SCIENCE · INVESTIGATE AND EXPLAIN

Separate traveling phase from local particle motion

A wave speed describes how a disturbance advances, not how fast a marked piece of medium moves. In the supplied transverse wave, y=.01 sin[2π(t/.5−x/λ)] metres, t is time in seconds and x is position in metres. The period is .5 s, the amplitude is .01 m and the wavelength λ is controlled. A crest has a constant phase, so increasing time must be accompanied by increasing x: the pattern travels toward +x.

Learning outcome: Predict the two observer displacements at t=0,.125,.25 s before reading them. Change wavelength while retaining period and amplitude. Explain which speed changes, which does not and which additional record would remove a whole-period delay ambiguity.

Starting recap

Separate local oscillation from propagation, amplitude from its square, and a calibration fit from a fresh prediction. Use declared phase, medium frame and boundary conditions.

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Try

At λ=2 m, propagation speed is 4 m/s and the .5 m observer spacing gives delay .125 s, a quarter period. Local transverse speed never exceeds .04π≈.125664 m/s. At t=.25 s the first displacement is zero while the second is +.01 m.

Open the experiment

Predict

Doubling wavelength while keeping period and amplitude fixed does what?

Your prediction and reason

Experiment

Predict the two observer displacements at t=0,.125,.25 s before reading them. Change wavelength while retaining period and amplitude. Explain which speed changes, which does not and which additional record would remove a whole-period delay ambiguity.

Separate traveling phase from local particle motion

Worked teaching model · synthetic values · Ideal transverse wave y=.01 sin[2pi(t/.5−x/lambda)] m on an infinite nondispersive medium. Period .5 s, small amplitude. Two fixed observers at x=0 and .5 m; no material drift or attenuation. Silent visual model.

Wavelength: 2 m Elapsed time: 0.25 s

Reset keeps your prediction, captures and practice history.

Simultaneous displacement snapshot

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: same time, different positions (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Simultaneous displacement snapshot

same time, different positions

medium position (m)	transverse displacement (m)
0	1.22465e-18
0.0375	0.00117537
0.075	0.00233445
0.1125	0.00346117
0.15	0.0045399
0.1875	0.0055557
0.225	0.00649448
0.2625	0.00734323
0.3	0.00809017
0.3375	0.00872496
0.375	0.0092388
0.4125	0.00962455
0.45	0.00987688
0.4875	0.00999229
0.525	0.00996917
0.5625	0.00980785
0.6	0.00951057
0.6375	0.00908143
0.675	0.0085264
0.7125	0.00785317
0.75	0.00707107
0.7875	0.00619094
0.825	0.00522499
0.8625	0.0041866
0.9	0.00309017
0.9375	0.0019509
0.975	0.000784591
1.0125	-0.000392598
1.05	-0.00156434
1.0875	-0.0027144
1.125	-0.00382683
1.1625	-0.00488621
1.2	-0.00587785
1.2375	-0.00678801
1.275	-0.00760406
1.3125	-0.0083147
1.35	-0.00891007
1.3875	-0.00938191
1.425	-0.0097237
1.4625	-0.00993068
1.5	-0.01
1.5375	-0.00993068
1.575	-0.0097237
1.6125	-0.00938191
1.65	-0.00891007
1.6875	-0.0083147
1.725	-0.00760406
1.7625	-0.00678801
1.8	-0.00587785
1.8375	-0.00488621
1.875	-0.00382683
1.9125	-0.0027144
1.95	-0.00156434
1.9875	-0.000392598
2.025	0.000784591
2.0625	0.0019509
2.1	0.00309017
2.1375	0.0041866
2.175	0.00522499
2.2125	0.00619094
2.25	0.00707107
2.2875	0.00785317
2.325	0.0085264
2.3625	0.00908143
2.4	0.00951057
2.4375	0.00980785
2.475	0.00996917
2.5125	0.00999229
2.55	0.00987688
2.5875	0.00962455
2.625	0.0092388
2.6625	0.00872496
2.7	0.00809017
2.7375	0.00734323
2.775	0.00649448
2.8125	0.0055557
2.85	0.0045399
2.8875	0.00346117
2.925	0.00233445
2.9625	0.00117537
3	2.44929e-18

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Two fixed observer histories

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: observer x=0 m (solid).2: observer x=.5 m (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Two fixed observer histories

observer x=0 m

time (s)	transverse displacement (m)
0	0
0.0125	0.00156434
0.025	0.00309017
0.0375	0.0045399
0.05	0.00587785
0.0625	0.00707107
0.075	0.00809017
0.0875	0.00891007
0.1	0.00951057
0.1125	0.00987688
0.125	0.01
0.1375	0.00987688
0.15	0.00951057
0.1625	0.00891007
0.175	0.00809017
0.1875	0.00707107
0.2	0.00587785
0.2125	0.0045399
0.225	0.00309017
0.2375	0.00156434
0.25	1.22465e-18
0.2625	-0.00156434
0.275	-0.00309017
0.2875	-0.0045399
0.3	-0.00587785
0.3125	-0.00707107
0.325	-0.00809017
0.3375	-0.00891007
0.35	-0.00951057
0.3625	-0.00987688
0.375	-0.01
0.3875	-0.00987688
0.4	-0.00951057
0.4125	-0.00891007
0.425	-0.00809017
0.4375	-0.00707107
0.45	-0.00587785
0.4625	-0.0045399
0.475	-0.00309017
0.4875	-0.00156434
0.5	-2.44929e-18
0.5125	0.00156434
0.525	0.00309017
0.5375	0.0045399
0.55	0.00587785
0.5625	0.00707107
0.575	0.00809017
0.5875	0.00891007
0.6	0.00951057
0.6125	0.00987688
0.625	0.01
0.6375	0.00987688
0.65	0.00951057
0.6625	0.00891007
0.675	0.00809017
0.6875	0.00707107
0.7	0.00587785
0.7125	0.0045399
0.725	0.00309017
0.7375	0.00156434
0.75	3.67394e-18
0.7625	-0.00156434
0.775	-0.00309017
0.7875	-0.0045399
0.8	-0.00587785
0.8125	-0.00707107
0.825	-0.00809017
0.8375	-0.00891007
0.85	-0.00951057
0.8625	-0.00987688
0.875	-0.01
0.8875	-0.00987688
0.9	-0.00951057
0.9125	-0.00891007
0.925	-0.00809017
0.9375	-0.00707107
0.95	-0.00587785
0.9625	-0.0045399
0.975	-0.00309017
0.9875	-0.00156434
1	-4.89859e-18

