Interference of Waves Simulator

What Is the Interference of Waves Simulator?

How to use the interference of waves simulator

1. Read the four cards before you touch anything. They are already answering: Phase difference 1.57080 rad, Resultant amplitude 7.07107 units, Intensity vs the two waves alone 1.00000 x, and a verdict reading A quarter cycle apart - the intensity is exactly what the two waves would give without interfering. That middle figure is the most illuminating number on the panel: overlapping the two waves has given nothing and taken nothing.
2. Look at the drawing as one scene in two bands, not as two pictures. Wave 1 is gold and wave 2 is blue along the top, with the blue one shoved along the axis by the path difference, and the view always holds exactly three wavelengths. The cream curve beneath is their sum, point by point, drawn at the same pixels-per-unit as the band above — which is what makes a doubling look like a doubling, and what the caption both bands share one vertical scale is promising. The caption at the top right, cos of the phase = 0.000000, is the factor the arithmetic turns on.
3. Now drag Path difference alone, from 0 towards half a wavelength. Start at 0 and the two crests sit on top of each other: 10.0000 units, 2.00000 x. Push to 1.000 m on the opening 2.000 m wavelength and the cream curve flattens onto the axis — 0.00000 units, 0.00000 x. One slider, the whole argument.
4. Keep going, a whole wavelength further, and watch it come back. At 2.000 m the resultant is 10.0000 units again; at 6.000 m, three wavelengths on, it is still 10.0000 units at 2.00000 x. The card that notices is Phase difference, which has climbed to 18.8496 rad, while Phase difference within one cycle has returned to 0.00000 rad of 6.28319 rad.
5. Now drop Amplitude of wave 2 and watch the cancellation stop being complete. Put the path difference back at 1.000 m and pull the second amplitude to 3.0: the resultant is 2.00000 units, not zero, and the intensity is 0.117647 x. Out of step leaves the difference between the two amplitudes behind, which is the single most common thing people get wrong here.
6. Check what that amplitude move did not do. Phase difference, The path difference in wavelengths and Phase difference within one cycle are byte-identical before and after it. The amplitudes own how big the answer is allowed to be; the two lengths own where inside that band you land. Neither pair ever reaches into the other.
7. Watch the bracket under the two waves, and watch it give up. It runs from a crest of wave 1 to the matching crest of wave 2 and carries the figure with it: path difference 0.250000 wavelengths at the opening state. Push past two of the three wavelengths in view and there is no matching pair left on screen, so the lab drops the bracket and prints a caption in its place — path difference 3.000000 wavelengths - wider than this view on the fifth preset, and path difference 100.000000 wavelengths - wider than this view at the far corner.
8. Read the one-column strip under the canvas as a decomposition, not as more evidence. The path difference in wavelengths is the quantity everything turns on, Possible range of the resultant is the band the amplitudes fix, Phase difference within one cycle is the card above with the whole turns dropped, and The resultant, term by term restates the second card on purpose with the arithmetic showing.
9. Take the amplitude unit at face value: it is the bare word units. A wave is measured in whatever it displaces, so a string is in millimetres and a sound is in pascals, and the panel refuses to pretend otherwise. Press Both amplitudes doubled, same geometry and the resultant goes to 14.1421 units while 1.00000 x does not move: only the ratio of the two amplitudes changes anything.
10. 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 reading the envelope and the bracket. The button then reads Play. The travel rate means nothing at all — this lab publishes no wave speed and no frequency — so no reading changes either way.
11. Reset is the way back when too many sliders have moved to remember. It returns all four to 5.0, 5.0, 2.000 m and 0.500 m, zeroes the travelling phase and re-lights the opening preset. Dragging any slider by hand drops the preset highlight, so no button can claim a state the cards do not show.

That the view always holds exactly three wavelengths is a drawing decision rather than anything about the waves. A view fixed instead at the widest one the lab ever needs, 15.000 m, would draw 150 wavelengths at one end of the wavelength slider and three at the other, so the scale follows the wavelength and the bottom-right caption prints how many metres three of them is.

