Interference of Waves Calculator

What Is the Interference of Waves Calculator?

How to use the interference of waves calculator

1. Choose the unknown. The Solve for menu opens on Path difference in wavelengths, and the other two choices are Path difference and Wavelength of both waves. Any two of the three fix the third.
2. Type the count of wavelengths. It is dimensionless and has no unit menu, and it opens on 0.25. Whole wavelengths in it change nothing, so 0.25, 1.25 and 7.25 all describe the same pair.
3. Type the path difference. It takes m, cm, mm or in and opens on 0.500 m. It is the extra distance one wave travels, not the gap between the two sources.
4. Type the wavelength. It takes m, cm, mm or nm and opens on 2.000 m. One box serves both waves, and that is an assumption rather than a law.
5. Type the two amplitudes. Both are inputs in every mode, so neither ever leaves the form. The unit is the bare word units, because a wave amplitude is measured in whatever the wave displaces.
6. 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.
7. Read the phase chip first. It is 2 pi x cycles, and at the defaults it reads 1.57080 rad. Three whole wavelengths further on it reads 18.8496 rad while nothing else on the page moves at all.
8. Read the resultant chip. At the defaults it reads 7.07107 units. It can never leave the band the two amplitudes fix on their own, from the difference between them up to their sum.
9. Read the intensity chip. At the defaults it reads 1.00000 x — interference has moved nothing. In step it reads 2.00000 x, out of step 0.00000 x, and with both amplitudes at zero it reads a hyphen.
10. Read the verdict chip. One of exactly seven fixed phrases, with the two amplitude cases tested before the geometry so that one wave switched off is never reported as a quarter cycle apart.
11. Open Show working. The steps split the count of wavelengths into whole ones plus the fraction, build the resultant term by term with the cosine printed to six decimals, form the intensity ratio and give the band the two amplitudes allow.

This page exists for the one step that turns a distance into a verdict: dividing the path difference by the wavelength. The interference of waves guide is the prose half of it, with the two conditions, the worked problems and the misconceptions set out in full. Where the wavelength 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 in a given medium, and nothing on this calculator derives it.

Three mistakes account for most wrong answers here. The first is typing the distance between the two sources into the path difference box, which is a different quantity and usually a much larger one; the second is expecting a complete cancellation from unequal amplitudes, when out of step they leave the difference behind. The third is reading the resultant chip as a measure of how much energy is present, when the intensity chip is the figure that actually compares like with like.

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 path difference, only the wavelength and only an amplitude, walks both unit menus, runs the two back-solves worth doing, and finally reaches the entries the calculator declines. Rows 1 to 3 are one state 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 pair of waves 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 wavelength is the unknown still has a wavelength to divide by. The residual is exactly zero in all three directions, because no rounding enters this relation anywhere — which is the whole reason the first box counts wavelengths rather than radians.

Rows 4 to 6 are the two conditions and the honest version of the second. No extra path gives 10.0000 units and 2.00000 x; half a wavelength gives 0.00000 units and 0.00000 x; and the same half wavelength with the second wave at 3 instead of 5 leaves 2.00000 units behind, which is the difference between the two amplitudes rather than nothing. Out of step and nothing left are not the same claim, and only equal amplitudes make them agree.

Rows 7 and 8 are the point of the whole page. Three whole wavelengths and a hundred whole wavelengths give the same 10.0000 units and the same 2.00000 x as no path difference at all, byte for byte, while the phase chip climbs from 0.00000 rad to 18.8496 rad to 628.319 rad. The phase chip is the only one of the four that can tell those three states apart.

Rows 9 to 11 walk round the cycle. Three quarters of a cycle returns the identical 7.07107 units and 1.00000 x as a quarter of one, because the cosine does not care which side of the half you are on. An eighth of a cycle sits above that at 9.23880 units and 1.70711 x, and a third of one below it at 5.00000 units and 0.500000 x.

Rows 12 to 15 move the amplitudes alone. Unequal and in step gives exactly 7.00000 units, the two simply added, while one wave switched off leaves 5.00000 units and a verdict of its own — and through both the phase chip holds at 1.57080 rad, because the amplitudes never touch the geometry. Both switched off is the one row where a chip prints a hyphen: a1^2 + a2^2 is zero, so the ratio is undefined rather than a NaN.

Row 15 is the scale test. Doubling both amplitudes doubles the resultant to 14.1421 units and leaves 1.00000 x exactly where it was, which is what makes the bare word units an honest label on the two amplitude boxes.

