Fitting a Damped Oscillator with Noisy Data
A student records the displacement of a damped harmonic oscillator at 200 evenly spaced times over 10 s and fits x(t) = Aexp(-gammat)cos(omegat + phi) using a least-squares routine. The fit returns gamma = 0.48 +/- 0.02 s^-1, omega = 6.28 +/- 0.01 rad/s, and A = 2.0 cm. The residual plot shows clear structure: residuals oscillate with the same period as the data but with growing amplitude toward the end of the record. The student's lab partner suggests the fit is fine because chi-squared per degree of freedom is about 1.0. The student wants to report gamma as the damping constant and use it to compute the quality factor Q = omega/(2*gamma).
1. Interpret the structured residuals and decide whether the reported gamma is trustworthy.
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Structured residuals that oscillate at the signal period and grow toward the end indicate a systematic model mismatch, not random noise. A chi-squared per dof near 1.0 can still occur if the residuals are correlated (the dof count assumes independent Gaussian errors), so it does not validate the model. The growing residual amplitude suggests the assumed functional form is wrong-e.g. the true decay is not a pure exponential (perhaps the damping is amplitude-dependent, or there is a second mode, or the envelope is being fit with the wrong phase). The reported gamma = 0.48 +/- 0.02 s^-1 is therefore not trustworthy: the quoted uncertainty assumes white noise and underestimates the true systematic error. The correct move is to inspect the residuals, try alternative models (e.g. add a second frequency or a non-exponential envelope), and only then quote gamma with a realistic uncertainty.
2. Estimate the quality factor from the fit and discuss how the residual structure affects that estimate.
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Using the fitted values, Q = omega/(2gamma) = 6.28/(20.48) = 6.54. Propagating the quoted errors: relative error in Q is sqrt((0.01/6.28)^2 + (0.02/0.48)^2) ≈ sqrt(2.5e-6 + 1.7e-3) ≈ 0.041, so Q ≈ 6.5 +/- 0.3 from the fit alone. However, because the residuals show a systematic mismatch, the true uncertainty is larger than this statistical estimate; the systematic error from model misfit likely dominates. The student should report Q with a caveat or re-fit with a better model before quoting a value. This tests whether the student distinguishes statistical from systematic uncertainty and recognizes that a good chi-squared does not certify a model.
3. The student proposes to improve the fit by discarding the last 50 data points where residuals are largest. Evaluate this proposal.
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Discarding data solely because the residuals are large is data dredging and biases the result: it removes precisely the points that carry information about the late-time behavior (where damping is most evident) and can artificially shrink the residuals, making the fit look better without being more correct. A legitimate reason to exclude points would be a documented instrumental artifact (e.g. a known glitch at late times), not the size of the residual itself. The proper approach is to keep all data, diagnose the model mismatch (e.g. plot the envelope on a log scale to check exponentiality), and if a subset is genuinely corrupted, justify the exclusion independently of the fit residuals. The proposal should be rejected as stated.