Dragging is the right way to see which reading answers to which slider, and a poor way to hit an exact figure. When you already have two of the three geometry quantities and want the third typed out, the interference of waves calculator solves the same division for whichever one is missing, with unit menus on every box. The interference of waves guide is the prose half of the subject, with the two conditions, beats and the worked problems set out in full.

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 second prints the strip rows for the same five, so the band and the arithmetic can be read beside the answers rather than inferred from them.

Rows one and two are the headline, and only the path difference separates them. Same wavelength, same two amplitudes — and the resultant goes from 10.0000 units to 0.00000 units while the intensity goes from 2.00000 x to 0.00000 x. Half a wavelength of extra path is the entire difference between the two rows.

Rows two and three are the pair worth pressing yourself. Only the second amplitude moved, and the first two reading columns are byte-identical down the two rows: the same 0.500000 wavelengths, the same 3.14159 rad. The resultant goes from nothing to 2.00000 units and the verdict does not change a character, because the resultant is the difference of the amplitudes is true of both.

Rows one and four are the pair that proves only the fraction matters. Three whole wavelengths of extra path, and the resultant, the intensity and the verdict are byte-identical across them. The count of wavelengths has gone from 0.000000 to 3.000000 and the phase from 0.00000 rad to 18.8496 rad, and nothing else on the panel has noticed.

The strip, for the same five states

The four cards answer what; the strip underneath shows the band the answer had to land in and the arithmetic it came out of. Read the range column down the first two rows and it never moves, because the two amplitudes never moved. Read the reduced-phase column down rows one and four and it does not move either, which is the whole claim of the page in one cell.

The third row is the only one whose band is narrower than the others. At 5.0 against 3.0 the resultant can never leave 2.00000 to 8.00000 units, which is why a complete cancellation is not available to it at any path difference you can set. The rows above and below it, with two equal waves, run from 0.00000 to 10.0000 units and can reach both ends.

The last column is where the cosine does its work. It swings between 1.000000 in step and -1.000000 out of step and sits at 0.000000 a quarter of a cycle apart, and the middle term of the sum follows it. In the last row every term has quadrupled while the printed cosine has not changed at all, which is scale invariance written out in one line.

Formula and symbol reference

The lab prints two of its relations on itself, over the cards they belong to. Written out in the plain-digit form the panel uses:

cycles = dx / lam, then frac = cycles - floor(cycles), then dphi = 2 pi cycles and dphiR = 2 pi frac. The resultant is A = sqrt(a1^2 + a2^2 + 2 a1 a2 cos(dphiR)) and the intensity ratio is ratio = A^2 / (a1^2 + a2^2). The band runs from the difference between a1 and a2 up to a1 + a2.

One route computes the resultant, and it runs through dphiR rather than dphi. That matters less for the answer than you might think and more for the trust: with two routes, one of them could forget to drop the whole turns and no printed amplitude would reveal it. The reduced-phase row is the only reading on the panel where that mistake becomes visible, which is the measured reason it is there.

A single slider sets the wavelength of both curves, and the panel takes that on trust. Two waves whose wavelengths do not match settle into no steady pattern whatever, and nothing on screen would warn you. Where the wavelength itself comes from is a property of the material and a different page: the guide to wave speed in different media is what sets it in a given medium, and the wavelength formula and units guide is where a reader who arrives holding a frequency instead should start. Neither quantity is derived anywhere on this page.

Twelve rows follow and only four of them are things you drag. The rest are cards, strip rows or quantities the panel uses without showing. For the sliders the right-hand column gives the end stops, the step and the published positions; 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 dphiR. It is the only entry that is openly a stripped-down version of another reading, and the panel prints it against the full turn so that the stripping is visible rather than asserted. Everything the lab computes about the resultant goes through that figure and not through the card above it.

The physics: why the phase difference decides everything

Two waves on the same line add displacement by displacement, and the only thing that decides what the sum looks like is how far out of step they are when they meet. One whole wavelength of extra path is one whole cycle of phase, so the quantity that settles it is the path difference measured in wavelengths. Everything else on the panel is a consequence of that one division.

