University Physics III · topic UPIII-03.07
Doppler geometry, reflected shifts, and model limits
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.
Doppler geometry and reflected shifts: predict a finite change
Only the component of a velocity along the line joining source and observer shifts a frequency, so a velocity u at angle θ to that line acts through u cos θ alone — although in a medium the source component and the observer component still enter the formula in different places. Use the supplied topic model Y_out = 100*tand(theta)*pi/180, where Y_out is ANGLE SENSITIVITY (% per degree). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Beam angle to flow θ, measured in °. Keep u (Blood speed u) = 0.6 m s⁻¹ fixed. For this exercise use only 26.375 ° ≤ theta ≤ 73.625 ° 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 theta = 36.77 ° and 43.385 °. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δtheta across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
ANGLE SENSITIVITY against theta
Predictions from the supplied Doppler geometry and reflected shifts 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): ANGLE SENSITIVITY / % per degree
Horizontal axis (x): theta: Beam angle to flow θ / °
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| theta: Beam angle to flow θ / ° | ANGLE SENSITIVITY / % per degree |
|---|---|
| 26.375 | 0.8654399692628872 |
| 38.1875 | 1.3728222325816377 |
| 50 | 2.0800024063240072 |
| 61.8125 | 3.2567311637243885 |
| 73.625 | 5.939687385366014 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of theta separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At theta = 36.77 °, substitute into Y_out = 100*tand(theta)*pi/180: Y₀ = 1.30425 % per degree.
- At theta = 43.385 °, the same fixed-parameter model gives Y₁ = 1.64961 % per degree.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.345362 % per degree; the input change is Δtheta = 6.615 °.
- Average sensitivity is ΔY_out/Δtheta = 0.0522089 % per degree per °. Its positive sign means that this response increases 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 = 1.30425 % per degree, Y₁ = 1.64961 % per degree; ΔY_out = 0.345362 % per degree; average sensitivity = 0.0522089 % per degree per °.
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.
Doppler geometry and reflected shifts: infer an input with uncertainty
Only the component of a velocity along the line joining source and observer shifts a frequency, so a velocity u at angle θ to that line acts through u cos θ alone — although in a medium the source component and the observer component still enter the formula in different places. Use the supplied topic model Y_out = 100*tand(theta)*pi/180, where Y_out is ANGLE SENSITIVITY (% per degree). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Beam angle to flow θ, measured in °. Keep u (Blood speed u) = 0.6 m s⁻¹ fixed. For this exercise use only 26.375 ° ≤ theta ≤ 73.625 ° 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 = 2.5109659 % per degree, with a bounded reading uncertainty ±0.24683 % per degree. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown theta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.004725 °, 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.
ANGLE SENSITIVITY against theta
Predictions from the supplied Doppler geometry and reflected shifts 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): ANGLE SENSITIVITY / % per degree
Horizontal axis (x): theta: Beam angle to flow θ / °
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| theta: Beam angle to flow θ / ° | ANGLE SENSITIVITY / % per degree |
|---|---|
| 26.375 | 0.8654399692628872 |
| 38.1875 | 1.3728222325816377 |
| 50 | 2.0800024063240072 |
| 61.8125 | 3.2567311637243885 |
| 73.625 | 5.939687385366014 |
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(theta) = 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(26.375) = 0.86544 % per degree and Y_out(73.625) = 5.93969 % per degree. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives theta_line = 26.375 + (2.51097 − 0.86544)(73.625 − 26.375)/(5.93969 − 0.86544) = 41.6977 °. 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 theta = 55.1975 °; substituting back gives Y_out = 2.5109659 % per degree.
- The interpolation error in the input is |theta_line − theta| = 13.4998 °. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.2641359 % per degree and 2.7577959 % per degree. Sort the roots even if the response decreases: 52.3728 ° ≤ theta ≤ 57.6715 °.
- The lower and upper input deviations are 2.82465 ° and 2.47396 °. 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 theta ≈ 55.1975 °; compatible interval [52.3728, 57.6715] °. Endpoint interpolation gives 41.6977 °.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.