Wave equation (hyperbolic) 1D — derivation: Foundations

4.7.10 · D1 · Maths › Partial Differential Equations › Wave equation (hyperbolic) 1D — derivation

Yeh page har ek notation ka piece build karta hai jo parent derivation mein use hota hai, un cheezon se shuru karke jo ek curious 12-saal-ka bachcha pehle se jaanta hai, aur ek waqt mein exactly ek nayi idea add karke. Yahan kuch bhi assumed nahi hai. Agar main note ne koi symbol use kiya hai, toh woh neeche define kiya gaya hai pehle se ki aap wahan use milein.

1 · String aur uski shape: $u (x, t)$ kya hai?

Ek guitar string ko socho jo do pegs ke beech flat stretch ki gayi hai. Uski resting line ke saath ek ruler rakho — ruler ke saath ki direction ko $x$ -direction kaho (horizontal, left–right).

Ab use pluck karo. String ka har point resting line se thoda upar uth jaata hai. Koi point kitna upar ya neeche gaya hai woh uski displacement hai.

Do inputs kyun chahiye. String ki ek akeli photo (ek frozen instant) sirf $x$ ki function hai — ek curve. Lekin string move karti hai, toh thodi der baad uski poori curve alag hoti hai. Motion describe karne ke liye humein ek aisi height chahiye jo dono par depend kare — kahan ( $x$ ) aur kab ( $t$ ). Yahi exactly ek two-variable function $u (x, t)$ hai.

2 · Greek aur symbol dictionary

Kisi bhi calculus se pehle, yahan parent note ke har letter ke plain-word meanings hain. Is table ko apne paas rakho.

Ab hum un do ideas ko earn karte hain jo in letters ko physics mein badal deti hain: slope/tangent aur derivatives.

3 · Angle, slope, aur $θ$ kyun aata hai

String pluck karne ke baad flat nahi rehti; har point par woh tilt karti hai. Do words usi tilt ko describe karte hain.

Yeh sab ek right triangle se connect hote hain. Thoda aage step karo aur string kuch amount rise karti hai; tangent hypotenuse hai. Us triangle par: $tan θ = \frac{rise}{run} = slope .$

$tan$ kyun aur $sin$ ya $cos$ kyun nahi? $tan θ$ opposite over adjacent hai — yeh literally rise-over-run hai, isliye yeh hai hi slope. Isliye parent note likhta hai $tan θ = u_{x}$ : string ka slope aur uski tilt-angle ka tangent ek hi number hain.

4 · Derivative: "steepness" ko ek symbol mein daalna

Sabse important tool neeche derivative hai. Yahan yeh zero se explain hai.

"Partial" kyun aur curly $\partial$ kyun? Kyunki $u$ ke do inputs hain, $x$ aur $t$ . Jab hum $x$ -direction mein slope measure karte hain toh hum $t$ ko frozen rakhte hain — hum sirf ek variable wiggle karte hain. Curly $\partial$ ek flag hai jo kehta hai "aur bhi variables hain; main sirf yeh wala change kar raha hoon."

Limit kyun chahiye. Real slope ek point par define hoti hai, lekin ek point ki koi width nahi hoti — aap ek single spot par rise-over-run compute nahi kar sakte. Trick yeh hai ki ek chote chunk par compute karo aur phir chunk ko vanish karne do. Baad wala change of variables aur poori derivation is shrinking idea par depend karti hai; yeh calculus ka engine hai.

5 · Second derivative = curvature

Ab wohi slope-machine dobara apply karo, is baar slope ke upar hi.

Topic is par kyun jeeta-marta hai. Poora wave equation kehta hai $u_{tt} = c^{2} u_{xx}$ : acceleration curvature ke proportional hai. Ek point jo dip mein baitha hai ( $\cup$ , $u_{xx} > 0$ ) upar push hota hai; ek point hump par ( $\cap$ , $u_{xx} < 0$ ) neeche push hota hai. Curvature woh messenger hai jo har point ko batata hai ki uske neighbours use kis direction mein yank kar rahe hain.

