HomeConcept libraryOscillations and WavesWaves

OscillationsIntermediateStarter track

Concept module

Wave Interference

Superpose two coherent sources, trace their path difference to phase difference, and watch bright and dark regions emerge on the same live screen.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 4.00 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s4.00 s

Wave Interference

Two coherent paths feed one screen point, so geometry, phase, and amplitude stay visibly linked.

Live setup

Graphs

Switch graph views without breaking the live stage and time link.

Probe motion

Tracks the two arrivals and the live superposition at the selected probe point.

time (s): 0 to 4displacement (a.u.): -4 to 4

Source ASource BResultant

Hover or scrub to link the graph back to the stage.time (s) / displacement (a.u.)

Controls

Adjust the physical parameters and watch the motion respond.

Source A amplitude

A_{1}

1 a.u.

Controls how strongly source A contributes to the combined motion.

Source B amplitude

A_{2}

1 a.u.

Controls how strongly source B contributes to the combined motion.

Wavelength

λ

1.6 m

Shorter wavelengths make the same path difference produce a larger phase shift.

Source phase offset

ϕ_{0}

0 rad

Adds a direct phase shift between the two sources without changing the geometry.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Hide

Probe height

y_{p}

0.8 m

Moves the screen point up or down so the path difference changes.

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 1

Notice that the probe dot and the screen band are telling the same story: a larger envelope at the probe means a brighter region on the screen.

Try this

Keep the amplitudes equal, move the probe slowly, and watch the screen strip brighten and dim as the resultant envelope changes.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

A_{1}

Source A amplitude

1 a.u.

Sets how strongly source A contributes to the final superposition at every screen point.

Graph: Probe motionGraph: Screen patternOverlay: Result envelopeOverlay: Phase wheel

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use one cue at a time. The prompt changes when the graph, control change, or inspection mode makes a different interference pattern easier to read honestly.

ObservationPrompt 1 of 1

Graph: Probe motion

Notice that the probe dot and the screen band are telling the same story: a larger envelope at the probe means a brighter region on the screen.

Try this

Keep the amplitudes equal, move the probe slowly, and watch the screen strip brighten and dim as the resultant envelope changes.

Why it matters

The pattern is not separate from the live oscillation. It is the long-run result of the same superposition happening at each screen point.

Control: Probe heightGraph: Probe motionGraph: Screen patternOverlay: Result envelope

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

2 visible

Overlay focus

Path-difference guide

Labels the two travel distances and their difference.

What to notice

Moving the probe changes r1 and r2 together, but it is the difference between them that matters for interference.

Why it matters

It keeps the geometry-to-phase step visible instead of treating bright and dark regions as magic.

Control: Probe heightControl: WavelengthGraph: Screen patternEquation

y_{p}

Equation

λ

Challenge mode

Turn the screen pattern and probe motion into small interference hunts that update from the same live geometry.

0/2 solved

ConditionCore

1 of 3 checks

Find a dark band

Starting from Center bright, move the probe onto a dark region where the screen intensity almost vanishes.

Graph-linkedGuided start2 hints

Suggested start

Use the screen-pattern graph as the guide while you move the probe off the center line.

Pending

Open the Screen pattern graph.

Probe motion

Matched

Keep the Path-difference guide visible.

Pending

Drive the relative intensity below

0.1

0.76

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the probe is at y = 0.8 m. The path difference is 0.26 m, so the total phase difference is 1.02 rad and the interference is partial. The resultant displacement is 0.47 a.u., and the relative intensity at that screen point is 0.76.

Equation detailsDeeper interpretation, notes, and worked variable context.

Path difference

The extra distance one arrival travels before the two waves meet at the same screen point.

Δ r = r_{2} - r_{1}

The geometry fixes r1 and r2 once the probe height is chosen.

y_{p}

Probe height 0.8 m

Total phase difference

The screen point compares both the geometric path phase and any explicit source phase offset.

Δ ϕ = \frac{2 π Δ r}{λ} + ϕ_{0}

λ

Wavelength 1.6 m

ϕ_{0}

Source phase offset 0 rad

y_{p}

Probe height 0.8 m

Resultant amplitude

The time-varying probe motion sits inside this amplitude envelope after the two arrivals superpose.

A_{res} = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos (Δ ϕ)

Equal amplitudes can cancel completely only when the phase split is close to π.

A_{1}

Source A amplitude 1 a.u.

A_{2}

Source B amplitude 1 a.u.

