# Physics 203 at Portland State 2014

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rainbow_project

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======Modelling the rainbow====== ==By Britt Vedder, Delvin Akins, and Jessica Ngin== ==in collaboration with Prof. Nicholas Kuzma== =====Introduction===== //Feel free to expand and contribute// ====Observations==== ...[[wp>Rainbow]] can be observed when... ===Local observations by authors=== In the [[wp>Portland, OR]] area, rainbows are bright and quite frequent. This spring, Prof. Kuzma observed a double rainbow three times in south-west Portland, all between the hours of 2pm and 6pm. | {{:projects:rainbow:rainbowbeaverhighway.jpg?nolink|}} | {{ :projects:rainbow:rainbowohsutram.jpg?direct&573 |}} | ^ Figure 1. Double rainbow observed from the Beaverton-Hillsdale highway by N.K. ^ Figure 2. View from the OHSU tram landing (N.K.) ^ ====History==== ... Scientific research on [[wp>Rainbow#Scientific_history|rainbow]] goes back to [[wp>Aristotle]], [[wp>Shen Kuo]] (China, 1031–1095), [[wp>Kamāl al-Dīn al-Fārisī]] (Persia, 1267–1319) and [[wp>Rene Descartes]] (Europe, 1596-1650)... | {{:projects:rainbow:descartes_rainbow.png?nolink|}} | ^ Figure 3. René Descartes' sketch of how primary and secondary rainbows are formed (//Courtesy Wikipedia//) ^ ====Cultural and cinematographic references==== Rainbow is mentioned in ... In some cultures, it can be a symbol of * Good luck * ... =====Theory===== ====Assumptions==== A simple, most commonly observed single or double rainbow can be explained by assuming the following: - Rainbow is formed by sun rays scattered by the water droplets of the rain - The droplets are much larger than the wavelengths of visible light, so diffraction effects can be mostly neglected - The droplets are perfect spheres (most cases can be explained this way) - The light enters each sphere, refracts, has multiple internal reflections, and exits via another refraction * <color red>Important</color>: The light enters the sphere not at a specific point, but at all the points, from hitting head-on to barely grazing it * In our model, we calculate what happens to **each** point of entry, then do the analysis of how much power is associated with each point * We parametrize the point of entry by the variable $d$, the distance from the path of the incident ray to the path of the central ray - Each ray of light will remain within a two-dimensional plane formed by that ray and the center of the droplet. - The droplets are sparse enough that the scattering of the same ray by multiple droplets can be neglected - The droplets are falling much slower than the speed of light, and their motion can be ignored - It can be shown, that most light intensity will exit at a specific angle - The refraction index of air can be assumed to be 1 or very close to one - The refraction index of water depends on the wavelength $\lambda$ (measured in air) in a known way - For each number of internal reflections, the intensity of scattered light as a function of angle can be modeled - For each number of internal reflections, there can be a separate rainbow ====Refraction==== * Snell's law: * $n_1\sin\theta_i=n_2\sin\theta_r$ ===From air into water=== The initial incident ray satisfies the equation: * $\arcsin\left(\frac{n_\text{air}\sin\theta_i}{n}\right)=\theta_r$ * Here $n_\text{air}=1.000283$ is the refraction index of air at 10$\,^\circ$C ([[rainbow project#references|Ref. [3]]]) * $n$ is the refraction index of water that depends on the vacuum wavelength of light, $\lambda$ ===From water into air=== The final emerging ray satisfies the equation:... ====Internal reflections==== * The law of reflection: * $\theta_1=\theta_2$ =====Methods===== ====Computational goals==== The main goal of this project is to predict, for each color, the intensity distribution of the scattered light as a function of the scattering angle. In addition, the following observable parameters can be elucidated: - The