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SLIDE 1

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❚✉r❜✉❧❡♥❝❡

❙❖❊✸✷✶✶✴✷ ❋❧✉✐❞ ▼❡❝❤❛♥✐❝s ❧❡❝t✉r❡ ✺

slide-2
SLIDE 2

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❚✉r❜✉❧❡♥❝❡ ✭❆✮

❚✉r❜✉❧❡♥❝❡ ✐s ❞✐✍❝✉❧t t♦ ❞❡☞♥❡ ♣r❡❝✐s❡❧② ④ ❡❛s✐❡r t♦ ❞✐s❝✉ss ✐ts ♣r♦♣❡rt✐❡s ✿

✎ st❛t❡ ♦❢ ✌✉✐❞ ♠♦t✐♦♥ ❝❤❛r❛❝t❡r✐s❡❞ ❜② ❝♦♠♣❧❡①✱ ❝❤❛♦t✐❝

♠♦t✐♦♥

✎ q✉❛s✐✲r❛♥❞♦♠ ♠♦t✐♦♥s ✎ ♦❢t❡♥ ❞❡s❝r✐❜❡❞ ✐♥ t❡r♠s ♦❢ t✉r❜✉❧❡♥t ❡❞❞✐❡s ♦❢ ❞✐☛❡r❡♥t

s❝❛❧❡s ✐♥ t❤❡ ✌♦✇

✎ ✈♦rt✐❝✐t② r❡♣r❡s❡♥ts str❡♥❣t❤ ♦❢ ❡❞❞✐❡s

❲❤❡r❡ ❞♦❡s ✐t ♦❝❝✉r❄

✎ ❲❛❧❧ t✉r❜✉❧❡♥❝❡ ✿ ✇❛❧❧s ✭t✉r❜✉❧❡♥t ❜✳❧✳✮✱ ♣✐♣❡s ❡t❝ ✎ ❋r❡❡ t✉r❜✉❧❡♥❝❡ ✿ ✇❛❦❡s✱ ❥❡ts

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SLIDE 3

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

■♥❞✐✈✐❞✉❛❧ ❡❞❞✐❡s ♦❜❡② ◆❙ ❡q✉❛t✐♦♥s ✭♦♥❡ s♦❧✉t✐♦♥ t❡❝❤♥✐q✉❡ ✐s t♦ ❝♦♠♣✉t❡ t❤❡♠ ④ ❉✐r❡❝t ◆✉♠❡r✐❝❛❧ ❙✐♠✉❧❛t✐♦♥✱ ❉◆❙✮✳ ■♥❞✐✈✐❞✉❛❧ ❡❞❞✐❡s ✐♥t❡r❛❝t ✰ t❡♥❞ t♦ ❜r❡❛❦ ✉♣✳

slide-4
SLIDE 4

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❊♥❡r❣② ✐♥ t✉r❜✉❧❡♥❝❡

✶ st❛rts ♦☛ ✐♥ ❧❛r❣❡ s❝❛❧❡ ❡❞❞✐❡s ✷ ✐s tr❛♥s♠✐tt❡❞ t♦ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ❡❞❞✐❡s ✸ ✉♥t✐❧ ✐t ❡♥❞s ✉♣ ✐♥ t❤❡ s♠❛❧❧❡st ♣♦ss✐❜❧❡ ❡❞❞✐❡s

❚❤❡ s♠❛❧❧❡st ❡❞❞✐❡s ❛r❡ t❤♦s❡ ❞♦♠✐♥❛t❡❞ ❜② ✈✐s❝♦✉s ❡☛❡❝ts ✿ t❤❡✐r ❡♥❡r❣② ✐s ❞✐ss✐♣❛t❡❞ ❛s ❤❡❛t✳ ❚❤✐s ♣❛tt❡r♥ ✐s ❦♥♦✇♥ ❛s t❤❡ t✉r❜✉❧❡♥t ❝❛s❝❛❞❡✳