Time ko bhi wohi treatment milti hai: $u_{tt} = \frac{\partial ^{2} u}{\partial t ^{2}}$ acceleration hai — ek fixed point ki up/down velocity $u_{t}$ kitni tezi se change ho rahi hai. Yeh $F = ma$ ka " $a$ " hai.

6 · Tension $T$ , density $ρ$ , aur speed $c$

String ki do physical properties cast ko complete karti hain.

Recall

$c$ metres per second mein kyun aata hai? $T / ρ$ ki units ::: $\frac{N}{kg/m} = \frac{kg \cdot m / s ^{2}}{kg / m} = \frac{m ^{2}}{s ^{2}} = \frac{m}{s}$ — ek genuine speed.

7 · Foundations topic ko kaise feed karte hain

Isse upar se neeche padho: geometry (slope, curvature) plus physics (tension, mass) $F=ma$ ke andar milte hain, aur bahar nikalta hai wave equation. Yahi skeleton — ek balance law jo ek PDE feed karta hai — heat equation aur Laplace's equation mein bhi milta hai, isliye second-order PDEs ka classification unhe ek saath group karta hai.

8 · Ek combined coordinate preview: $ξ$ aur $η$

Parent ka d'Alembert step $(x, t)$ ko $ξ = x - c t$ aur $η = x + c t$ se swap karta hai. Aapko ise yahan master nahi karna, bas idea pehchanna hai:

Equipment checklist

Self-test: right side cover karo aur reveal karne se pehle har ek ka answer do.

u (x, t)

physically kya matlab hai?

Position

x

par time

t

par string ki vertical height.

Displacement ko do inputs kyun chahiye, ek nahi?

Ek input frozen snapshot deta hai; string time mein bhi change hoti hai, isliye humein dono chahiye — position

x

aur time

t

String ki slope symbols mein kya hai, aur uska trig meaning kya hai?

u_{x} = tan θ

= rise over run = tangent triangle ka opposite over adjacent.

Slope ke liye

tan θ

use karte hain

sin θ

ki jagah, kyun?

Kyunki

tan θ

opposite/adjacent = rise/run hai, jo exactly slope hai.

Curly

\partial

kya signal karta hai?

Ek partial derivative — ek variable ke respect mein change jab baaki sab fixed hon.

Derivative ko shrinking limit

Δ x \to 0

kyun use karni chahiye?

Ek single point par slope ki koi width nahi hoti; hum ek chunk par rise/run lete hain aur use zero shrink karte hain exactly us point par slope paane ke liye.

u_{xx}

kya represent karta hai, aur "steepness" kyun nahi?

Curvature — string ka bending; steepness

u_{x}

hai, aur ek straight steep line mein

u_{xx} = 0

hota hai.

u_{tt}

kya represent karta hai?

Ek fixed point ka vertical acceleration —

F = ma

ka "

a

Length

Δ x

ke ek chunk ka mass kya hai?

ρ Δ x

Tension ka kaunsa component string ko vertically actually accelerate karta hai?

Vertical component

T sin θ

(chunk ke across uska difference).

c

T

aur

ρ

ke terms mein do aur ek sanity fact.

c = T / ρ

; tighter string → tez, bhaari string → dheemi, aur units m/s aati hain.

Recall Feynman check: poora page ek saanson mein bolo

String ki height $u (x, t)$ hai; uski slope $u_{x} = tan θ$ hai; us slope ka bend curvature $u_{xx}$ hai; mass $ρ Δ x$ ka har chunk vertical tension $T sin θ$ se upar pull hota hai, aur Newton's $F = ma$ bend ko acceleration $u_{tt}$ mein badal deta hai, deta hai $u_{tt} = c^{2} u_{xx}$ with $c = T / ρ$ .