overall angles of the primary and secondary rainbows - The ordering of colors for each rainbow - The angular spread between the <color darkred>dark-red</color> and <color darkviolet>violet</color> colors in each rainbow - The relative brightness of the sky "background" inside and outside the rainbow - //feel free to suggest more// ====Software==== Microsoft Excel software will be used. The following functions are needed: - Creating a "grid" of numbers, e.g. a row or column of 400,410,420,...800 for the model wavelength (typing such an array is tedious, but it's quite easy using a "drag the corner" trick) * See the [[white noise project]] page for details - Creating a formula in the neighboring column, taking the preceding column as an input, and dragging it all the way to the end of the input column - Plotting (using scatter chart) the output column versus the input column * with main title * and with axis labels (called "axis titles" in Excel) - Automatically counting the number of cells in a certain range in an array, using ''COUNTIF()'' functions * See the [[white noise project]] page for details | {{ :projects:rainbow:rainbowdrawing1.png?direct |}} | ^ Figure 4. Ray diagram in the plane containing the droplet's center and the incoming ray ^ ====Coding tasks==== This is the detailed list of tasks to be accomplished: - Find and import the data on the refraction index of water as a function of wavelength measured in air * Create two columns: - one titled $\lambda$, for the input (from 400 nm to 800 nm in steps of 10 or 5 nm) - the other, titled //n//, for the refraction index values corresponding to the wavelengths to the left - create a scatter-plot of $n(\lambda)$ (choose "smooth lines" option) - Define the ray offset parameter - $\delta=\frac{d}{R}$ where //d// is the shortest distance from the line of the incident ray to the droplet's center, and //R// is the radius of the droplet - Find the incident angle $\theta_i$ for the first refraction in terms of the parameter $\delta$ - From the diagram in Fig. 4 above, figure out by what angle the light deviates from the original direction during - refraction from air into the droplet (angle $\beta_1$) - internal reflection (angles $\beta_2$, $\beta_3$, etc.). //Hint//: are these angles different or the same? - refraction from the water into the air (angles $\beta_2'$, $\beta_3'$, $\beta_4'$, etc.). //See hint above//. - Pick several rainbow colors and look up their wavelengths. For example, these classic seven: - <color red>Red</color> - <color orange>Orange</color> - <color gold>Yellow</color> - <color green>Green</color> - <color blue>Blue</color> - <color indigo>Indigo</color> - <color violet>Violet</color> - For each color, create a column of offset parameters from $\delta\!=\!0$ (central ray) to $\delta\!=\!1$ (grazing ray) with a fine step of 0.005 or even 0.001 - For each number of reflections //N// (0,1,2, and maybe 3?), calculate the total scattering angle $\gamma$ due to two refractions and //N// reflections. * Note that the total scattering angle is the sum of individual deviations in each process: * For //N// = 0, the total scattering angle is $\gamma_0$ $=\beta_1+\beta_2'$ * For //N// = 1, the total scattering angle is $\gamma_1$ $=\beta_1+\beta_2+\beta_3'$ * For //N// = 2, the total scattering angle is $\gamma_2$ $=\beta_1+\beta_2+\beta_3+\beta_4'$ - For //N// = 1 and 2, plot the scattering angles $\gamma_1$ and $\gamma_2$ as a function of $\delta$ - Calculate the relative scattered intensity for each color, each //N//, and each offset $\delta$ * Referring to Fig. 5 below, we can assume that the output power is some fixed fraction $s$ of the input power (since some light is lost to scattering corresponding to other values of //N//, and to initial reflection): * $P_\text{inp}$ $\,=\,I_\text{inp}\!\cdot\!{\text{Area}}_\text{inp}$ $\,\approx\,I_\text{inp}\!\cdot\!2\pi d\,\Delta d$ * $P_\text{sc}$ $\,=\,I_\text{sc}\!\cdot\!{\text{Area}}_\text{sc}$ $\,\approx\,I_\text{sc}\!