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SLIDE 5

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

■❢ ✇❡ ♣❧♦t t❤❡ ❡♥❡r❣② ❧♦❣ ❊ ✈❡rs✉s ❛ ❝❤❛r❛❝t❡r✐st✐❝ ❧❡♥❣t❤ ♦❢ t❤❡ ❡❞❞② ❧♦❣ ✕ ✇❡ ❣❡t ❛♥ ❡♥❡r❣② s♣❡❝tr✉♠ ✿

log λ log E

η

  • 5/3
slide-6
SLIDE 6

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

◆♦t❡

✶ ❑♦❧♠♦❣♦r♦✈ ❧❡♥❣t❤ s❝❛❧❡ ✑ ✇❤❡r❡ ✈✐s❝♦✉s ❡☛❡❝ts ❞♦♠✐♥❛t❡✱

✐✳❡✳ ❘❡✑ ❂ ✉✵✑

✗ ✘ ✶ ✷ ❙❧♦♣❡ ♦❢ ❡♥❡r❣② ❝❛s❝❛❞❡ ✐s ✺❂✸

slide-7
SLIDE 7

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ✭❆✮

◗✉❡st✐♦♥ ✿ ❤♦✇ ❛r❡ ✇❡ ❣♦✐♥❣ t♦ ❝❤❛r❛❝t❡r✐s❡ s✉❝❤ ❛ ❝♦♠♣❧❡① ✌♦✇❄ ■♠❛❣✐♥❡ ♠❡❛s✉r✐♥❣ ✉① ✐♥ ❛ ❧❛♠✐♥❛r ✌♦✇ ✿

t u (t) x

■♥ ❛ t✉r❜✉❧❡♥t ✌♦✇ t❤❡ ❣r❛♣❤ ✇♦✉❧❞ ❧♦♦❦ ❧✐❦❡ t❤✐s ✿

t u (t) x

slide-8
SLIDE 8

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❉❡s❝r✐❜❡ t❤✐s ❛s ✬✌✉❝t✉❛t✐♦♥s✬ ❛r♦✉♥❞ ❛♥ ✬❛✈❡r❛❣❡✬ ✈❛❧✉❡

t u (t) x ux

❉❡☞♥❡ t❤❡ t✐♠❡ ❛✈❡r❛❣❡ ♦❢ ✉① ✉① ❂ ✶ ✁t ❩ t✰✁t

t

✉①✭t✮❞t ❍♦✇ ❜✐❣ ✐s ✁t❄ ❉❡♣❡♥❞s ♦♥ t❤❡ ❝❛s❡ ✿

✎ ■❢ t❤❡ ✌♦✇ ✐s q✉❛s✐✲st❡❛❞②✱ ✁t ❝❛♥ ❜❡ ❛s ❧♦♥❣ ❛s ♣r❛❝t✐❝❛❧✳ ✎ ■❢ t❤❡ ✌♦✇ ✈❛r✐❡s ✇✐t❤ t✐♠❡s❝❛❧❡ t✈❛r ✭❡❣✳ ❛ ♣❡r✐♦❞✐❝ ✌♦✇✮

✁t ✜ t✈❛r

slide-9
SLIDE 9

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❉❡s❝r✐❜❡ t❤✐s ❛s ✬✌✉❝t✉❛t✐♦♥s✬ ❛r♦✉♥❞ ❛♥ ✬❛✈❡r❛❣❡✬ ✈❛❧✉❡

t u (t) x ux

❉❡☞♥❡ t❤❡ t✐♠❡ ❛✈❡r❛❣❡ ♦❢ ✉① ✉① ❂ ✶ ✁t ❩ t✰✁t

t

✉①✭t✮❞t ❍♦✇ ❜✐❣ ✐s ✁t❄ ❉❡♣❡♥❞s ♦♥ t❤❡ ❝❛s❡ ✿

✎ ■❢ t❤❡ ✌♦✇ ✐s q✉❛s✐✲st❡❛❞②✱ ✁t ❝❛♥ ❜❡ ❛s ❧♦♥❣ ❛s ♣r❛❝t✐❝❛❧✳ ✎ ■❢ t❤❡ ✌♦✇ ✈❛r✐❡s ✇✐t❤ t✐♠❡s❝❛❧❡ t✈❛r ✭❡❣✳ ❛ ♣❡r✐♦❞✐❝ ✌♦✇✮

✁t ✜ t✈❛r

slide-10
SLIDE 10

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

◆♦♥✲st❡❛❞② ✌♦✇ ✿

t u (t) x ux

❍♦✇❡✈❡r t❤❡ t✐♠❡ ❛✈❡r❛❣❡ ♦♥❧② ❞❡s❝r✐❜❡s ♣❛rt ♦❢ t❤❡ ✌♦✇✳ ■♥tr♦❞✉❝❡ t❤❡ ✌✉❝t✉❛t✐♦♥ ✉✵

