1 gives the fundamental, f_1 = v / (2L), and every other mode is a whole-number multiple of it.

The wave speed is the medium's, not the mode's, and this page never derives it — the wave speed calculator is the tool that relates a speed to a frequency and a wavelength directly, and the speed of sound calculator is where the 343 m/s in air comes from. For a reader who arrives with a period rather than a frequency, the frequency formula guide is the conversion. Every particle on the pattern is meanwhile doing simple harmonic motion about its own rest position, which is a separate story told on its own page.

The page solves the both-ends case, and says so out loud. f = n v / (2L) covers a string fixed at both ends and a pipe open at both ends alike. A pipe closed at one end sounds at (2m-1) v / (4 Lc): to get that from this calculator, enter twice the pipe's length and use only the odd mode numbers — n = 1, 3, 5, 7 — because (2m-1) v / (4 Lc) is exactly n v / (2 x 2Lc) with n = 2m-1. That is an identity, not an approximation, and the lab draws the picture with its third boundary button.

One chip needs reading carefully when you use that trick. With a 0.500 m closed pipe entered as 1.000 m the ladder chip prints 171.500, 343.000, 514.500, 686.000 Hz, because those are the first four modes of the doubled length. Only the odd-numbered members of that row are modes of the closed pipe — 171.500 and 514.500 Hz — and 343.000 Hz is not one of its modes at all. The even harmonics of a closed pipe are not quiet; they do not exist.

The physics: why only certain frequencies fit

The ends are the whole story. A fixed end cannot move, so the pattern must have a node there; an open end is free to move as much as it likes, so the pattern has an antinode there. Either way the length is pinned to a whole number of half wavelengths, and a wavelength that does not satisfy that condition cancels itself out on the return trip instead of building up.

The mechanism is two waves running in opposite directions and adding — the outgoing wave and its own reflection — which is why a standing wave needs a boundary and never forms on an endless string. The pattern itself does not travel: the nodes never move, and no net energy is carried along the length, which is what separates it from the travelling waves of the transverse and longitudinal waves guide. That guide is also where the relation between a speed, a frequency and a wavelength is set out properly.

Four quantities on this page respond to four different things, and the chips are arranged so you can see it in one change at a time. Move the length and everything moves. Move the mode number and the headline and the wavelength move while the fundamental and the ladder stand still. Move the wave speed and every frequency moves while the wavelength and the counts do not change at all.

That last one is worth loading rather than reading. A one metre column open at both ends has a fundamental wavelength of 2.00000 m in water and 2.00000 m in steel — byte for byte the same chip — while its pitch goes from 741.000 Hz to 2.98000 kHz. Same shape, same nodes, same spacing; a four-times-faster medium only raises the pitch. The travelling wave lab is the contrast: there the pattern moves and there are no fixed nodes at all.

Where the standing wave formula breaks down

The arithmetic here is exact to the last digit a double can hold. What still goes wrong is physical: a boundary the boxes assume and the real object does not have, a wave speed that is not the same all along the length, and a loudness question put to a condition that only sorts shapes.

Nothing you can type describes a medium with no end

A wave that never meets a boundary never forms a standing pattern, because it is the return from one that builds the pattern up — a long cable with an absorbing termination is exactly that case, and nothing on this page applies to it. One boundary does better than none: the reflection off a single end gives a real pattern with its nodes half a wavelength apart, which is what a cable shorted at one end shows. What it does not give is a ladder of modes, because the length box wants the distance between two ends and there is no second one to measure to.

Four boxes describe a shape, and none of them describes a push

L = n lam / 2 says which modes fit. It says nothing about how a mode is set going, how strongly it responds, how loud it is or how long it rings on after you stop driving it. Those are separate questions with separate physics, and nothing on this page answers any of them — which is also why no amplitude appears anywhere, a point the amplitude guide takes up properly.

The speed box holds one number, and it has to be true everywhere

A string of varying thickness, a wound string over part of its length, or an air column whose temperature changes along it has no single wavelength and therefore no clean ladder. Pick a stretch over which the medium really is the same throughout, or treat the answer as an estimate and verify before use.

The ladder chip prints exact multiples that a real string will not quite hold

Bending stiffness is the part this model drops: a real string pushes back against being bent, not just against being stretched, and that stiffening lifts its short-wavelength modes a little above the whole-number multiples the ladder chip prints. Quantifying the shift takes a measurement, and this site carries none for it, on any instrument, so no figure is quoted here. Expect the ladder to be very good low down and progressively less exact high up.

A pipe sounds as though it were a little longer than the length you type

The displacement antinode of an open pipe sits slightly beyond the physical opening, so a real pipe's fundamental is a little lower than v / (2L) or v / (4 Lc) predicts. The mechanism is honest; the size of the correction depends on the bore and is not published on this site, so no figure for it appears here. Treat a pipe answer as slightly sharp of reality.

Every state this relation reaches is arithmetically right, including the inaudible ones

Nothing in f = n v / (2L) knows whether anything can hear the mode or whether the medium could carry it. A short length at a high speed and a high mode number returns a frequency far above the roughly 20 Hz to 20,000 Hz a healthy young ear covers, as the guide to how sound waves travel sets out, and a long slow one returns a frequency far below it. Both are arithmetically right and worth a second look.