ϕ_{0}

Source phase offset 0 rad

Screen intensity

Bright and dark regions follow the square of the resultant amplitude rather than the instantaneous displacement.

I \propto A_{res}^{2}

A_{1}

Source A amplitude 1 a.u.

A_{2}

Source B amplitude 1 a.u.

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 7 of 90 / 9 complete

Waves

Earlier steps still set up Wave Interference.

1. Simple Harmonic Motion2. Wave Speed and Wavelength3. Sound Waves and Longitudinal Motion4. Pitch, Frequency, and Loudness / Intensity+5 more steps

Previous step: Doppler Effect.

Start from track beginning

Short explanation

What the system is doing

Wave interference is what happens when two oscillating disturbances reach the same place at the same time. The result is not decided by one wave alone. It depends on how their amplitudes and phases combine at that point.

This lab keeps the source spacing and screen distance fixed so you can focus on one honest chain of ideas: geometry sets path difference, path difference sets phase difference, and phase difference controls whether the screen point brightens, dims, or lands somewhere in between.

Key ideas

01Constructive interference happens when the total phase difference is close to a whole-number multiple of 2π, so the arrivals reinforce.

02Destructive interference happens when the total phase difference is close to an odd multiple of π, so equal-amplitude arrivals can nearly cancel.

03Changing wavelength or probe position changes the path-difference phase term, while changing source phase shifts the whole pattern without moving the geometry.

Live worked example

Solve the exact state on screen.

These examples read the current probe geometry and source settings directly from the live state, so the steps stay tied to the same bright or dark region you are inspecting.

Live valuesFollowing current parameters

For the current probe position at y = 0.8\,\mathrm{m}, what total phase difference reaches the screen point?

y_{p}

Probe height

0.8 m

Δ r

Path difference

0.26 m

λ

Wavelength

1.6 m

ϕ_{0}

Source phase offset

0 rad

1. Identify the relation

Use

Δ ϕ = \frac{2 π Δ r}{λ} + ϕ_{0}

2. Substitute the live geometry

Δ ϕ = \frac{2 π ( 0.26 )}{1} .6 + 0

3. Wrap it to the useful comparison angle

That gives a wrapped phase difference of

1.02 rad

, or about 0.16 wavelengths of extra travel before the source phase offset is added.

Current phase difference

Δ ϕ = 1.02 rad

The phase difference is between the bright and dark limits, so the probe shows only partial reinforcement.

Common misconception

If the path lengths are different, the interference must always be destructive.

Different path lengths only matter through the phase they produce relative to the wavelength.

A nonzero path difference can still be constructive if it equals one wavelength, two wavelengths, or another whole-number multiple.

Mini challenge

You are sitting on a bright region and you are not allowed to change either source amplitude. What is one reliable way to turn that same probe point dark?

Prediction prompt

Decide whether changing probe position, wavelength, or source phase would work and why.

Check your reasoning

Shift the total phase difference by about π, either by moving the probe so the path difference changes by roughly λ/2 or by adding a source phase offset of π.

Destructive interference depends on the total phase difference, not on one control in isolation. Geometry and source phase both feed the same phase sum.

Quick test

Graph reading

Question 1 of 4

Use geometry, phase, and amplitude together. Each question asks what the screen point must do, not what a formula looks like in isolation.

A bright peak on the screen pattern means which condition is most nearly true at that screen height?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows two coherent sources on the left, a shared screen on the right, and one adjustable probe point on that screen. Two wavy paths run from the sources to the probe so the user can compare the path lengths, the local phase split, and the resulting probe motion in the same picture.

The probe point also appears on a vertical screen strip that brightens or darkens according to the time-averaged resultant amplitude. Optional overlays label the path difference, the resultant envelope, and a phasor-style phase map.

Graph summary

The probe-motion graph plots Source A, Source B, and the live resultant at one selected screen point as functions of time.

The screen-pattern graph plots relative intensity against screen height, so it stays position-based even while the time rail continues to inspect the instantaneous probe motion.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Single-slit spreadOptics

Wave Interference

See each variable before you move it.

Path-difference guide

Find a dark band

Waves

What the system is doing

Solve the exact state on screen.

For the current probe position at y = 0.8\,\mathrm{m}, what total phase difference reaches the screen point?

A bright peak on the screen pattern means which condition is most nearly true at that screen height?

Keep moving through the concept map.

Diffraction

Double-Slit Interference

Standing Waves