✉①✭t✮ ❂ ✉① ✰ ✉✵

❋❛✐r❧② ♦❜✈✐♦✉s❧②✱ ✉✵

① ❂ ✵

❍♦✇❡✈❡r ✉✵

① ✷ ✻❂ ✵

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SLIDE 11

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

◆♦♥✲st❡❛❞② ✌♦✇ ✿

t u (t) x ux

❍♦✇❡✈❡r t❤❡ t✐♠❡ ❛✈❡r❛❣❡ ♦♥❧② ❞❡s❝r✐❜❡s ♣❛rt ♦❢ t❤❡ ✌♦✇✳ ■♥tr♦❞✉❝❡ t❤❡ ✌✉❝t✉❛t✐♦♥ ✉✵

✉①✭t✮ ❂ ✉① ✰ ✉✵

❋❛✐r❧② ♦❜✈✐♦✉s❧②✱ ✉✵

① ❂ ✵

❍♦✇❡✈❡r ✉✵

① ✷ ✻❂ ✵

slide-12
SLIDE 12

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

◆♦♥✲st❡❛❞② ✌♦✇ ✿

t u (t) x ux

❍♦✇❡✈❡r t❤❡ t✐♠❡ ❛✈❡r❛❣❡ ♦♥❧② ❞❡s❝r✐❜❡s ♣❛rt ♦❢ t❤❡ ✌♦✇✳ ■♥tr♦❞✉❝❡ t❤❡ ✌✉❝t✉❛t✐♦♥ ✉✵

✉①✭t✮ ❂ ✉① ✰ ✉✵

❋❛✐r❧② ♦❜✈✐♦✉s❧②✱ ✉✵

① ❂ ✵

❍♦✇❡✈❡r ✉✵

① ✷ ✻❂ ✵

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SLIDE 13

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

✶ ✷✉✵ ① ✷ ✐s ♦♥❡ ❝♦♠♣♦♥❡♥t ♦❢ ❛ ❦✐♥❡t✐❝ ❡♥❡r❣②

❦ ❂ ✶ ✷ ✏ ✉✵

① ✷ ✰ ✉✵ ② ✷ ✰ ✉✵ ③ ✷✑

④ t❤❡ t✉r❜✉❧❡♥t ❦✐♥❡t✐❝ ❡♥❡r❣② ■❢ t❤❡ t✉r❜✉❧❡♥❝❡ ✐s ✐s♦tr♦♣✐❝✱ t❤❡♥ ✉✵

① ✷ ❂ ✉✵ ② ✷ ❂ ✉✵ ③ ✷ ❂ ✉✵✷❀

❛♥❞ ❦ ❂ ✸ ✷✉✵✷ ❲❡ ❝❛♥ ❛❧s♦ ❝♦♥s✐❞❡r t❤❡ r❛t❡ ♦❢ ❞✐ss✐♣❛t✐♦♥ ♦❢ t✉r❜✉❧❡♥t ❦✐♥❡t✐❝ ❡♥❡r❣②✳ ❚❤✐s ✐s ✉s✉❛❧❧② ❞❡♥♦t❡❞ ✎✳

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SLIDE 14

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮

✷ ♠❛✐♥ ❡☛❡❝ts ❤❡r❡ ✿

✶ ❉✐ss✐♣❛t✐♦♥ ♦❢ ✌♦✇ ❡♥❡r❣② ④ t❤❡ t✉r❜✉❧❡♥t ♠♦t✐♦♥ ❝♦♥t❛✐♥s

❦✐♥❡t✐❝ ❡♥❡r❣② ✉♥r❡❧❛t❡❞ t♦ t❤❡ ♠❡❛♥ ♠♦t✐♦♥ ♦❢ t❤❡ ✌✉✐❞

✷ ❉✐☛✉s✐✈❡ ❡☛❡❝ts

❈♦♥s✐❞❡r ❛ ♣❛rt✐❝❧❡ ✐♥ t❤❡ ✌♦✇✳ ❚❤✐s ♣❛rt✐❝❧❡ ✇✐❧❧ ❜❡ s✇❡♣t ❛❧♦♥❣ ❜② s✉❝❝❡ss✐✈❡ ❡❞❞✐❡s✱ ❛♥❞ t❤✉s ❜❡ tr❛♥s♣♦rt❡❞ ❢r♦♠ ✐ts st❛rt✐♥❣ ♣♦✐♥t✳ ◆❡✐❣❤❜♦✉r✐♥❣ ♣❛rt✐❝❧❡s ♠❛② s❡❡ ❞✐☛❡r❡♥t ❡❞❞✐❡s✱ ❛♥❞ ❡♥❞ ✉♣ ❛ ❧♦♥❣ ✇❛② ❛♣❛rt✳ ❙✐♠✐❧❛r t♦ t❤❡ ❵r❛♥❞♦♠ ✇❛❧❦✬ ♠♦❧❡❝✉❧❛r ♣r♦❝❡ss ❢♦r r❡❛❧ ❞✐☛✉s✐♦♥ ✐♥ ❣❛s❡s ④ t❤✉s t✉r❜✉❧❡♥❝❡ ♣r♦❞✉❝❡s ❛ ❞✐☛✉s✐✈❡ ❡☛❡❝t✳