Rows 16 to 18 are the two back-solves. Two equal waves of 1.400 m cancel at a path difference of 0.7 m, and again every 1.400 m after that — row 17 takes the next one at 2.1 m, where the phase has reached 9.42478 rad and the resultant is still 0.00000 units. Row 18 turns a measurement round: if 0.900 m of extra path is the first cancellation, the wavelength behind it is 1.8 m.

Rows 19 to 23 change nothing but the units, and every one returns the same strings: one path difference reads 0.500 m, 50 cm, 500 mm or 19.6850393701 in, and one wavelength reads 2.000 m or 200 cm. Row 23 is the best of them: an 800 nm wavelength with a path difference of 0.0000002 m — 200 nm, typed in metres because the nanometre is on the wavelength menu only — is a quarter cycle apart just as surely as the metre-scale default. Every chip there is byte for byte identical to the default row, because the geometry only ever cared about the ratio.

Rows 24 to 31 sit outside the domain and are declined there rather than answered. A zero or negative wavelength has no cycle behind it; a negative path difference and a negative count of wavelengths would hand back negative lengths; a negative amplitude is not a wave; and an entry that parses to infinity is refused rather than published, because Infinity > 0 is true and a sign test alone would let it through.

Row 29 is the one worth reading twice. A zero path difference is perfectly legal when you are solving for the count of wavelengths — row 4 answers it with 0 and the in-step verdict — and illegal when you are solving for the wavelength, where it would divide out to a zero wavelength. The same entry is answered in one direction and declined in the other.

The no answer rows are what the widget actually does with those entries: the result area prints That combination has no valid solution. Check your inputs.

No chips are drawn beside that message, which is why the last four columns of those rows are hyphens rather than figures. It is a refusal rather than a crash, and nothing on this page ever returns a NaN or an infinity.

Those seven phrases are the whole vocabulary of the fourth chip, and the order of the tests matters. The two amplitude cases are checked before the geometry, so the one wave switched off state is never reported as a quarter cycle apart even though its path difference is exactly that. Note also that there is no "they cancel completely" phrase: the destructive string says the resultant is the difference of the amplitudes, which is true whether that difference is zero or not.

Formula and symbol reference

One division does all the work here. The path difference in wavelengths is cycles = dx / lam, and because one whole wavelength of extra path is one whole cycle of phase, the total phase difference is dphi = 2 pi x cycles. Rearranged the other two ways that is dx = cycles x lam and lam = dx / cycles, and those are the three positions of the Solve for menu.

The resultant follows from the phase and the two amplitudes: A = sqrt(a1^2 + a2^2 + 2 a1 a2 cos dphi), worked out from the phase reduced into a single cycle. In step that collapses to a1 + a2; out of step it collapses to the difference between a1 and a2, which is zero only when the two are equal. Those are the two conditions, dx = n lam and dx = (n + 1/2) lam, and the page prints both halves rather than only the reinforcing one.

The wavelength is a property of the wave in its medium and this page never derives it — the wave speed calculator is the tool that relates a wavelength to a speed and a frequency, and the guide to wave speed in different media is what sets that speed in the first place. For a reader who arrives holding a frequency rather than a wavelength, the wavelength formula and units guide is the conversion. Interference needs no speed and no frequency at all, which is why neither has a box here.

The intensity figure is a ratio, and that is deliberate. It is A^2 / (a1^2 + a2^2): the resultant intensity against what the same two waves would give with no interference at all. Intensity goes as the square of the amplitude, which is the one reason the ratio is the honest comparison and is the whole of what this page says about it — the amplitude guide is where a wave amplitude and the intensity that follows from it belong, and averaged over one whole wavelength of path difference this ratio is exactly 1: interference moves energy about, it does not make any.

The physics: what the five boxes actually decide

Two waves on the same line add displacement by displacement, so at any instant the combined displacement is just the sum of the two. Shift one of them along by a distance dx and every point of it arrives a fraction of a cycle later; that fraction, and nothing else, decides whether the two reinforce or fight each other. The page is built around the observation that the fraction is all that survives.

The five boxes split cleanly into two jobs, and the chips are arranged so you can watch the split. The two amplitude boxes set how big the answer is allowed to be — from the difference between them up to their sum — and never touch the phase or the count of wavelengths at all. The wavelength and the path difference decide where inside that band you land, and never move the band.

