大学物理自定义实验A rough simulation of rainbow.docx

A rough simulation of rainbow1351956 Thomas Xia, School of Automotive Studies, Tongji University 1. AbstractIn this project, I would like to explore how rainbows come into being and simulate a rainbow by software.Equipment: Laptop (ASUS A450E47JF-SL), Software(Matlab R2014a、Photoshop CC 2014).2. Analysis & Calculation2.1 After a shower of rain or nearby a waterfall, thin fog consist of tiny water drops will be widespread in the air. The parallel sunlight cast on a single drop will have two refractions and one reflection on the surface. Since water has different indexes of refraction to lights of different frequency, as a result, the drop bends light of different colors by different angles.2.2 Suppose the sunlight is horizontal, relative index of reflection of light with frequency v is n21. The radium of the drop is r. A beam has a distance h from the center of the drop. Angle of incidence is i. Angle of refraction is r。 The angle that light is bent .Rr”i”hirrisinisinr=n21 , r=i,=r,=i, ,sini,sinr,=1n21 , i=arcsinhR ,=2-2-i,-r,-i-r-r,-i, , (h)=4arcsinhRn21-2arcsinhR Because parallel beams of sunlight are evenly distributed in 0hR, for two near points on function (h), independent variable difference h can represent number of beams shoot out by angle from 0 to (0 + )：h0= 0 , h0+h= 0+ ,then，lim0h=dhd=-1(0) can represent the density of lights shoot out by angle 0.【In Matlab, we can also use function (h) to get many random data (i,hi) first, and then use function “ksdensity” to calculate s distribution to represent light density】Chapter 2.2 calculated the function between light density and angle by two ways, we mark this function as (). The function fits every single drop in the fog.Sydyd2.3 We simplify the thin water fog described in 2.1 into a very tall cube, which is from observers eyes by distance S. And only drops on its surface can sparkle directly into observers eyes.y=Stan dy=Ssec2tand Luminance received by the observerL=Cnumberdrop =C()(dxdy)=C()sec2tan (C、Care constants,is drops number density on the surface, dx is unit width) Here we ignore the energy loss when light travel long distance through the air.3. Simulation process3.1 Look up index of refraction of different lights on the Internet.wavelengthOptical glassWaterZFQKred656nm1.64441.51451.3311yellow589.3nm1.64991.51701.3330cyan486.1nm1.66571.52301.3371purple404.7nm1.68521.53181.3428for k=1:1:N h=rand(); alpha(1,k)=4*asin(h/n21)-2*asin(h);endy3=ksdensity(alpha*360/pi,0:0.1:100);for k=1:1:1001alpha=(0.1*k)*pi/360; y3(1,k)=10000*pi*(sec(alpha)2*tan(alpha)*y3(1,k);endplot(0:0.1:100,y3,g)hold on;n21=1.3428;alpha=0*rand(1,N);for k=1:1:N h=rand(); alpha(1,k)=4*asin(h/n21)-2*asin(h);endy4=ksdensity(alpha*360/pi,0:0.1:100);for k=1:1:1001 alpha=(0.1*k)*pi/360;y4(1,k)=10000*pi*(sec(alpha)2*tan(alpha)*y4(1,k);endplot(0:0.1:100,y4,m)clear;clc;N=50000000;n21=1.3311;alpha=0*rand(1,N);%随机取N个(h,)for k=1:1:N h=rand(); alpha(1,k)=4*asin(h/n21)-2*asin(h);end%单个水滴的出射光线分布y1=ksdensity(alpha*360/pi,0:0.1:100);%观察者眼中的光照强度for k=1:1:1001 alpha=(0.1*k)*pi/360;y1(1,k)=10000*pi*(sec(alpha)2*tan(alpha)*y1(1,k);endplot(0:0.1:100,y1,r)hold on;n21=1.3330;alpha=0*rand(1,N);for k=1:1:N h=rand(); alpha(1,k)=4*asin(h/n21)-2*asin(h);endy2=ksdensity(alpha*360/pi,0:0.1:100);for k=1:1:1001alpha=(0.1*k)*pi/360; y2(1,k)=10000*pi*(sec(alpha)2*tan(alpha)*y2(1,k);endplot(0:0.1:100,y2,y)hold on;n21=1.3371;alpha=0*rand(1,N);3.2 Make use of the calculation in chapter2 to design algorithm, run the program below in Matlab:y1(x), y2(x), y3(x), y4(x) are the luminance function of light with wavelength 656nm, 589.3nm, 486.1nm, 404.7nm, which are stored in Matlab Workspace now. This is a simulated “rainbow” in data form.3.3 Input function y1(x) y4(x) into Photoshop Gradient Tool, using the change of transparency to simulate the change of light luminance.3.4 Use several single-color layers to simulate light with different wavelengths in sunlight. Apply gradient styles we get in 3.3 to corresponding layers mask. At last, Photoshop can draw out the rainbow simulated in Matlab.Relative luminance / unit4. ConclusionVision angle / degree4.1. Simulation in MATLAB4.2. Simulation in PhotoshopFrom the simulation we know, Rainbow do appear when sunlight is cast on thin fog in assumed situation. It has a vision angle of approximate eighty degrees, when sunlight is horizontal. The red light, with the longest wavelength