Because a whole cycle brings a wave back exactly where it started, the whole wavelengths fall out and only the fraction survives. That is why the lab can print 10.0000 units at a path difference of nothing, of three wavelengths and of a hundred, byte for byte the same, while the phase card reads 0.00000 rad, 18.8496 rad and 628.319 rad. The resultant genuinely cannot tell them apart.

Four readings answer to four different things, and the sliders are arranged so you can see it one change at a time. From the test runs rather than from a card: that dependency grid was checked at a tolerance of zero on every non-mover, over 3,024 amplitude pairs and 2,940 geometry pairs, so these are byte-identical strings rather than figures that agree to six digits.

The two amplitudes move the resultant, the verdict, the intensity and the range, and never the phase or the count of wavelengths. The wavelength and the path difference move the other six — the total phase, the reduced phase, the count, the resultant, the verdict and the intensity — and never the range; that is seven readings rather than the eight the panel prints, because the term-by-term row restates the resultant card instead of answering on its own. Multiply both amplitudes by the same factor and the resultant and the range scale by exactly that factor while the intensity and the verdict hold their strings.

Two neighbouring subjects get a clause apiece here and no more. Two waves running in opposite directions along one line give a pattern that stays put instead of travelling, which is a subject of its own and is not drawn here. And the same adding-up, taken 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 is derived anywhere on this page.

Where interference of waves breaks down

Every position these four sliders can reach is answered, and answered to the full precision of a double. The arithmetic is not where this panel can let you down; the model behind it is. Seven assumptions sit under every reading on the page, and the lab compresses all seven into one line under the strip because they belong here, at length.

It needs coherence, and that is by far the biggest one

Every figure here assumes the two waves hold a fixed phase relationship for as long as you care about. The panel holds it by construction, which is exactly what two independent real sources will not do: they drift against each other, the fraction of a cycle wanders, and the pattern washes out into nothing in particular. There is no slider for this and no card that would warn you. Read every verdict on the page as conditional on it.

A huge path difference needs that coherence over the whole distance

The sixth preset 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 describe anything real the source would have to hold its phase over the whole of that run, and a real one loses it long before. That preset is a test fixture rather than a physical case, and it appears on this page only with that label attached.

Both waves must share a wavelength, or you get beats instead

One slider sets the wavelength and the drawing applies it to both curves. If the two do not match there is no steady pattern to report at all: what you get instead is a slow swelling and fading in time, governed by the difference between the two frequencies rather than by anything on this panel. Beats are named here and quantified in the interference of waves guide; anyone arriving holding a period rather than a frequency needs the frequency formula guide first.

Superposition is linear, and that is an assumption about the medium

Adding the two displacements point by point — which is literally what the lower band draws — is itself a model. Real materials obey it up to a point and then stop obeying it, so a very big displacement in an actual string, membrane or air column is not just the two added together. The panel will happily add 10.0 units to 10.0 units and draw the result; whether the thing being displaced would is a question about the material.

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

There is one axis in the drawing and there will not be a second. Two waves crossing at an angle give a two-dimensional pattern, which belongs to the diffraction guide and to the wave-particle duality guide rather than to this panel; the diffraction lab is where that geometry is actually drawn. One axis, two curves, one displacement along it: that is all the geometry this panel has.

Interference moves intensity about; it never makes any

Reading 2.00000 x and stopping there would be dishonest. That doubling is borrowed from the 0.00000 x half a wavelength away, and averaged over one whole wavelength of path difference the ratio is exactly 1. The quarter-cycle state the lab opens in is the cleanest view of it: 1.00000 x, nothing gained and nothing lost. How loud the result is is a separate question that the decibel calculator handles and this panel never touches.

The amplitude unit is arbitrary, and only the ratio of the two matters

The sliders say units rather than millimetres or pascals on purpose, because the relation does not know what is being displaced. That is the amplitude guide’s subject rather than this panel’s. Press Both amplitudes doubled, same geometry: the resultant and the range scale by exactly two while 1.00000 x and the verdict hold. A resultant of 7.07107 units only means something next to the two amplitudes it came from.

