ProblemProblem 5.5.

MirageMirage

MirageMirage

Create a mirage like a road or Create a mirage like a road or desert mirage in the laboratory and desert mirage in the laboratory and study its parameters.study its parameters.

IntroductionIntroduction

• What is mirage?What is mirage?

• How is it formed?How is it formed?

• WWhihich are parameters of a mirage?ch are parameters of a mirage?

ExperimentExperiment

EffectEffect::heated heated surfacesurface

observedobserved objectobject

Formation of mirageFormation of mirage

SnellSnell’s’s lawlaw:: 2211 sinsin nn

φφ11

φφ22

nn11

nn22

nn11 – index of refraction of first medium – index of refraction of first medium

nn22 – index of refraction of second – index of refraction of second

mediummedium

φφ11 – ray angle in first medium – ray angle in first medium

φφ22 – ray angle in second medium – ray angle in second medium

nn11 > n > n2 2 → → ray is refracting away ray is refracting away

fromfrom vertical vertical

Formation of mirageFormation of mirage

- applying Snell’s law on sequence of parallel - applying Snell’s law on sequence of parallel planes of infinitesimal thickness whose index of planes of infinitesimal thickness whose index of refraction is changing monotonously:refraction is changing monotonously:

sin...sinsin 1100 nnn

x

y

Formation of mirageFormation of mirage

In one moment (sooner or later), if In one moment (sooner or later), if sequence of nsequence of nii isn’t convergent, will be: isn’t convergent, will be:

1sin...sinsin 11100 nnn nnnn

Total reflection occurs and ray is Total reflection occurs and ray is returning.returning.

Mirage parametersMirage parameters

• index of refraction dependence of temperature index of refraction dependence of temperature

• temperature dependence of shift in direction temperature dependence of shift in direction parallel with temperature gradientparallel with temperature gradient

nn11

nn22

nn33

……nnNN

temperature temperature index of index of refractionrefraction

EExxperimentperiment

horizontal linehorizontal line

vesselvessel withwith flouresceinflourescein

termomettermometeerr

EExxperimentperiment

• temperature distribution was measured with temperature distribution was measured with thermometer put in the waterthermometer put in the water

• rays are photographed and deviation from rays are photographed and deviation from horizontal line is measuredhorizontal line is measured

Theoretical modelTheoretical model

It is sufficient to derive ray equation.It is sufficient to derive ray equation.

• assumptions:assumptions:

• linear dependence of temperature linear dependence of temperature of depth i.e.:of depth i.e.:

• linear dependence of index of linear dependence of index of refraction of temperature i.e.:refraction of temperature i.e.:

Tkdy

dT

nkdT

dn

Theoretical modelTheoretical model

We will use following coordinate system and We will use following coordinate system and following equations:following equations:

yy

xx

dxdx

φφ -dy-dy

tandy

dx

)sin()()sin( ddnnn …………..

Snell’s law in differential form:Snell’s law in differential form:

Theoretical modelTheoretical model

qe

rrepyy p

x

p

x1

2

10

nTkk

np 00 sin

Ray equation:Ray equation:

0sin

1

q

12 qqr

yy00 – initial depth – initial depth

nn00 – initial index of – initial index of

refractionrefraction

φφ00 – initial ray angle – initial ray angle

dy

dTkT

dT

dnkn

ResultsResultsTemperature gradient 01

depth [cm]

0 5 10 15 20 25 30

Tem

pera

ture

[K]

290

300

310

320

330

340

350

360

370

ResultsResults

kkT1T1=-0.78Kmm=-0.78Kmm-1-1

kkT3T3=-0.40Kmm-1=-0.40Kmm-1

kkT2T2=-0.32Kmm-1=-0.32Kmm-1

kkT4T4=-0.27Kmm-1=-0.27Kmm-1

Mirage gradient 02

depth [cm]

0 5 10 15 20 25 30

Te

mp

era

ture

[K

]

290

300

310

320

330

340

Mirage gradient 03

depth [cm]

0 5 10 15 20 25 30

Te

mp

era

ture

[K]

290

300

310

320

330

340

Mirage gradient 04

depth [cm]

0 5 10 15 20 25 30

Te

mp

era

ture

[K

]

295

300

305

310

315

320

325

330

ResultsResultsMirage gradient

dubina [cm]

0 5 10 15 20 25 30

Tem

pera

tura

[K

]

290

300

310

320

330

340

350

360

370

gradient 1

gradient 3

gradient 2

gradient 4

ResultsResults

deviation [mm]

0 100 200 300 400 500

de

pth

[mm

]

12

14

16

18

20

22

24

measurementsray equation fit

ResultsResults

kkn1n1=-4.86*10=-4.86*10-4-4KK-1-1

kkn3n3=-4.58*10=-4.58*10-4-4KK-1-1

kkn2n2=-4.68*10=-4.68*10-4-4KK-1-1

kkn4n4=-4.75*10=-4.75*10-4-4KK-1-1

Mirage 2

deviation [cm]

0 20 40 60

de

pth

[m

m]

12

14

16

18

20

22

24

26

Mirage 3

deviation [mm]

0 200 400 600

de

pth

[m

m]

18

20

22

24

26

28

30

32

Mirage 4

deviation [cm]

0 10 20 30 40 50 60

dep

th [m

m]

16

18

20

22

24

26

28

30

32

ResultsResults

For coefficient of dependence of index of refraction For coefficient of dependence of index of refraction of water of water on temperature on temperature we got (in interval we got (in interval ≈ 318K – ≈ 318K – 358K358K):):

1410)06.072.4( KkdT

dnn

According to According to literature: literature:

14105.4 KdT

dn

ConclusionConclusion

• we created mirage in a laboratorywe created mirage in a laboratory

• we constructed theoretical model whose results are we constructed theoretical model whose results are good compared to literaturegood compared to literature

• mirage parameters are dependence of temperature mirage parameters are dependence of temperature oonn depth and dependence of index of refraction o depth and dependence of index of refraction onn temperaturetemperature

• for water:for water:

• for practical reasons we photographedfor practical reasons we photographed and and measuredmeasured ray in water but we have also created ray in water but we have also created mirage in airmirage in air

1410)06.072.4( KdT

dn