1.6.22 · D1Shock waves — Mach number, Mach cone — - CRITICAL for rocke
Ek chalta hua sound-maker hamesha ripples phenkta rehta hai jo hamesha fixed speed of sound par travel karti hain; agar maker us fixed speed se tez chale, toh woh apni ripples ko peeche chhod deta hai aur woh ek cone-shaped wall of squeezed air mein stack ho jaati hain. Parent page par jo bhi hai woh sirf do speeds aur unse banne wale triangle ka bookkeeping hai.
Yeh page assume karta hai ki aapko parent note ke symbols ke baare mein kuch bhi nahi pata. Hum har ek cheez banate hain — pehle plain words mein, phir ek picture, phir yeh kyun zaroori hai — ek aisi tartib mein jahan har naya piece sirf unhi cheezoon par depend karta hai jo pehle se aapke paas hain.
Kisi bhi symbol se pehle, aapko ek mental image chahiye. Ek pathar ko shaant talab mein phenken. Ek ring baahir ki taraf phailti hai. Woh ring ek wavefront hai: un sab points ka set jahan disturbance ek pal mein pahunch gayi hai.
Wavefront ek disturbance ka chalta hua front edge hota hai — us par har point ne source ko ek hi waqt chhoda hota hai aur tab se utni hi doori taiy ki hoti hai. Pond mein yeh ek circle hota hai; khuli hawa mein (3D mein) yeh ek sphere hota hai.
Intuition Hum kyun parwah karte hainSound exactly yahi hoti hai, lekin hawa mein. Source har pal pressure ka ek naya sphere banata hai. Poori shock-wave story bas yahi hai ki jab source chalta hai toh in spheres ki family ka kya hota hai .
Prerequisite: yeh pond-ripple idea Speed of sound in a medium se aata hai aur, zyada broadly, wave motion se.
t t elapsed time hai — kitne seconds guzre hain jab se humne clock shuru kiya (jaise, jab se ek ripple release hui). Iske units seconds (s t
Speed = taiy ki gayi doori ÷ laga hua time. Iska notation ek letter hota hai (hum v m/s
Ek ghonghey ki picture socho jo chal raha hai. Agar woh 2 3 3/2 = 1.5 m/s m/s
Intuition Topic ko yeh kyun chahiyePoora subject do cheezon ki race hai. Race do speeds ki tulna hai — isliye hamen pehle ek speed ko measure karna aur name karna aana chahiye.
Ek algebra fact jo hum baar baar use karenge:
Reveal check:
Agar koi cheez speed v t v × t
Definition Speed of soundv s v s 340 m/s
Subscript "s s ound" — yeh kuch multiply nahi karta. v s
Intuition Zaroori mental hookv s parwah nahi ki source kitni tez chal raha hai. Har ripple, ek baar release hone ke baad, v s
v s v s Speed of sound in a medium mein hai. Is page ke liye isse ek jaana-pehchaana fixed number maano.
Common mistake "Plane sound ko tez push karta hai."Kyun sahi lagta hai: ek pheka hua ball tez jaata hai agar phenkne wala bhaage. Fix: sound hawa se carry hoti hai, plane se nahi. Ek baar ripple nikal jaaye, woh source ko bilkul bhool jaati hai aur v s aage nikalne deti hai.
v obj v obj m/s
Ab hamare paas do racers ek saath khade hain:
Symbol
Kaun
Fixed?
v s ripple (sound)
haan, hawa se set hoti hai
v obj source (jet/rocket)
nahi, pilot choose karta hai
Intuition Dono kyun, sirf ek kyun nahiShock wave purely is baare mein hai ki "kaun tez hai". Aap physically sirf ek speed se yeh sawaal nahi pooch sakte. Agla symbol bas inhe dono ko ek single number mein glue karta hai.
M M M = v s v obj ratio hai, isliye iske koi units nahi — upar aur neeche ke m/s
Aise padhein: M = 2
Intuition Ratio kyun, subtraction kyun nahi?Hum pooch sakte the "kitna tez hai" (v obj − v s geometry (jo aage aane wali hai) proportion par depend karti hai, gap par nahi. Saath hi v s 500 m/s M kya aapne apni ripples ko beat kiya? — ek clean number mein:
M < 1 Subsonic. M = 1 Transonic — "sound barrier".M > 1 Supersonic — shock form hoti hai.
Reveal check:
Ek jet 680 m/s v s = 340 m/s M = 680/340 = 2
Ab do racers same elapsed time t v t
Sabse purani ripple radius v s t ×
Source apni seedhi track par v obj t
Jab v obj > v s baahir khatam hota hai, aur sphere cone ki slanted wall ko sirf touch karta hai. Woh touch point, source ka start, aur source ki abhi ki position ek right-angled triangle banate hain.
Definition Right triangle aur uske teen sidesEk right triangle mein ek 9 0 ∘ relative name dete hain:
opposite side angle ko face karti hai (use touch nahi karti),adjacent side angle ko touch karti hai lekin sabse lamba nahi hoti,hypotenuse sabse lamba side hota hai, right angle ko face karta hua .
Hamaari picture mein:
Hypotenuse = v obj t Apex angle ka opposite side = v s t
Apex angle ka adjacent side = bacha hua side jo flight track ke saath hai,
Apex par chhota corner woh angle hai jo hum chahte hain.