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

observer x=.5 m

time (s)	transverse displacement (m)
0	-0.01
0.0125	-0.00987688
0.025	-0.00951057
0.0375	-0.00891007
0.05	-0.00809017
0.0625	-0.00707107
0.075	-0.00587785
0.0875	-0.0045399
0.1	-0.00309017
0.1125	-0.00156434
0.125	0
0.1375	0.00156434
0.15	0.00309017
0.1625	0.0045399
0.175	0.00587785
0.1875	0.00707107
0.2	0.00809017
0.2125	0.00891007
0.225	0.00951057
0.2375	0.00987688
0.25	0.01
0.2625	0.00987688
0.275	0.00951057
0.2875	0.00891007
0.3	0.00809017
0.3125	0.00707107
0.325	0.00587785
0.3375	0.0045399
0.35	0.00309017
0.3625	0.00156434
0.375	1.22465e-18
0.3875	-0.00156434
0.4	-0.00309017
0.4125	-0.0045399
0.425	-0.00587785
0.4375	-0.00707107
0.45	-0.00809017
0.4625	-0.00891007
0.475	-0.00951057
0.4875	-0.00987688
0.5	-0.01
0.5125	-0.00987688
0.525	-0.00951057
0.5375	-0.00891007
0.55	-0.00809017
0.5625	-0.00707107
0.575	-0.00587785
0.5875	-0.0045399
0.6	-0.00309017
0.6125	-0.00156434
0.625	-2.44929e-18
0.6375	0.00156434
0.65	0.00309017
0.6625	0.0045399
0.675	0.00587785
0.6875	0.00707107
0.7	0.00809017
0.7125	0.00891007
0.725	0.00951057
0.7375	0.00987688
0.75	0.01
0.7625	0.00987688
0.775	0.00951057
0.7875	0.00891007
0.8	0.00809017
0.8125	0.00707107
0.825	0.00587785
0.8375	0.0045399
0.85	0.00309017
0.8625	0.00156434
0.875	3.67394e-18
0.8875	-0.00156434
0.9	-0.00309017
0.9125	-0.0045399
0.925	-0.00587785
0.9375	-0.00707107
0.95	-0.00809017
0.9625	-0.00891007
0.975	-0.00951057
0.9875	-0.00987688
1	-0.01

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Frequency	2	Hz
Propagation speed	4	m/s
Travel delay between observers	0.125	s
Displacement at origin	1.22465e-18	m
Displacement at second observer	0.01	m
Maximum local transverse particle speed	0.125664	m/s
Local transverse velocity at origin	-0.125664	m/s

A delayed phase travels along the medium while each modeled element oscillates locally.

The constant-phase condition t/T−x/lambda=constant gives speed lambda/T. At a fixed location the displacement oscillates with frequency 1/T; its maximum transverse speed is the supplied sinusoidal result 2pi A/T. These are different velocities with different directions. The second observer history is the first shifted by .5/speed; a repeated sine alone identifies phase delay only modulo a period unless spacing/direction or a pulse onset is independently known.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

A delayed phase travels along the medium while each modeled element oscillates locally.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the observable

Build the bounded model

Keeping t/T−x/λ constant gives a propagation speed λ/T. At a fixed x, the medium element oscillates transversely with frequency 1/T. The supplied sinusoidal particle-speed amplitude is 2πA/T; this is generally neither equal to the propagation speed nor directed along it. The two plotted observer histories at x=0 and x=.5 m describe different elements, not one element being carried from the first location to the second.

Interpret the evidence

The second history lags the first by distance divided by propagation speed. A sine repeats, however, so phase matching alone gives possible delays differing by whole periods. Independent spacing, a known propagation direction or a tagged pulse onset is needed to select a unique travel delay. The snapshot and time-history axes also answer different questions: a snapshot shows spatial shape at one instant, while a history shows one element at many times. This ideal small-amplitude model excludes dispersion, damping and material drift.

Worked original case

Predict and investigate

Changed-case practice

Try the changed case before feedback

For wavelength 3 m and period .5 s, find the travel delay across .75 m in seconds.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original prose and cases checked 30 October 2026 against the primary source. These are silent, synthetic wave models: no sound playback, exposure prescription, hearing test or physical apparatus is requested. A correct public number is not independent mastery; retain first explanations and a changed-case review in the course evidence workspace.

Compare transverse and longitudinal disturbances

The displayed string-like displacement is transverse to propagation. Ordinary sound in a fluid instead involves longitudinal pressure/density changes and local particle motion along propagation; it requires a material medium and does not propagate through a vacuum. Neither picture requires each particle to travel from source to receiver. Predict/check: a compression pattern moving 3 m while a fluid element oscillates through a tiny distance is consistent with wave transport; identifying the two distances is not. Electromagnetic light, introduced later, is a different field wave that does not require a material carrier.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which fresh observation separates a failed physical model from a fitted calibration parameter?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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