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SLIDE 15

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❲❡ ♥♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣

✶ ❚❤❡r❡ ❤❛s t♦ ❜❡ ❛ ❧❛♠✐♥❛r r❡❣✐♦♥ ❝❧♦s❡ t♦ t❤❡ ✇❛❧❧

✎ ✇❛❧❧ ❧❛②❡r✴✈✐s❝♦✉s s✉❜❧❛②❡r ✎ ✜✈✐s❝ ✢ ✜t✉r❜

✷ ❋❛r ❢r♦♠ t❤❡ ✇❛❧❧ t❤❡r❡ ✇✐❧❧ ❜❡ ❛ t✉r❜✉❧❡♥t r❡❣✐♦♥ ✇❤❡r❡ ✉

r❡❣❛✐♥s t❤❡ ❢r❡❡ str❡❛♠ ✈❡❧♦❝✐t②

✎ ❢r❡❡ t✉r❜✉❧❡♥t✴❧♦❣✲❧❛✇ r❡❣✐♦♥ ✎ ✜✈✐s❝ ✜ ✜t✉r❜

✸ ■♥ ❜❡t✇❡❡♥ ✿ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ r❡❣✐♦♥

✎ ✇❛❧❧ t✉r❜✉❧❡♥t✴tr❛♥s✐t✐♦♥ r❡❣✐♦♥ ✎ ✜✈✐s❝ ✘ ✜t✉r❜

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SLIDE 16

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❖❢t❡♥ ✇r✐t❡ t❤❡ ❣r♦✉♣ ②✉✜ ✗ ❂ ②✰❀ ✉✜ ❂ r✜✵ ✚ ❛♥❞ r❡❢❡r t♦ ❞✐st❛♥❝❡s ♠❡❛s✉r❡❞ ✐♥ ✇❛❧❧ ✉♥✐ts

✎ ❱✐s❝♦✉s s✉❜❧❛②❡r ✵ ❁ ②✰ ❁ ✺ ✎ ❚r❛♥s✐t✐♦♥ ✺ ❁ ②✰ ❁ ✸✵ ✎ ❋r❡❡ t✉r❜✉❧❡♥t ②✰ ❃ ✸✵

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SLIDE 17

❚✉r❜✉❧❡♥❝❡ ❚✉r❜✉❧❡♥❝❡ ✭❆✮ ❉❡s❝r✐♣t✐♦♥ ♦❢ t✉r❜✉❧❡♥❝❡ ✭✈✷✮ ❊☛❡❝ts ♦❢ t✉r❜✉❧❡♥❝❡ ✭❆✮ ❚✉r❜✉❧❡♥t ❇♦✉♥❞❛r② ▲❛②❡rs ✭❆✮

❚❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ♠❛② st❛rt ❧❛♠✐♥❛r ✰ ❜❡❝♦♠❡ t✉r❜✉❧❡♥t

Laminar Log Law Transition Laminar Sublayer

❚r❛♥s✐t✐♦♥ ✇✐❧❧ ❞❡♣❡♥❞ ♦♥ ❘❡①✱ ❛s ❞❡☞♥❡❞ ❜❡❢♦r❡ ✿ ❘❡① ❂ ❯✶① ✗

✎ ❆❜♦✈❡ ❘❡① ✘ ✺ ✂ ✶✵✺ t❤❡ ❜✳❧✳ ✐s t✉r❜✉❧❡♥t✳ ✎ ❇❡❧♦✇ ❘❡① ✘ ✶✵✺ ✐t ✐s ❧❛♠✐♥❛r ✎ ❚r❛♥s✐t✐♦♥❛❧ ❢♦r ✐♥t❡r♠❡❞✐❛t❡ ✈❛❧✉❡s ✎ ❚✉r❜✉❧❡♥❝❡ ❝❛♥ ❜❡ tr✐❣❣❡r❡❞ ❡❛r❧② ❜② r♦✉❣❤ s✉r❢❛❝❡s ✎ ❖r ❝❛♥ r❡♠❛✐♥ ❧❛♠✐♥❛r ✐❢ t❤❡ s✉r❢❛❝❡ ✐s ✈❡r② s♠♦♦t❤