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
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