Load out of step, but one wave is weaker straight after exactly out of step to see that split happen. The phase chip holds at 3.14159 rad while the resultant goes from 0.00000 to 2.00000 units, which is one control moving one half of the answer and nothing else.

Whole wavelengths are the other half of the lesson, and the phase chip is the only place they show. Press exactly in step and then three whole wavelengths further on: the resultant, the intensity and the verdict are byte for byte the same, while the phase climbs from 0.00000 rad to 18.8496 rad. That is why the chip prints the total phase rather than the phase within one cycle — drop the whole cycles and no chip on the page could separate those two states at all.

One clause for each of the two neighbours this page deliberately stops short of. Two waves running in opposite directions along the same line give a standing wave, which is a pattern that does not travel and is a subject of its own. And the same adding-up, seen at an angle through a pair of slits, is what the diffraction guide calls diffraction — the diffraction grating calculator solves that angular condition, and no angle, slit or fringe spacing is derived anywhere here.

Where the interference of waves calculator breaks down

The arithmetic here is exact to the last digit a double can hold. What still goes wrong is physical: a phase relationship the boxes assume and two real sources will not hold, a wavelength box that has to be true of both waves, and a question about energy put to a relation that only sorts geometry.

Nothing you can type tells the page whether the two sources are coherent

Every answer here assumes the two waves hold a fixed phase relationship for as long as you care about, and that is the assumption most likely to be false. Two independent sources drift against each other, the fraction of a cycle wanders, and the pattern washes out into nothing in particular — the chips keep printing a steady verdict regardless, because they only ever see the numbers in the boxes. This is the single biggest caveat on the page and there is no box for it.

A hundred wavelengths of path is arithmetic the boxes will do and a real source will not sustain

Row 8 of the table is reachable and arithmetically correct: a hundred whole wavelengths of extra path returns the same 10.0000 units as none at all. For that to be true of anything real, the source has to hold its phase over the whole of that distance, and a real one loses it long before. Treat that row as a test of the arithmetic rather than as a description of an experiment.

One wavelength box serves both waves, and two different wavelengths are a different calculation

If the two waves do not share a wavelength there is no steady pattern to report at all: what you get instead is a slow swelling and fading in time rather than a fixed verdict in space, and it is governed by the difference between the two frequencies rather than by anything in these boxes. This page has no second wavelength box and will not acquire one, because that is a different relation.

The two displacements only add while the medium stays linear

Adding the displacements is itself a model assumption. A real medium responds linearly over some range and not beyond it, so a very large amplitude in a real string, membrane or column does not simply superpose. The page will happily add 10 units to 10 units; whether the thing being displaced would is a question about the material, not about this relation.

Both waves are assumed to run along the same line, with the same polarisation

There is no angle box here and there will not be one. Two waves crossing at an angle give a two-dimensional pattern, which is the subject of the wave-particle duality guide and of the diffraction guide rather than of this page, and the diffraction grating lab is where that geometry is drawn. Everything here is one line, two waves and one shift along it.

The 2.00000 x is borrowed, not created

Reading 2.00000 x and stopping there would be dishonest. The extra intensity where two waves reinforce is paid for by the 0.00000 x half a wavelength away, and over one whole wavelength of path difference the average ratio is exactly 1. Interference redistributes intensity; it never adds any. The quarter-cycle default is the cleanest view of that: 1.00000 x, nothing gained and nothing lost.

The amplitude unit is whatever you put in, and only the ratio matters

The boxes say units rather than millimetres or pascals on purpose, because the relation does not know what is being displaced. Multiply both amplitudes by any factor and the resultant and the band scale by exactly that factor while the intensity chip and the verdict do not move at all. That is also why a resultant of 7.07107 units is only meaningful next to the two amplitudes it came from.

The cosine is printed to six decimals rather than to six significant figures

At half a cycle the cosine of the half-angle is 6.1e-17 in floating point rather than 0, and printing it raw would put a 23-character number on the page. Every displayed cosine here is rounded to six decimals first, and every threshold in the verdict is tested on that printed figure rather than on the raw float. A negative zero is stripped too, because cos(3 pi / 2) is a small negative number and a readout of -0.000000 would be nonsense.

A hyphen in the intensity chip is an answer, not a fault

With both amplitudes set to zero there is no intensity to compare anything with: a1^2 + a2^2 is zero, so the ratio is genuinely undefined and the chip prints a hyphen rather than a NaN. The geometry chips still read normally in that state, because the path difference and the wavelength do not depend on the amplitudes at all.

