4.7.11 · D1Solving wave equation — D'Alembert's solution
Wave equation kehta hai "string par ek bump bina shape badle chalti rehti hai," aur poora topic ek hi trick se bana hai: ek partial derivative measure karta hai ki koi cheez kitni tezi se change hoti hai jab tum sirf EK input ko thoda hilao. Woh ek picture master kar lo — ek slope jise tum padhte ho jab baaki sab variables frozen hon — aur parent page ka har symbol (∂ ξ , η
Is page par koi assumption nahi ki. Agar parent note mein koi symbol tha, toh hum use pehle ek picture se build karte hain. Upar se neeche padho; har block agle ko earn karta hai.
f ( x ) machine jo ek number x f ( x ) curve hai: horizontal axis = input x
Parent page par f ( x ) = e − x 2
Intuition Hum kyun care karte hainString, time zero par frozen, AISE hi ek curve hai: har point x f
Hum baar baar "slope" bol rahe hain. Ise earn karte hain.
Definition Slope / derivativeCurve par ek point par khado aur ek chota seedha ruler uske saath seedha lagao — woh tangent line hai. Woh ruler jitna teda ho, woh slope hai. Likha jata hai f ′ ( x ) d x df f
Intuition Slope ek real sawaal ka jawab kyun deta haiSlope answer deta hai "agar main x Tezi se upar = bada positive slope; flat = zero; neeche = negative. Woh "rate of change" hi woh ek cheez hai jo derivatives hamesha encode karte hain.
Hum f ′
u ( x , t ) do numbers khaati hai: position x t surface hai — ek aisa landscape jiska floor point ( x , t ) u
Intuition Physically kya matlab haix t u
Yahan se baad ke har symbol is landscape par rehta hai.
Poora subject Partial Differential Equations isliye kehlata hai:
Definition Partial derivativeDo-input landscape par tum "the slope" nahi pooch sakte — kis direction mein slope? Toh hum ek input ko freeze karte hain aur sirf doosre ko slide karte hain, phir ordinary slope padhte hain. t x partial derivative u x ∂ x u ∂ x ∂ u x t u t
Landscape ko ek vertical knife se kaato. x u x t u t velocity hai.
Yahi wajah hai ki parent ki doosri initial condition u t ( x , 0 ) = g ( x ) initial velocity kehlati hai: u t time per unit rise hai.
∂ x d x d Kyun sahi lagta hai: dono x Reality: curly ∂ doosra variable (t d
u xx u x x curvature , string kitni tezi se moodti hai. Usi tarah u tt acceleration hai.
Intuition Wave equation ko ek sentence ki tarah padhnau tt = c 2 u xx "ek point accelerate karta hai (u tt u xx Ek tezi se curved jagah zor se wapis snap hoti hai — yahi waves ko ripple karaata hai. Number c 2
c c > 0 speed jis par ek bump chalta hai, length-per-time mein. Picture par yeh woh tilt hai characteristic lines ka jo tum aage miloge.
Definition Differential operator∂ x verb hai: "jo kuch aage aaye use x ∂ t + c ∂ x ∂ t c ∂ x ∂ tt − c 2 ∂ xx = ( ∂ t − c ∂ x ) ( ∂ t + c ∂ x ) .
allowed kyun hai
Kyunki ye verbs ordinary algebra ki tarah behave karte hain: a 2 − b 2 = ( a − b ) ( a + b ) a , b commute karein (order matter nahi karta). Smooth u u x t = u t x ∂ x ∂ t transport equations mein badal deta hai.
ξ = x − c t , η = x + c t naye coordinates jo purane se bane hain. ξ η ξ x − c t = characteristic line trace karta hai — x t
Space-time floor par, x − c t = const t x c x + c t = const x , t u ξ η = 0
Common mistake "Minus sign matlab woh baayein chalta hai."Kyun sahi lagta hai: minus backwards jaisa feel hota hai. Fix: f ( x − c t ) x − c t t x badhna chahiye use constant rakhne ke liye → shape +x (daayein) direction mein slide karta hai. Minus = daayein chalta hai.
Parent page par akhri anjaana mark D'Alembert's formula mein hai:
2 c 1 ∫ x − c t x + c t g ( s ) d s .
Definition Definite integral∫ a b g ( s ) d s signed area hai jo curve g a b s dummy variable hai — bas ek placeholder naam "us point ke liye jise hum abhi add kar rahe hain"; yeh answer mein nahi bachtaa.
Intuition Integral kyun aata haiInitial velocity g displacement (ek height) mein convert karna hoga. Ek interval par velocity add karna = total distance = velocity curve ke neeche area. Yahi exactly ∫ g d s [ x − c t , x + c t ] domain of dependence .
Recall Units check (kyun2 c 1 2 1 g ∫ g d s c length = ek displacement bachta hai. Sawaal — agar tum c
Function of one variable f of x
Partial derivative u sub x and u sub t
Function of two variables u of x t surface
Second partials u xx and u tt curvature and acceleration
Wave equation u tt equals c squared u xx
Operators partial t plus or minus c partial x
Characteristic coordinates xi and eta
Definite integral area under g
Har arrow kehta hai "right box samajhne ke liye left box chahiye." Inhe follow karo aur tum parent note line by line padh sakte ho.
Kya tum ek bump f ( x ) Haan — slope f ′ ( x )
Curly ∂ Ek aur variable frozen hai jab tum chosen ek mein differentiate kar rahe ho.
Shabd mein, u t String par ek fixed point ki velocity — height rise per unit time, x
u tt = c 2 u xx Ek point ki acceleration equals c 2
Hum ∂ tt − c 2 ∂ xx a 2 − b 2 Kyunki u x t = u t x ∂ x ∂ t
f ( x − c t ) Daayein (+x); jaise t x − c t x
∫ a b g ( s ) d s s a b g s
Velocity integral par 2 c 1 2 1 Speed c
Parent topic 4.7.11 → Transport Equation — woh first-order pieces jisme operator factor hota hai.Method of Characteristics — ξ , η Domain of Dependence and Influence — integral ke interval ka matlab.Classification of Second-Order PDEs — kyun u tt = c 2 u xx Separation of Variables — Wave Equation — bounded strings par Fourier alternative.Heat Equation — contrast: no characteristics, infinite speed. 