The counts chip is the both-ends case

The n + 1 nodes and n antinodes it prints are those of a string fixed at both ends; a pipe open at both ends has exactly the same wavelength, frequency, fundamental and spacing with the counts the other way round. A pipe closed at one end needs its own mode number recovered from the typed one as m = (n + 1) / 2, because the doubled-length trick types n = 2m-1 for the pipe's own m-th mode, and that mode has m nodes and m antinodes: n = 1 is 1 node, 1 antinode rather than the 2 nodes, 1 antinode on the chip, n = 3 is two of each, n = 5 three of each and n = 7 four of each. The frequencies hold at every mode number for the two both-ends cases and at the odd mode numbers only for a closed pipe, and the lab's third boundary button prints the closed-pipe counts directly.

A hyphen or a refusal is an answer, not a fault

A zero or negative length, a zero or negative wave speed, a zero or negative frequency in the length mode and a mode number below 1 are all declined in words rather than answered with an infinity or a NaN. A zero frequency would need an infinitely long string and a negative one a negative length, so neither is published.

The calculator measures nothing — you supply the numbers

Two quantities and a mode number go in, one condition is rearranged, and the answer describes the string or column you described. The preset names are shorthand for the four figures behind them and are not claims about any particular instrument or installation.

Where the standing wave formula is actually used

Tuning a string to a pitch

Point Solve for at the length and the question inverts: a pitch you want and the speed the medium gives you return the length that sounds it. At 320 m/s a string must be 0.363636 m long to sound 440 Hz in its fundamental — and because a longer length sounds lower, building it long gives a flat note rather than a sharp one.

Getting a wave speed out of a pitch you can measure

Point Solve for at the wave speed and a measured frequency on a known length gives the speed in the medium. A 0.900 m length whose second harmonic is 300 Hz is carrying waves at 270.000 m/s. Get the mode number wrong in that measurement and the speed is wrong by exactly that factor, which is the one real trap in this direction.

Wind instruments, and the doubled length box that drops the octave

Load the half-metre air column preset and then the closed at one end preset: 343 Hz against 171.5 Hz, exactly a factor of two. A pipe closed at one end fits only a quarter of a wavelength in its fundamental rather than a half, so it sounds an octave below an open pipe of the same length — and its even harmonics do not merely go quiet, they do not exist.

Changing the speed box alone, with the length left exactly as it was

Load the water preset against the steel preset: the same 1.000 m column, the same 2.00000 m wavelength, and 741.000 Hz against 2.98000 kHz. That is how one piece of apparatus can be used to compare two media, and it is why the wavelength chip and the frequency chip are printed separately.

When the headline answer lands outside the audible band

Because the relation is arithmetic, it reaches well past hearing in both directions, and the audible band this site publishes is roughly 20 Hz to 20,000 Hz. Pushing the length down and the speed and mode number up walks a mode straight into the ultrasound range, which is the region sonar and medical imaging work in; pulling the other way reaches infrasound. The decibel calculator is the one for how loud rather than how high.

Why the ladder chip prints four modes when you asked for one

Any resonator with two ends has a mode condition of this shape, which is why the first four modes are printed rather than just the one you asked for. The Helmholtz resonator calculator is the different case — a cavity and a neck rather than a length — and the two are worth comparing before assuming a bottle behaves like a pipe.

Recognising the same mathematics somewhere else entirely

The condition that only whole numbers of half waves fit between two walls is not unique to sound. The infinite-well lab plots exactly the same eigenvalue problem for a quantum particle, with the nodes marked, which is why the pictures look so alike. Its subject is a probability density rather than a pitch, and it prints no wavelength or frequency, but the shapes are the same shapes.

Where to go next

For what a standing wave is, where the nodes sit, the harmonic ladder, the worked problems and the misconceptions, read the standing waves and harmonics guide, which this tool is the arithmetic half of. The transverse and longitudinal waves guide is where a travelling wave and the relation between speed, frequency and wavelength belong, the simple harmonic motion guide is what each particle on the pattern is doing, and the guide to wave speed in different media is what sets the speed this page takes as given.

Four more posts pick up what this page deliberately leaves out. The wavelength formula and units guide is the wavelength on its own, the frequency formula guide is the period-to-frequency conversion, the amplitude guide is where the height of a vibration belongs, and the guide to how sound waves travel is where the audible range comes from. The speed of sound guide is the 343 m/s the air-column presets use.

Four tools sit next to this one. The wave speed calculator relates a speed, a frequency and a wavelength directly; the speed of sound calculator supplies a speed for a gas; the Helmholtz resonator calculator is a cavity rather than a length; and the simple harmonic motion calculator is the oscillation each particle performs. The decibel calculator answers how loud instead of how high.

The travelling wave lab, the transverse and longitudinal lab, the lab for a wave crossing into a new medium and the simple harmonic motion lab are the neighbouring simulations, and the infinite-well lab is the same eigenvalue problem on a different subject. The full physics lab library and calculator index are open too.