\cdot\!2\pi\,(D\sin\gamma)\,D\Delta \gamma\;$, where $\,D$ is the distance from the droplet to the observer. * $P_\text{sc}$ $=s\!\cdot\!P_\text{inp}$ * $I_\text{sc}$ $=s\!\cdot\!I_\text{inp}\frac{d}{D^2\sin\gamma}\frac{\Delta d}{\Delta\gamma}$ $=s \cdot I_\text{inp}\frac{R^2\delta}{D^2\sin\gamma}\frac{\Delta \delta}{\Delta\gamma}$ $\sim I_\text{inp}\frac{\delta}{\sin\gamma}\frac{\Delta \delta}{\Delta\gamma}\;$: here the factors //s//, //R//, and //D// are pretty constant (or too complicated to calculate) and do not contribute to the relative intensity at various angles and wavelengths | {{ :projects:rainbow:rainbowdrawing2.png?nolink |}} | ^ Figure 5. Input power in the annulus between offset distances $d$ and $d\!+\!\Delta d$ (hatched ring on the left) scatters into the conical annulus between the corresponding scattering angles $\gamma$ and $\gamma\!+\!\Delta\gamma$ (gray ring). ^ =====Data===== Give yourself 20 minutes to google or search the literature for the index of refraction of water as a function of wavelength. Paste references and data (formulas and raw numbers if compact) below. **Index of refraction for water** = 1.33. This is the average value. We need its dependence on the wavelength $\lambda$! **Average radius of water droplet** = 3 mm ([[rainbow project#references|Ref. [4]]]) {{visible_spectrum.jpg}} | **Table 1**. Refraction index of pure water at $T=10\,^\circ$C, $\rho=999.7\,\frac{\text{kg}}{ {\text m}^3}$ \\ //Calculated by N.K. from Ref. 2// ||| ^ $\lambda$ (nm) ^ $(n^2−1)/(n^2\!+2)$ ^ $n$ ^ | 400 | 0.211798 | 1.343924521 | | 410 | 0.211276 | 1.342985396 | | 420 | 0.210789 | 1.342112754 | | 430 | 0.210336 | 1.341300053 | | 440 | 0.209913 | 1.340541623 | | 450 | 0.209518 | 1.339832528 | | 460 | 0.209147 | 1.339168446 | | 470 | 0.208799 | 1.338545577 | | 480 | 0.208472 | 1.337960559 | | 490 | 0.208165 | 1.337410411 | | 500 | 0.207875 | 1.336892472 | | 510 | 0.207602 | 1.33640436 | | 520 | 0.207344 | 1.335943933 | | 530 | 0.207101 | 1.335509257 | | 540 | 0.206871 | 1.33509858 | | 550 | 0.206654 | 1.334710307 | | 560 | 0.206448 | 1.334342982 | | 570 | 0.206253 | 1.33399527 | | 580 | 0.206069 | 1.333665945 | | 590 | 0.205894 | 1.333353873 | | 600 | 0.205728 | 1.333058007 | | 610 | 0.205571 | 1.332777373 | | 620 | 0.205421 | 1.332511064 | | 630 | 0.205279 | 1.332258233 | | 640 | 0.205145 | 1.332018084 | | 650 | 0.205017 | 1.331789872 | | 660 | 0.204895 | 1.331572894 | | 670 | 0.204779 | 1.331366485 | | 680 | 0.204669 | 1.331170018 | | 690 | 0.204564 | 1.330982896 | | 700 | 0.204464 | 1.330804551 | | 710 | 0.204368 | 1.330634445 | | 720 | 0.204277 | 1.33047206 | | 730 | 0.204190 | 1.330316905 | | 740 | 0.204106 | 1.330168507 | | 750 | 0.204027 | 1.330026414 | | 760 | 0.203950 | 1.329890191 | | 770 | 0.203877 | 1.32975942 | | 780 | 0.203806 | 1.3296337 | | 790 | 0.203738 | 1.329512641 | | 800 | 0.203672 | 1.329395872 | =====References===== - http://refractiveindex.info/legacy/?group=LIQUIDS&material=Water - I. Thormählen, J. Straub, and U. Grigull,\\ //[[http://www.nist.gov/data/PDFfiles/jpcrd282.pdf|"Refractive Index of Water and Its Dependence on Wavelength, Temperature, and Density"]]//,\\ J. Phys. Chem. Ref. Data **14**, 933 (1985)\\ Note, that Eq.(5) <color red>contains a typo</color>: the left-hand side, typeset as $\frac{n^2-1}{n^2+1}\!\cdot\!\frac{1}{\rho^*}=$, should read $\frac{n^2-1}{n^2+2}\!\cdot\!\frac{1}{\rho^*}=$ instead. - "Handbook of Optical Metrology: Principles and Applications", edited by Toru Yoshizawa, Ch. 16: "Displacement", A. Hirai et al., p. 409. CRC Press, 2009. ISBN: [[http://books.google.com/books?id=DdzBQsqPbzcC&pg=PA408&dq=refractive+index+of+air+temperature&hl=en&sa=X&ei=-iZ8U5e9Es3eoASEo4KgBQ&ved=0CC0Q6AEwAA#v=onepage&q=FIGURE%2016.15&f=false|978-0-8493-3760-4]] - Emmanuel Villermaux, Benjamin Bossa.\\ "//Single-drop fragmentation distribution of raindrops.//"\\ Nature Physics **5** (9): 697-702 (September 2009).\\ [[http://doi.org/10.1038/NPHYS1340|doi:10.1038/NPHYS1340]]. Lay summary