SCIENCE · INVESTIGATE AND EXPLAIN

Infer wave parameters without confusing propagation with material motion

A traveling wave transports a disturbance and energy; the medium’s particles need not travel with the wave pattern. Transverse displacement is perpendicular to propagation, while longitudinal displacement is parallel. A spatial snapshot gives wavelength λ, and a time record at one position gives period T or frequency f=1/T. Phase speed is v=fλ; neither record alone determines both frequency and wavelength.

Learning outcome: Infer wavelength and period separately, then compare in-phase, intermediate-phase and opposite-phase sums. Sketch a separate fixed-end standing mode and label nodes without treating the traveling sum as a standing pattern.

Starting recap

Declare temperature and time references, system boundaries and model regimes. Use absolute temperatures for ratios; distinguish state, path, oscillator phase and wave propagation.

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Try

At 3 Hz, wavelength is 2 m. Equal coherent waves in phase have amplitude 0.02 m and intensity four times one wave; they are not twice the single-wave intensity under fixed phase coherence.

Open the experiment

Predict

At fixed wave speed, doubling frequency:

Your prediction and reason

Experiment

Infer wavelength and period separately, then compare in-phase, intermediate-phase and opposite-phase sums. Sketch a separate fixed-end standing mode and label nodes without treating the traveling sum as a standing pattern.

Connect wave snapshots, clocks and coherent superposition

Worked teaching model · synthetic values · Ideal linear nondispersive string wave speed 6 m/s, individual amplitude .01 m. Two waves have same frequency and direction with selected relative phase. Displacement graph at t=0; fixed medium and frequency when comparing intensity to one wave. No loud-sound or physical-string activity.

Frequency: 4 Hz Relative phase of equal second wave: 60 degrees

Reset keeps your prediction, captures and practice history.

Two coherent traveling snapshots and their sum

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: first wave at t=0 (solid).2: coherent sum at t=0 (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Two coherent traveling snapshots and their sum

first wave at t=0

position (m)	transverse displacement (m)
0	0.01
0.0375	0.00987688
0.075	0.00951057
0.1125	0.00891007
0.15	0.00809017
0.1875	0.00707107
0.225	0.00587785
0.2625	0.0045399
0.3	0.00309017
0.3375	0.00156434
0.375	6.12323e-19
0.4125	-0.00156434
0.45	-0.00309017
0.4875	-0.0045399
0.525	-0.00587785
0.5625	-0.00707107
0.6	-0.00809017
0.6375	-0.00891007
0.675	-0.00951057
0.7125	-0.00987688
0.75	-0.01
0.7875	-0.00987688
0.825	-0.00951057
0.8625	-0.00891007
0.9	-0.00809017
0.9375	-0.00707107
0.975	-0.00587785
1.0125	-0.0045399
1.05	-0.00309017
1.0875	-0.00156434
1.125	-1.83697e-18
1.1625	0.00156434
1.2	0.00309017
1.2375	0.0045399
1.275	0.00587785
1.3125	0.00707107
1.35	0.00809017
1.3875	0.00891007
1.425	0.00951057
1.4625	0.00987688
1.5	0.01
1.5375	0.00987688
1.575	0.00951057
1.6125	0.00891007
1.65	0.00809017
1.6875	0.00707107
1.725	0.00587785
1.7625	0.0045399
1.8	0.00309017
1.8375	0.00156434
1.875	1.19434e-17
1.9125	-0.00156434
1.95	-0.00309017
1.9875	-0.0045399
2.025	-0.00587785
2.0625	-0.00707107
2.1	-0.00809017
2.1375	-0.00891007
2.175	-0.00951057
2.2125	-0.00987688
2.25	-0.01
2.2875	-0.00987688
2.325	-0.00951057
2.3625	-0.00891007
2.4	-0.00809017
2.4375	-0.00707107
2.475	-0.00587785
2.5125	-0.0045399
2.55	-0.00309017
2.5875	-0.00156434
2.625	-4.28626e-18
2.6625	0.00156434
2.7	0.00309017
2.7375	0.0045399
2.775	0.00587785
2.8125	0.00707107
2.85	0.00809017
2.8875	0.00891007
2.925	0.00951057
2.9625	0.00987688
3	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.