Eight entries are declined in words rather than answered

A zero wavelength, a negative wavelength, a negative path difference, a negative count of wavelengths, a negative amplitude, and — in the wavelength mode only — a zero path difference or a zero count of wavelengths. Each would produce an infinity, a negative length or a zero wavelength, so the page says so instead. The eighth is anything that parses to infinity, which needs testing explicitly: Infinity > 0 is true, so a sign test on its own would publish it.

The calculator measures nothing — you supply the numbers

Two quantities go in, one division is rearranged, and the answer describes the pair of waves you described. The preset names are shorthand for the five figures behind them and are not claims about any particular experiment, instrument or installation.

Where the interference of waves calculator is actually used

Working out the extra path that will cancel two sources

Point Solve for at the path difference and the question inverts: a fraction of a cycle you want and the wavelength you have return the distance that produces it. Two equal waves of 1.400 m cancel at 0.7 m of extra path, and again at 2.1 m, and at every 1.400 m after that. The repeat is the useful part — there is never only one answer.

Getting a wavelength out of a cancellation you can measure

Point Solve for at the wavelength and a measured path difference becomes a wavelength. If 0.900 m of extra path is the first place a pair cancels, the wavelength is 1.8 m. Get the order wrong in that measurement — use the second cancellation as though it were the first — and the answer is wrong by exactly the factor you misread, which is the one real trap in this direction.

Noise cancelling, which inverts the phase instead of adding path

Adding half a wavelength of path is one way to put two waves out of step; electronically inverting one of them is another, and it reaches the same place without any geometry. Load exactly out of step for the ideal case and then drop one amplitude to see what imperfect matching costs: at 5 against 3 the residue is 2.00000 units, which is why the useful figure is the amplitude ratio rather than either amplitude alone.

Quarter-wave layers, where a round trip doubles the path

A layer a quarter of a wavelength thick gives a wave that crosses it and comes back half a wavelength of extra path, because the round trip counts twice. That is row 5 of the table: 0.5 of a cycle, 0.00000 units for equal amplitudes. The reason real coatings do not cancel perfectly is the reason row 6 exists — the two reflected amplitudes are not equal, and the resultant is their difference.

Checking two sources are driven in step rather than inverted

Swapping the connections on one of a pair of loudspeakers inverts its displacement, which is exactly half a cycle, and the pair that should have been adding is now subtracting. With matched sources the verdict chip goes from exactly in step at 10.0000 units to exactly out of step at 0.00000 units: everything gone rather than merely quieter, since what the pair forfeits is the doubling of one source's own 5.00000 units. That is a wiring fault worth catching before anyone else hears it.

Deciding whether the resultant chip or the intensity chip answers your question

The resultant is in whatever unit you typed, so it is the right figure when you are comparing against a displacement you can measure. The intensity ratio is dimensionless and scale-free, so it is the right figure when you want to know what interference actually did: 1.00000 x means nothing moved, 2.00000 x is the ceiling for two equal waves, and 0.117647 x says most of it has gone somewhere else.

Sizing the next quantity on a neighbouring tool

This page takes the wavelength as given, so the usual next step is somewhere else. The wave speed calculator turns a frequency into that wavelength or back again, and the de Broglie wavelength calculator is the one for a particle rather than a sound or a ripple. Neither adds two waves together, which is the one thing this page does.

Where to go next

For what the interference of waves is, the two conditions in full, the worked problems, beats and the misconceptions, read the interference of waves guide, which this tool is the arithmetic half of. The diffraction guide is the same adding-up seen at an angle through slits, the guide to wave speed in different media is what sets the wavelength this page takes as given, and the amplitude guide is where the height of a single wave belongs.

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 a reader arriving at beats will want, the transverse and longitudinal waves guide is what is actually doing the displacing, and the guide to how sound waves travel is the everyday case of two overlapping waves. The wave-particle duality guide is the double slit, and what it does to single electrons.

Four tools sit next to this one. The diffraction grating calculator solves the angular condition for slits; the wave speed calculator relates a speed, a frequency and a wavelength; the de Broglie wavelength calculator is the matter-wave case; and the decibel calculator answers how loud rather than how far out of step.

The travelling wave lab is one wave on its own, with nothing to add to. The transverse and longitudinal lab shows the two particle motions, and the lab for a wave crossing into a new medium is where a wavelength changes. The diffraction grating lab and the double-slit lab are the same adding-up drawn as an angular pattern through slits, which is honestly a different geometry from this one.

The full physics lab library and calculator index are open too.