University Physics II · unit UPII-PREFLIGHT-W
Unnumbered preflight · Waves for E&M and Optics
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.
Extra challenging questions
A selection from this pathway. Open a topic below for its complete practice and challenge set.
Wave quantities and representations: infer an input with uncertainty
A periodic travelling wave is a disturbance whose pattern repeats over a distance λ in space and a time T at any fixed point, and which carries that pattern forward at speed v = λ/T. Use the supplied topic model Y_out = 2*pi/lam, where Y_out is WAVE NUMBER k (rad m⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. lam is Wavelength λ, measured in m. Keep amp (Amplitude A) = 6 mm; per (Period T) = 0.05 s fixed. For this exercise use only 0.435937 m ≤ lam ≤ 1.81406 m 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 = 4.9218362 rad m⁻¹, with a bounded reading uncertainty ±0.21874 rad m⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lam by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000137812 m, 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.
WAVE NUMBER k against lam
Predictions from the supplied Wave quantities and representations 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): WAVE NUMBER k / rad m⁻¹
Horizontal axis (x): lam: Wavelength λ / m
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| lam: Wavelength λ / m | WAVE NUMBER k / rad m⁻¹ |
|---|---|
| 0.4359375 | 14.4130415648564 |
| 0.7804687499999999 | 8.050527720910782 |
| 1.125 | 5.585053606381854 |
| 1.4695312499999997 | 4.275639124502856 |
| 1.8140625 | 3.4635991357406852 |
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(lam) = 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(0.435937) = 14.413 rad m⁻¹ and Y_out(1.81406) = 3.4636 rad m⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives lam_line = 0.435937 + (4.92184 − 14.413)(1.81406 − 0.435937)/(3.4636 − 14.413) = 1.63053 m. 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 lam = 1.27659 m; substituting back gives Y_out = 4.9218362 rad m⁻¹.
- The interpolation error in the input is |lam_line − lam| = 0.353931 m. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 4.7030962 rad m⁻¹ and 5.1405762 rad m⁻¹. Sort the roots even if the response decreases: 1.22227 m ≤ lam ≤ 1.33597 m.
- The lower and upper input deviations are 0.0543212 m and 0.0593741 m. 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 lam ≈ 1.27659 m; compatible interval [1.22227, 1.33597] m. Endpoint interpolation gives 1.63053 m.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Phase, Superposition & Coherence: infer an input with uncertainty
Superposition is the rule that two waves crossing at a point add their displacements, and the phase difference — how far out of step they are, in radians — decides whether that sum is larger or smaller than either wave alone. Use the supplied topic model Y_out = sqrt(1+a2*a2+2*a2*cosd(360*d+phi0)), where Y_out is RESULTANT A_R (× A₁). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; sind/cosd/tand use degrees. phi0 is Source phase offset Δφ₀, measured in °. Keep d (Path difference Δr) = 0.25 λ; a2 (Amplitude ratio A₂/A₁) = 1 × A₁ fixed. For this exercise use only 94.95 ° ≤ phi0 ≤ 265.05 ° 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 = 1.6252968 × A₁, with a bounded reading uncertainty ±0.055926 × A₁. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown phi0 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.01701 °, 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.
RESULTANT A_R against phi0
Predictions from the supplied Phase, Superposition & Coherence 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): RESULTANT A_R / × A₁
Horizontal axis (x): phi0: Source phase offset Δφ₀ / °
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| phi0: Source phase offset Δφ₀ / ° | RESULTANT A_R / × A₁ |
|---|---|
| 94.94999999999999 | 0.08636693241118089 |
| 137.475 | 0.8050939805765227 |
| 180 | 1.414213562373095 |
| 222.525 | 1.8307986460666417 |
| 265.05 | 1.99813431805419 |
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(phi0) = 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(94.95) = 0.0863669 × A₁ and Y_out(265.05) = 1.99813 × A₁. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives phi0_line = 94.95 + (1.6253 − 0.0863669)(265.05 − 94.95)/(1.99813 − 0.0863669) = 231.877 °. 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 phi0 = 198.711 °; substituting back gives Y_out = 1.6252968 × A₁.
- The interpolation error in the input is |phi0_line − phi0| = 33.1657 °. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 1.5693708 × A₁ and 1.6812228 × A₁. Sort the roots even if the response decreases: 193.383 ° ≤ phi0 ≤ 204.409 °.
