Verify a traveling wave through derivatives and energy flux
The ideal small-slope string equation is ytt = (T/μ) yxx, where T is tension and μ is mass per unit length. It follows by balancing the small difference of transverse tension components on a short string segment. The approximation keeps tension constant and neglects terms beyond the small transverse slope.
Learning outcome: Check the wave equation by two derivatives, compare phase and material velocities and derive the amplitude-squared mean-power scaling.
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Derive equations with their initial or boundary conditions, integrate distributed profiles, and check model domains independently of numerical outputs.
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At k = 2 rad/m and A = 0.01 m, ω = 40 rad/s and maximum material speed remains 0.4 m/s. Mean power is again 0.16 W because ω²A² is unchanged, although wavelength is halved.
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For k = 1 and A = 0.02 in this model, maximum transverse particle speed is:
Experiment
Check the wave equation by two derivatives, compare phase and material velocities and derive the amplitude-squared mean-power scaling.
Verify a traveling solution and its energy flux
Worked teaching model · synthetic values · Ideal taut string tension40 N, mass density.1 kg/m, small transverse slope at most.06. y=A cos(kx−omega t), omega=20k, no dispersion or damping. Average power belongs to this traveling solution, not arbitrary superpositions.
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Small-slope traveling snapshot at t0
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: A cos(kx) (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Small-slope traveling snapshot at t0
A cos(kx)
| position (m) | transverse displacement (m) |
|---|---|
| 0 | 0.02 |
| 0.1570796 | 0.01975377 |
| 0.3141593 | 0.01902113 |
| 0.4712389 | 0.01782013 |
| 0.6283185 | 0.01618034 |
| 0.7853982 | 0.01414214 |
| 0.9424778 | 0.01175571 |
| 1.099557 | 0.00907981 |
| 1.256637 | 0.00618034 |
| 1.413717 | 0.003128689 |
| 1.570796 | 1.22465e-18 |
| 1.727876 | -0.003128689 |
| 1.884956 | -0.00618034 |
| 2.042035 | -0.00907981 |
| 2.199115 | -0.01175571 |
| 2.356194 | -0.01414214 |
| 2.513274 | -0.01618034 |
| 2.670354 | -0.01782013 |
| 2.827433 | -0.01902113 |
| 2.984513 | -0.01975377 |
| 3.141593 | -0.02 |
| 3.298672 | -0.01975377 |
| 3.455752 | -0.01902113 |
| 3.612832 | -0.01782013 |
| 3.769911 | -0.01618034 |
| 3.926991 | -0.01414214 |
| 4.08407 | -0.01175571 |
| 4.24115 | -0.00907981 |
| 4.39823 | -0.00618034 |
| 4.555309 | -0.003128689 |
| 4.712389 | -3.67394e-18 |
| 4.869469 | 0.003128689 |
| 5.026548 | 0.00618034 |
| 5.183628 | 0.00907981 |
| 5.340708 | 0.01175571 |
| 5.497787 | 0.01414214 |
| 5.654867 | 0.01618034 |
| 5.811946 | 0.01782013 |
| 5.969026 | 0.01902113 |
| 6.126106 | 0.01975377 |
| 6.283185 | 0.02 |
| 6.440265 | 0.01975377 |
| 6.597345 | 0.01902113 |
| 6.754424 | 0.01782013 |
| 6.911504 | 0.01618034 |
| 7.068583 | 0.01414214 |
| 7.225663 | 0.01175571 |
| 7.382743 | 0.00907981 |
| 7.539822 | 0.00618034 |
| 7.696902 | 0.003128689 |
| 7.853982 | 6.12323e-18 |
| 8.011061 | -0.003128689 |
| 8.168141 | -0.00618034 |
| 8.325221 | -0.00907981 |
| 8.4823 | -0.01175571 |
| 8.63938 | -0.01414214 |
| 8.796459 | -0.01618034 |
| 8.953539 | -0.01782013 |
| 9.110619 | -0.01902113 |
| 9.267698 | -0.01975377 |
| 9.424778 | -0.02 |
| 9.581858 | -0.01975377 |
| 9.738937 | -0.01902113 |
| 9.896017 | -0.01782013 |
| 10.0531 | -0.01618034 |
| 10.21018 | -0.01414214 |
| 10.36726 | -0.01175571 |
| 10.52434 | -0.00907981 |
| 10.68142 | -0.00618034 |
| 10.83849 | -0.003128689 |
| 10.99557 | -8.57253e-18 |
| 11.15265 | 0.003128689 |
| 11.30973 | 0.00618034 |
| 11.46681 | 0.00907981 |
| 11.62389 | 0.01175571 |
| 11.78097 | 0.01414214 |
| 11.93805 | 0.01618034 |
| 12.09513 | 0.01782013 |
| 12.25221 | 0.01902113 |
| 12.40929 | 0.01975377 |
| 12.56637 | 0.02 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Phase speed | 20 | m/s |
| Angular frequency | 20 | rad/s |
| Wavelength | 6.283185 | m |
| Maximum transverse particle speed | 0.4 | m/s |
| Maximum slope magnitude | 0.02 | ratio |
| Average traveling-wave power | 0.16 | W |
| Wave-equation coefficient tension over density | 400 | m²/s² |
The wave equation constrains spatial and temporal derivatives without making material speed equal phase speed.
For y=A cos(kx−omega t), ytt=−omega²y and yxx=−k²y. The string equation ytt=(T/mu)yxx requires omega²=(T/mu)k². Here sqrt(T/mu)=20, while maximum transverse particle speed is omega A. Mean traveling power is mu omega² A² v/2 under the small-slope linear approximation.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The wave equation constrains spatial and temporal derivatives without making material speed equal phase speed.
For y=A cos(kx−omega t), ytt=−omega²y and yxx=−k²y. The string equation ytt=(T/mu)yxx requires omega²=(T/mu)k². Here sqrt(T/mu)=20, while maximum transverse particle speed is omega A. Mean traveling power is mu omega² A² v/2 under the small-slope linear approximation.
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