Kyunki ek seedha cone wall jo ek badhte sphere ko touch karta hai touch point par right angle banata hai (wall tangent hai, radius ek tangent se 9 0 ∘
θ θ angle ka naam hai — ghoomne ki ek quantity, degrees mein measure ki gayi. Yahan yeh cone ka half-angle hai, apex (object ki naak) par baitha hua.
Ab hamen ek tool chahiye jo ek angle ko hamare triangle ke do sides se connect kare. Woh tool sine hai.
Ek right triangle mein ek angle ke liye,
sin θ = hypotenuse opposite . 0 1 kitna proportion hai.
Intuition Sine kyun, tangent ya cosine kyun nahi?Humne woh do sides choose kiye jo hamare paas actually hain : opposite (ripple radius v s t hypotenuse (source path v obj t Opposite over hypotenuse se bana hua trig ratio sine hai. Cosine adjacent side use karta hai, tangent bhi adjacent side use karta hai — lekin humne "adjacent" kabhi measure nahi kiya, isliye unhe extra kaam chahiye. Sine hamare paas ke sides ke liye natural fit hai.
Hamare sides feed karein:
sin θ = v obj t v s t = v obj v s = M 1 . t v s / v obj ulta hai — yahan se 1/ M Yeh ek line poora parent formula hai.
sin − 1 sin sin − 1 ulta chalaata hai: isse ratio do, yeh angle wapas deta hai. Isliye sin θ = 1/ M θ = sin − 1 ( 1/ M )
Intuition Ratio 1 se zyada kyun nahi ho sakta — aur woh kya batata haiOpposite side hypotenuse se lamba kabhi nahi ho sakta (slanted side hamesha sabse lamba hota hai). Isliye sin θ ≤ 1 hamesha . Agar M < 1 1/ M > 1 speed of sound se neeche koi cone nahi hoti , kyunki source apni ripples ke andar rehta hai. Mathematics exactly tab cone draw karne se mana karta hai jab nature cone banana se mana karti hai.
Worked example Boundary caseM = 1 M = 1 1/ M = 1 θ = sin − 1 ( 1 ) = 9 0 ∘ . 9 0 ∘ seedhi cross khadi rehti hai, ek flat wavefront motion ke perpendicular , object ki naak par. Saari ripples ek baar is single plane par pile ho jaati hain: yeh납작하게bani hui wall hi "sound barrier" hai. Kisi bhi M > 1 9 0 ∘
Reveal check:
Agar sin θ = 1/2 θ θ = sin − 1 ( 1/2 ) = 3 0 ∘
M = 1 θ sin − 1 ( 1 ) = 9 0 ∘
Parent par Worked example 3 ko ek aur ratio chahiye, kyunki wahan hum height jaante hain aur ground lag chahiye — ek triangle ke do chote sides (opposite aur adjacent, dono section 5 mein define kiye gaye), hypotenuse nahi.
tan θ = adjacent opposite . adjacent side woh hai jo angle ko touch karti hai aur hypotenuse nahi hai.
Intuition Wahan tangent aur yahan sine kyun?Alag knowns, alag tool. Cone angle ke liye hum opposite + hypotenuse jaante the ⇒ sine. Ground lag ke liye hum ek vertical height jaante hain aur horizontal distance chahiye — woh do chote sides hain (opposite aur adjacent) ⇒ tangent. Hamesha woh ratio choose karo jo un sides se bana ho jo aapke paas pehle se hain.
Arrows ko "feeds into" ke roop mein padhein: har box is page ka ek idea hai, aur arrows follow karne se exact order pata chalta hai jisme woh stack hokar shock-cone result banate hain. Top-left se shuru karo (wavefront picture aur speed formula) aur bottom par khatam karo (Mach cone).
Speed equals distance over time
Right triangle of the two distances
Sine equals opposite over hypotenuse
Mach angle sine theta equals one over M
Mach cone and shock waves
Right side cover karo aur khud test karo — agar koi line shaky lage, toh parent page se pehle uska section dobara padhein.
Wavefront kya hota hai? Kisi disturbance ka chalta hua front edge; 2D mein circle, 3D mein sphere — us par har point ne source ko ek hi pal chhoda hota hai.
t Elapsed time (seconds) jab se clock shuru hua, jaise jab se ek ripple release hui.
v s Hawa mein speed of sound (~340 m/s
v obj Sound banane wale object (jet, bullet, rocket) ki speed.
Mach number likho aur batao uske units kyun nahi hote. M = v obj / v s m/s
M = 1 Object exactly speed of sound par chal raha hai — transonic "sound barrier"; cone ek plane mein flat ho jaata hai (θ = 9 0 ∘
Yahan use kiye gaye right triangle ke teen sides ke naam batao. Opposite (angle ko face karta hai, v s t v obj t
sin θ Opposite ÷ hypotenuse.
Cone angle ke liye sine (cosine/tangent nahi) sahi tool kyun hai? Kyunki jo do sides hum jaante hain woh opposite aur hypotenuse hain, aur sine exactly opposite/hypotenuse hai.
sin − 1 Sine ko undo karta hai — isse ratio do, woh angle return karta hai.
M < 1 Tab 1/ M > 1 sin − 1 ( 1/ M )
tan θ Jab aap do chote sides (opposite aur adjacent) jaante/chahte ho, jaise sonic-boom timing problem mein height vs ground lag.
Speed of sound in a medium — woh fixed v s Doppler effect — wohi badhte wavefront ki picture, source ke unhe outrun karne se pehle.Superposition & constructive interference — kyun touching wavefronts ek single strong wall banate hain. 
