Annus Mirabilis · Interactive critical edition in preparation
Moving-water drag hypotheses
Compare no drag, full drag, and Fresnel drag under the same settings.
Modern SI calibration and illustrative settings. This is not a verified historical dataset or a strict 1904-mode instrument. No FrankenSim WASM execution is claimed.
Change one setting and compare the consequences
Static worked exampleBar range: −0.4526 to +0.4526 fringes. The center line is zero. The scale is recalculated after Apply.
Accepted calculation 0. Every model row shares these inputs.
Inputs used for these results
- Total moving-water path per beam
- 3 m
- Signed water speed
- 7 m/s
- Assumed refractive index
- 1.333
- Vacuum wavelength
- 5.5 × 10⁻⁷ m
- Compare opposite flow directions (flow reversal)
- false
- Show the separately labeled later relativistic speed comparison
- false
| Observable | No drag | Full drag | Fresnel drag |
|---|---|---|---|
| Signed fringe shift (fringes) | 0 | 0.4526 | 0.1979 |
| First-order fringe shift (fringes) | 0 | 0.4526 | 0.1979 |
| Drag coefficient | 0 | 1 | 0.4372 |
| Travel-time difference (s) | 0 | 8.304 × 10⁻¹⁶ | 3.631 × 10⁻¹⁶ |
| Forward path speed (m/s) | 224900000 | 224900000 | 224900000 |
| Backward path speed magnitude (m/s) | 224900000 | 224900000 | 224900000 |
The table gives the one-direction beam comparison. Selecting flow reversal doubles it; changing the sign of the water speed reverses it. Fresnel's drag-speed law is a first-order description: exact travel-time arithmetic within that law is not an exact theory of a moving dielectric.
Light sent through moving water is carried along by it, but only partly. Comparing beams that travel with the flow and against it shows by how much, and three old hypotheses give three different answers.
Light in still water travels at c/n. If moving water carries the light along with a fraction f of its own speed v, a beam going with the flow travels at c/n + fv and one going against it at c/n − fv, and over a path Lw in the water the two arrive out of step by ΔN = (cLw/λ)(1/(c/n − fv) − 1/(c/n + fv)) fringes. No drag, f = 0, gives no shift; full drag, f = 1, gives the most; Fresnel's partial drag, f = 1 − 1/n2, lies between. With 3 m of water per beam, a flow of 7 m/s, n = 1.333 and light of 550 nm, the three predict 0, 0.4526 and 0.1979 fringes, and reversing the flow doubles each. Fizeau's measurement of 1851 favoured Fresnel's coefficient. In relativity the same coefficient follows, to first order in v/c, from adding the speeds of light and water by Einstein's composition law, which the lab keeps as a separately labelled later comparison.
A beam crossing the water path Lw at speed u takes Lw/u. The beam against the flow takes Lw/(c/n − fv) and the beam with it Lw/(c/n + fv). The difference, multiplied by the light's frequency c/λ, is the shift in fringes: ΔN = (cLw/λ)(1/(c/n − fv) − 1/(c/n + fv)). For small v this is close to (cLw/λ)(2fv)/(c/n)2 = 2Lwn2fv/(λc). With full drag, f = 1: 2 × 3 × 1.777 × 7/(5.5 × 10−7 × 2.998 × 108) = 74.6/164.9 = 0.4526 fringes. Fresnel's coefficient is f = 1 − 1/n2, and with n2 = 1.3332 = 1.777 that is 1 − 1/1.777 = 0.4372, so 0.4372 × 0.4526 = 0.1979 fringes. With no drag the two beams see the same speed and nothing shifts. Reverse the flow and the beams swap speeds, so each shift changes sign; comparing the two directions therefore measures twice the shift, 0.905 fringes for full drag and 0.396 for Fresnel's. The exact expression and the small-v form differ only by terms of order (nv/c)2, about 10−15 here, which is why the owner can use the exact one without the difference showing. The relativistic sum of c/n and v is (c/n + v)/(1 + v/(nc)), which to first order in v is c/n + v(1 − 1/n2): Fresnel's coefficient, with no medium for light required.
Fresnel proposed partial drag in 1818 to explain why aberration and refraction showed no sign of the Earth's motion. Fizeau measured the effect in running water in 1851 and found it close to Fresnel's value, and Michelson and Morley repeated the measurement with higher precision in 1886. Lorentz derived the coefficient from his electron theory in 1895, and von Laue showed in 1907 that it follows from Einstein's addition of velocities. The instrument's settings are illustrative rather than Fizeau's apparatus.
The explanation
Full explanation
Keep the water path per beam and wavelength fixed. Reverse the signed flow and distinguish a single-direction comparison from a flow-reversal measurement protocol.
Show every step of the investigation
Inspect the assumed drag coefficient, both path speeds, and the predicted fringe difference. The opt-in relativistic comparison is a later development, not evidence used to justify the earlier hypotheses. No historical observation is supplied by this preview.
An explanatory model, not an observation of nature. This embed starts from the laboratory’s worked defaults, not a saved run. Presentation options change the surrounding guide, never the numerical inputs.