University Physics III · topic UPIII-03.09
Wave-phenomena data studio
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Wave-phenomena data studio: predict a finite change
A wave-phenomena data studio is the inverse problem of the section: starting from a sampled signal or its spectrogram, recover the wavelength, phase difference, beat rate or source speed that produced it, and state how uncertain each recovered number is. Use the supplied topic model Y_out = 1/df, where Y_out is BEAT PERIOD 1/Δf (s). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. df is Tone separation Δf, measured in Hz. Keep tw (Window length T_w) = 0.2 s fixed. For this exercise use only 2.57187 Hz ≤ df ≤ 17.9281 Hz on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at df = 5.95025 Hz and 8.10012 Hz. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δdf across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
BEAT PERIOD 1/Δf against df
Predictions from the supplied Wave-phenomena data studio model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): BEAT PERIOD 1/Δf / s
Horizontal axis (x): df: Tone separation Δf / Hz
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| df: Tone separation Δf / Hz | BEAT PERIOD 1/Δf / s |
|---|---|
| 2.571875 | 0.3888213851761847 |
| 6.410937499999999 | 0.15598342676090668 |
| 10.249999999999998 | 0.09756097560975611 |
| 14.089062499999999 | 0.07097704336253743 |
| 17.928124999999998 | 0.055778281331706474 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of df separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At df = 5.95025 Hz, substitute into Y_out = 1/df: Y₀ = 0.16806 s.
- At df = 8.10012 Hz, the same fixed-parameter model gives Y₁ = 0.123455 s.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0446053 s; the input change is Δdf = 2.14987 Hz.
- Average sensitivity is ΔY_out/Δdf = -0.0207478 s per Hz. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 0.16806 s, Y₁ = 0.123455 s; ΔY_out = -0.0446053 s; average sensitivity = -0.0207478 s per Hz.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Wave-phenomena data studio: infer an input with uncertainty
A wave-phenomena data studio is the inverse problem of the section: starting from a sampled signal or its spectrogram, recover the wavelength, phase difference, beat rate or source speed that produced it, and state how uncertain each recovered number is. Use the supplied topic model Y_out = 1/df, where Y_out is BEAT PERIOD 1/Δf (s). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. df is Tone separation Δf, measured in Hz. Keep tw (Window length T_w) = 0.2 s fixed. For this exercise use only 2.57187 Hz ≤ df ≤ 17.9281 Hz on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.083757793 s, with a bounded reading uncertainty ±0.0041969 s. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown df by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00153562 Hz, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
BEAT PERIOD 1/Δf against df
Predictions from the supplied Wave-phenomena data studio model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): BEAT PERIOD 1/Δf / s
Horizontal axis (x): df: Tone separation Δf / Hz
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| df: Tone separation Δf / Hz | BEAT PERIOD 1/Δf / s |
|---|---|
| 2.571875 | 0.3888213851761847 |
| 6.410937499999999 | 0.15598342676090668 |
| 10.249999999999998 | 0.09756097560975611 |
| 14.089062499999999 | 0.07097704336253743 |
| 17.928124999999998 | 0.055778281331706474 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(df) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(2.57187) = 0.388821 s and Y_out(17.9281) = 0.0557783 s. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives df_line = 2.57187 + (0.0837578 − 0.388821)(17.9281 − 2.57187)/(0.0557783 − 0.388821) = 16.638 Hz. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives df = 11.9392 Hz; substituting back gives Y_out = 0.083757793 s.
- The interpolation error in the input is |df_line − df| = 4.69883 Hz. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.079560893 s and 0.087954693 s. Sort the roots even if the response decreases: 11.3695 Hz ≤ df ≤ 12.569 Hz.
- The lower and upper input deviations are 0.569698 Hz and 0.629802 Hz. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer df ≈ 11.9392 Hz; compatible interval [11.3695, 12.569] Hz. Endpoint interpolation gives 16.638 Hz.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.