- The lower and upper input deviations are 5.32782 ° and 5.69848 °. 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 phi0 ≈ 198.711 °; compatible interval [193.383, 204.409] °. Endpoint interpolation gives 231.877 °.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Wave functions, speed & dispersion: infer an input with uncertainty
A wave function is a rule y(x, t) giving the displacement of every point of the medium at every instant, and it describes a travelling wave when x and t enter only through the combination x − vt, so the whole shape slides at speed v. Use the supplied topic model Y_out = vg/max(vp,0.01), where Y_out is SPEED RATIO v_g / v_p (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. vp is Phase speed v_p, measured in m s⁻¹. Keep vg (Group speed v_g) = 1.5 m s⁻¹; t (Time t) = 1.5 s fixed. For this exercise use only 0.871875 m s⁻¹ ≤ vp ≤ 3.62812 m s⁻¹ 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 = 0.58750092, with a bounded reading uncertainty ±0.02611. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown vp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000275625 m s⁻¹, 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.
SPEED RATIO v_g / v_p against vp
Predictions from the supplied Wave functions, speed & dispersion 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): SPEED RATIO v_g / v_p
Horizontal axis (x): vp: Phase speed v_p / m s⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| vp: Phase speed v_p / m s⁻¹ | SPEED RATIO v_g / v_p |
|---|---|
| 0.871875 | 1.7204301075268817 |
| 1.5609374999999999 | 0.9609609609609611 |
| 2.25 | 0.6666666666666666 |
| 2.9390624999999995 | 0.5103668261562999 |
| 3.628125 | 0.41343669250646 |
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(vp) = 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(0.871875) = 1.72043 and Y_out(3.62812) = 0.413437. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives vp_line = 0.871875 + (0.587501 − 1.72043)(3.62812 − 0.871875)/(0.413437 − 1.72043) = 3.26105 m s⁻¹. 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 vp = 2.55319 m s⁻¹; substituting back gives Y_out = 0.58750092.
- The interpolation error in the input is |vp_line − vp| = 0.707863 m s⁻¹. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.56139092 and 0.61361092. Sort the roots even if the response decreases: 2.44455 m s⁻¹ ≤ vp ≤ 2.67193 m s⁻¹.
- The lower and upper input deviations are 0.108642 m s⁻¹ and 0.118747 m s⁻¹. 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 vp ≈ 2.55319 m s⁻¹; compatible interval [2.44455, 2.67193] m s⁻¹. Endpoint interpolation gives 3.26105 m s⁻¹.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Wave Energy, Power and Intensity: infer an input with uncertainty
A wave carries energy at the wave speed. Average power is the energy crossing a point per second, ½μvω²A² on a string; intensity spreads that power over the wavefront area, I = P(av)/A(⊥), and both scale as amplitude squared. Use the supplied topic model Y_out = 1/10^((10*n*log(r)+0.0043429*a*(r-1))/20), where Y_out is AMPLITUDE A(r)/A(1 m) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; log(x) means log₁₀x. r is Microphone distance r, measured in m. Keep n (Spreading exponent n (I ∝ r⁻ⁿ)) = 2; a (Absorption α) = 10 ×10⁻³ m⁻¹ fixed. For this exercise use only 107.144 m ≤ r ≤ 893.856 m 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 = 0.000090945346, with a bounded reading uncertainty ±0.00001171. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0786712 m, 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.
AMPLITUDE A(r)/A(1 m) against r
Predictions from the supplied Wave Energy, Power and Intensity 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): AMPLITUDE A(r)/A(1 m)
Horizontal axis (x): r: Microphone distance r / m
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r: Microphone distance r / m | AMPLITUDE A(r)/A(1 m) |
|---|---|
| 107.14375 | 0.005489683563791768 |
| 303.821875 | 0.0007241331007762083 |
| 500.5 | 0.0001644207569246912 |
| 697.1781249999999 | 0.0000441509709101146 |
| 893.8562499999999 | 0.000012880704432217978 |
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(r) = 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(107.144) = 0.00548968 and Y_out(893.856) = 0.0000128807. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives r_line = 107.144 + (0.0000909453 − 0.00548968)(893.856 − 107.144)/(0.0000128807 − 0.00548968) = 882.643 m. 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 r = 587.038 m; substituting back gives Y_out = 0.000090945346.
- The interpolation error in the input is |r_line − r| = 295.604 m. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.000079235346 and 0.00010265535. Sort the roots even if the response decreases: 569.042 m ≤ r ≤ 607.691 m.
- The lower and upper input deviations are 17.9966 m and 20.6523 m. 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 r ≈ 587.038 m; compatible interval [569.042, 607.691] m. Endpoint interpolation gives 882.643 m.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.