Derive allowed modes without mistaking harmonic ratios for calibration
A string fixed at both ends must have zero displacement at x = 0 and x = L. A separated spatial solution proportional to sin(kx) meets the first condition; the second requires kL = nπ for positive integer n. Combining with the wave equation gives fn = nv/(2L).
Learning outcome: Derive fixed-end modes, compare a fixed-free boundary and propagate an elapsed-time gain through frequency and density inference.
Starting recap
Derive equations with their initial or boundary conditions, integrate distributed profiles, and check model domains independently of numerical outputs.
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With a correct clock and n = 4, frequency is 20 Hz and adjacent-node spacing is 0.5 m. A gain of 0.9 would make the inferred density 0.081 kg/m while leaving all harmonic ratios unchanged.
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A clock that reads 1.1 times true elapsed time makes the inferred string density:
Experiment
Derive fixed-end modes, compare a fixed-free boundary and propagate an elapsed-time gain through frequency and density inference.
Infer a standing-wave parameter while auditing the clock
Worked teaching model · synthetic values · Fixed-fixed ideal string length2 m,tension40 N,density.1 kg/m; true speed20 m/s. A clock reads gain times true elapsed time, so indicated frequency is true frequency/gain. Independent linear mass density is supplied as a separate check.
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Fixed-fixed spatial mode at maximum standing displacement
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: sin(n pi x/L) (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Fixed-fixed spatial mode at maximum standing displacement
sin(n pi x/L)
| position (m) | normalized displacement |
|---|---|
| 0 | 0 |
| 0.025 | 0.1175374 |
| 0.05 | 0.2334454 |
| 0.075 | 0.3461171 |
| 0.1 | 0.4539905 |
| 0.125 | 0.5555702 |
| 0.15 | 0.649448 |
| 0.175 | 0.7343225 |
| 0.2 | 0.809017 |
| 0.225 | 0.872496 |
| 0.25 | 0.9238795 |
| 0.275 | 0.9624552 |
| 0.3 | 0.9876883 |
| 0.325 | 0.999229 |
| 0.35 | 0.9969173 |
| 0.375 | 0.9807853 |
| 0.4 | 0.9510565 |
| 0.425 | 0.9081432 |
| 0.45 | 0.8526402 |
| 0.475 | 0.7853169 |
| 0.5 | 0.7071068 |
| 0.525 | 0.6190939 |
| 0.55 | 0.5224986 |
| 0.575 | 0.4186597 |
| 0.6 | 0.309017 |
| 0.625 | 0.1950903 |
| 0.65 | 0.0784591 |
| 0.675 | -0.03925982 |
| 0.7 | -0.1564345 |
| 0.725 | -0.2714404 |
| 0.75 | -0.3826834 |
| 0.775 | -0.4886212 |
| 0.8 | -0.5877853 |
| 0.825 | -0.6788007 |
| 0.85 | -0.760406 |
| 0.875 | -0.8314696 |
| 0.9 | -0.8910065 |
| 0.925 | -0.9381913 |
| 0.95 | -0.9723699 |
| 0.975 | -0.9930685 |
| 1 | -1 |
| 1.025 | -0.9930685 |
| 1.05 | -0.9723699 |
| 1.075 | -0.9381913 |
| 1.1 | -0.8910065 |
| 1.125 | -0.8314696 |
| 1.15 | -0.760406 |
| 1.175 | -0.6788007 |
| 1.2 | -0.5877853 |
| 1.225 | -0.4886212 |
| 1.25 | -0.3826834 |
| 1.275 | -0.2714404 |
| 1.3 | -0.1564345 |
| 1.325 | -0.03925982 |
| 1.35 | 0.0784591 |
| 1.375 | 0.1950903 |
| 1.4 | 0.309017 |
| 1.425 | 0.4186597 |
| 1.45 | 0.5224986 |
| 1.475 | 0.6190939 |
| 1.5 | 0.7071068 |
| 1.525 | 0.7853169 |
| 1.55 | 0.8526402 |
| 1.575 | 0.9081432 |
| 1.6 | 0.9510565 |
| 1.625 | 0.9807853 |
| 1.65 | 0.9969173 |
| 1.675 | 0.999229 |
| 1.7 | 0.9876883 |
| 1.725 | 0.9624552 |
| 1.75 | 0.9238795 |
| 1.775 | 0.872496 |
| 1.8 | 0.809017 |
| 1.825 | 0.7343225 |
| 1.85 | 0.649448 |
| 1.875 | 0.5555702 |
| 1.9 | 0.4539905 |
| 1.925 | 0.3461171 |
| 1.95 | 0.2334454 |
| 1.975 | 0.1175374 |
| 2 | 3.67394e-16 |
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 |
|---|---|---|
| True mode frequency | 15 | Hz |
| Indicated mode frequency | 14.28571 | Hz |
| Adjacent-node spacing | 0.6666667 | m |
| Inferred fundamental from indicated harmonic | 4.761905 | Hz |
| Density inferred without clock correction | 0.11025 | kg/m |
| Independent linear mass density | 0.1 | kg/m |
| Separate fixed-free same-index frequency | 12.5 | Hz |
Correct harmonic ratios can coexist with an incorrect absolute frequency calibration.
Fixed ends require sin(kL)=0, so k=npi/L and fn=nv/(2L). An elapsed-time readout multiplied by gain g reports frequency divided by g, causing inferred mu=T/(2L f1read)² to scale by gain². A separately fixed-free boundary instead has odd quarter-wave modes(2n−1)v/(4L); it is not the boundary drawn in this graph.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Correct harmonic ratios can coexist with an incorrect absolute frequency calibration.
Fixed ends require sin(kL)=0, so k=npi/L and fn=nv/(2L). An elapsed-time readout multiplied by gain g reports frequency divided by g, causing inferred mu=T/(2L f1read)² to scale by gain². A separately fixed-free boundary instead has odd quarter-wave modes(2n−1)v/(4L); it is not the boundary drawn in this graph.
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Which independent measurement would distinguish a density change from a clock error?
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