Where interference of waves is actually used

Noise cancelling, which inverts a wave instead of adding path to it

Adding half a wavelength of extra path is one way to put two waves out of step; electronically inverting one of them reaches the same place with no geometry at all. Press Exactly out of step: nothing left at all for the ideal, then Out of step, but one wave is weaker for what imperfect matching costs: 2.00000 units of residue from a 5.0 against a 3.0. The useful figure is the amplitude ratio, which is why the panel prints the band as well as the answer.

Quarter-wave coatings, where the round trip counts the thickness twice

A layer a quarter of a wavelength thick gives a wave that crosses it and comes back 0.500000 wavelengths of extra path, because the trip there and the trip back each count once. That is the state the third preset loads, and with two equal amplitudes it reads 0.00000 units. Real coatings do not cancel perfectly for the reason the fourth preset shows: the two reflected amplitudes are not equal, so what is left is their difference.

Checking two loudspeakers are wired the same way round

Swapping the connections on one of a matched pair inverts its displacement, which is exactly half a cycle of phase and nothing else. On this panel that is the step from 10.0000 units to 0.00000 units with no slider moved but the path difference: not a halving but the whole lot gone, since the factor of two sits between the correctly wired pair and the 5.00000 units of either one on its own. The guide to how sound waves travel is the everyday case of two overlapping waves written out.

Beats, as the method people actually tune instruments by

Two notes that are close but not equal do not settle into a verdict at all; they swell and fade, and the rate they do it at is the difference between the two frequencies. That is the one case this panel deliberately cannot show you, because it has a single wavelength slider. It is named here and quantified in the guide, and the absence is the honest version of the limit above.

Deciding whether the resultant or the intensity answers your question

The resultant is in whatever unit you set the amplitudes in, so it is the figure to use when comparing against a displacement you can measure. The intensity ratio is dimensionless and scale-free, so it is the figure to use when you want to know what the overlap actually did. At 5.0 against 3.0 out of step, 2.00000 units and 0.117647 x are the same state described two ways, and only the second one is comparable between experiments.

Reading a cancellation backwards to get a wavelength

Set two equal amplitudes, then drag the path difference until the lower band goes flat: the first place it does is half a wavelength, so doubling the reading on Path difference gives the wavelength. It will go flat again every whole wavelength after that, which is the trap — there is never only one answer, and using the second cancellation as though it were the first is wrong by exactly the factor you misread.

Seeing why a single wave has nothing to report

Press One wave switched off. The resultant is 5.00000 units, the band collapses to 5.00000 to 5.00000 units, and the verdict stops describing geometry and says there is nothing to interfere with — even though the path difference still reads 0.250000 wavelengths. The travelling wave lab is that case on its own terms, with nothing to add to.

Where to go next

A panel can show you which readings a slider is allowed to move, and that is the end of what a panel is for. Interference of Waves is where the subject itself is written out: what interference is, the two conditions in full, beats with the arithmetic behind them, the worked problems and the misconceptions. The calculator that takes these figures typed heads the list of related tools below.

Three things this lab leans on but never explains have pages of their own. What sets the wavelength in a given medium is the guide to wave speed in different media. The wavelength taken on its own, for a reader who arrives holding a frequency, is the wavelength formula and units guide. The height of a single wave, which this panel uses as an input twice over, is the amplitude guide.

Further out, the transverse and longitudinal waves guide is what is actually doing the displacing in the two bands, the guide to how sound waves travel is the everyday case of two overlapping waves, and the frequency formula guide is the conversion a reader arriving at beats will want first.

Two labs on this site draw the same adding-up as an angular pattern through slits, and the resemblance is real rather than loose. The diffraction grating lab and the double-slit lab are both already published and are not changing; what they add is a geometry this panel does not have, with two waves arriving from different directions instead of along one line. The diffraction guide is the prose for it.

Nearer home, the travelling wave lab is one wave with nothing to add to it, 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. 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.