coherent sum at t=0

position (m)	transverse displacement (m)
0	0.015
0.0375	0.0134606
0.075	0.0115897
0.1125	0.00943342
0.15	0.00704489
0.1875	0.00448288
0.225	0.00181049
0.2625	-0.000906485
0.3	-0.00360114
0.3375	-0.00620711
0.375	-0.00866025
0.4125	-0.0109001
0.45	-0.0128716
0.4875	-0.0145262
0.525	-0.0158231
0.5625	-0.0167303
0.6	-0.0172256
0.6375	-0.0172968
0.675	-0.016942
0.7125	-0.0161701
0.75	-0.015
0.7875	-0.0134606
0.825	-0.0115897
0.8625	-0.00943342
0.9	-0.00704489
0.9375	-0.00448288
0.975	-0.00181049
1.0125	0.000906485
1.05	0.00360114
1.0875	0.00620711
1.125	0.00866025
1.1625	0.0109001
1.2	0.0128716
1.2375	0.0145262
1.275	0.0158231
1.3125	0.0167303
1.35	0.0172256
1.3875	0.0172968
1.425	0.016942
1.4625	0.0161701
1.5	0.015
1.5375	0.0134606
1.575	0.0115897
1.6125	0.00943342
1.65	0.00704489
1.6875	0.00448288
1.725	0.00181049
1.7625	-0.000906485
1.8	-0.00360114
1.8375	-0.00620711
1.875	-0.00866025
1.9125	-0.0109001
1.95	-0.0128716
1.9875	-0.0145262
2.025	-0.0158231
2.0625	-0.0167303
2.1	-0.0172256
2.1375	-0.0172968
2.175	-0.016942
2.2125	-0.0161701
2.25	-0.015
2.2875	-0.0134606
2.325	-0.0115897
2.3625	-0.00943342
2.4	-0.00704489
2.4375	-0.00448288
2.475	-0.00181049
2.5125	0.000906485
2.55	0.00360114
2.5875	0.00620711
2.625	0.00866025
2.6625	0.0109001
2.7	0.0128716
2.7375	0.0145262
2.775	0.0158231
2.8125	0.0167303
2.85	0.0172256
2.8875	0.0172968
2.925	0.016942
2.9625	0.0161701
3	0.015

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
Wavelength	1.5	m
Period	0.25	s
Phase speed	6	m/s
Combined displacement amplitude	0.0173205	m
Combined intensity relative to one traveling wave	3	ratio
Fixed-end 3 m string fundamental frequency	1	Hz
Selected fixed-end harmonic index	4	integer

Frequency and wavelength encode propagation, while a coherent phase difference changes the resultant amplitude.

For a nondispersive string, v=f lambda is fixed by the medium and tension assumptions, not by frequency alone. Linear superposition adds signed displacements, not intensities. Equal same-direction waves give amplitude 2A|cos(phi/2)| and intensity ratio 4cos²(phi/2) relative to one wave at the same frequency/medium. A separate fixed-end length 3 m supports fn=n v/(2L); its standing pattern is not the same-direction traveling pair plotted here.

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

Frequency and wavelength encode propagation, while a coherent phase difference changes the resultant amplitude.

Observation to attach to a capture

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

Understand

Declare the state, system and observation

Build the algebraic model

The ideal string medium has phase speed 6 m/s. At frequency 4 Hz, wavelength is 1.5 m and period 0.25 s. The supplied traveling form y=Acos(2πx/λ−2πft) propagates toward positive x, while reversing the sign before the time term reverses propagation. A particle at fixed x oscillates transversely rather than advancing at 6 m/s. In this nondispersive model changing f changes λ while the stipulated medium speed remains fixed.

Check the original evidence packet

Two equal same-direction waves of amplitude 0.01 m and phase difference φ add by signed displacement. The resultant amplitude is 2A|cos(φ/2)|. At 60° it is 0.01√3 m, giving intensity three times that of one wave at the same frequency and medium; at 180° the pair cancels exactly. Intensities cannot simply be added for coherent overlapping waves. Destructive interference does not destroy energy; the complete source and spatial interference account determines energy redistribution.

Challenge the domain and transfer

A separate pair traveling in opposite directions can form a standing wave. For fixed ends separated by 3 m, allowed wavelengths are 2L/n and frequencies n×6/(2×3)=n Hz. Nodes have zero displacement; antinodes have maximum oscillation. A fixed-end reflection inverts displacement whereas an ideal free-end reflection does not. Two close frequencies 4 and 4.5 Hz give beat frequency 0.5 Hz; changing the second to 5 Hz makes it 1 Hz. These standing and beat examples are separate from the equal-frequency same-direction graph, not claims about what that graph alone demonstrates.

Worked original case

At 3 Hz, wavelength is 2 m. Equal coherent waves in phase have amplitude 0.02 m and intensity four times one wave; they are not twice the single-wave intensity under fixed phase coherence.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

A wave has wavelength 0.8 m and frequency 5 Hz. Find phase speed in m/s.

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 limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. Packets are supplied or generated teaching evidence, not measured apparatus. No heating, pressure vessels, physical oscillators, sound exposure or construction is requested. Public checks and saved reports do not establish supervised practical skill, equipment safety or independent mastery.

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

Which assumptions must remain fixed when comparing intensity by amplitude squared?

Your explanation and limitations

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