Solving wave equation — D'Alembert's solution

4.7.11 · Maths › Partial Differential Equations

WHAT are we solving?

WHY does the solution split into two travelling waves?

Key idea: operator ko factor karo. Equation ko aise likho: $(\partial_{tt} - c^{2} \partial_{xx}) u = 0.$ $\partial_{t}, \partial_{x}$ ko algebra ki tarah treat karo (difference of squares): $\partial_{tt} - c^{2} \partial_{xx} = (\partial_{t} - c \partial_{x}) (\partial_{t} + c \partial_{x}) .$ Yeh step kyun? Kyunki $a^{2} - b^{2} = (a - b) (a + b)$ , aur yeh operators commute karte hain (smooth $u$ ke liye mixed partials equal hote hain). Toh wave equation actually do first-order transport equations hai jo ek saath chained hain.

Ek transport equation $(\partial_{t} + c \partial_{x}) w = 0$ kehta hai: $w$ lines $x - c t = const$ ke along constant hai → $w =$ function of $x - c t$ , ek wave right move karti hai. Isi tarah $\partial_{t} - c \partial_{x}$ waves ko left move karta hai. Yahi neeche aane wale change of variables ka seed hai.

HOW to derive D'Alembert (change of characteristics)

Step 1 — derivatives convert karo. Chain rule use karo with $x = \frac{ξ + η}{2}$ , $t = \frac{η - ξ}{2 c}$ : $\partial_{x} = \partial_{ξ} + \partial_{η}, \partial_{t} = - c \partial_{ξ} + c \partial_{η} .$ Kyun? Har purana variable dono naye variables pe depend karta hai; chain rule contributions ko sum karta hai.

Step 2 — plug in karo. $u_{xx} = (\partial_{ξ} + \partial_{η})^{2} u = u_{ξ ξ} + 2 u_{ξ η} + u_{η η},$ $u_{tt} = c^{2} (- \partial_{ξ} + \partial_{η})^{2} u = c^{2} (u_{ξ ξ} - 2 u_{ξ η} + u_{η η}) .$ Phir: $u_{tt} - c^{2} u_{xx} = c^{2} (u_{ξ ξ} - 2 u_{ξ η} + u_{η η}) - c^{2} (u_{ξ ξ} + 2 u_{ξ η} + u_{η η}) = - 4 c^{2} u_{ξ η} = 0.$ Yeh magic kyun hai: saare messy terms cancel ho jaate hain, aur bas yeh simple cheez bachti hai: $u_{ξ η} = 0.$

Step 3 — do baar integrate karo. $u_{ξ η} = 0$ matlab $u_{η}$ , $η$ pe depend nahi karta... $η$ mein integrate karo: $u_{ξ}$ sirf $ξ$ ka function hai; $ξ$ mein integrate karo: $u = F (ξ) + G (η) = F (x - c t) + G (x + c t) .$ Do arbitrary functions kyun? Ek 2nd-order PDE in two variables ko do "constants of integration" chahiye hote hain — yahaan yeh poori functions hain.

HOW to fit the initial data → the D'Alembert formula

Hume $F$ aur $G$ ko $f$ aur $g$ se pin down karna hai.

$t = 0$ pe: $u(x,0)=F(x)+G(x)=f(x).\tag{1}$ $u_{t} = - c F^{'} (x - c t) + c G^{'} (x + c t)$ differentiate karo aur $t = 0$ set karo: $u_t(x,0)=-cF'(x)+cG'(x)=g(x)\ \Rightarrow\ -F'(x)+G'(x)=\frac{g(x)}{c}.\tag{2}$ (2) ko $0$ se $x$ tak integrate karo: $-F(x)+G(x)=\frac1c\int_0^x g(s)\,ds + K.\tag{3}$ (1) aur (3) add/subtract karo:

F(x)=\tfrac12 f(x)-\tfrac1{2c}\int_0^x g\,ds-\tfrac{K}{2}.$$ Ab $u=F(x-ct)+G(x+ct)$ banao; constants $K$ cancel ho jaate hain aur integral limits combine ho jaati hain: > [!formula] **D'Alembert's solution** > $$\boxed{\,u(x,t)=\frac{1}{2}\Big[f(x-ct)+f(x+ct)\Big]+\frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds\,}$$ > *Term 1:* initial shape do halves mein split hoti hai, har half opposite direction mein travel karti hai. > *Term 2:* initial velocity ==domain of dependence== $[x-ct,\,x+ct]$ pe spread out hoti hai. ![[4.7.11-Solving-wave-equation-—-D'Alembert's-solution.png]] > [!intuition] Domain of dependence & influence > $u(x,t)$ SIRF $f$ ke do points $x\pm ct$ pe aur $g$ ke unke beech ke interval pe depend karta hai. Us interval ke bahar kuch bhi $(x,t)$ ko affect nahi kar sakta — **finite propagation speed $c$**. --- ## Worked examples > [!example] 1 — Pure displacement, zero velocity > $f(x)=e^{-x^2}$, $g=0$, $c=1$. > Kyunki $g=0$ hai, integral zero ho jaata hai: > $$u(x,t)=\tfrac12 e^{-(x-t)^2}+\tfrac12 e^{-(x+t)^2}.$$ > *Yeh step kyun?* Koi initial velocity nahi hone se bump simply do half-height bumps mein split ho jaata hai jo ek doosre se door jaate hain. **Forecast-then-verify:** $t=0$ pe yeh overlap karte hain → $u=e^{-x^2}=f$. ✓ > [!example] 2 — Zero displacement, a velocity kick > $f=0$, $g(x)=\cos x$, $c=2$. > $$u=\frac1{4}\int_{x-2t}^{x+2t}\cos s\,ds=\frac1{4}\big[\sin(x+2t)-\sin(x-2t)\big] > =\frac12\cos x\,\sin 2t.$$ > *Kyun?* $\sin A-\sin B=2\cos\frac{A+B}2\sin\frac{A-B}2$ use kiya. Check karo $t=0$: $u=0$ ✓; $u_t|_0=\cos x\cdot\cos(2t)\cdot... \to \cos x$ ✓. > [!example] 3 — Both data present > $f(x)=\sin x$, $g(x)=1$, $c=1$. > $$u=\tfrac12[\sin(x-t)+\sin(x+t)]+\tfrac12\int_{x-t}^{x+t}1\,ds > =\sin x\cos t + t.$$ > *Kyun?* Sum-to-product se $\sin x\cos t$ milta hai; $1$ ka integral interval length $2t$ hai, halved → $t$. String oscillate bhi karti hai aur unit speed pe upar bhi drift karti hai (constant velocity input). --- ## Steel-manned mistakes > [!mistake] "$F$ aur $G$ same function hain." > *Kyun sahi lagta hai:* formula $f$ mein symmetric lagta hai. *Reality:* yeh velocity integral se alag hote hain; sirf jab $g=0$ ho tab half-shapes ek jaisi lagti hain. **Fix:** hamesha (1) aur (3) ka system solve karo; symmetry assume mat karo. > [!mistake] $x-ct$ vs $x+ct$ mein sign bhool jaana. > *Kyun sahi lagta hai:* dono "$x$ aur $t$" hain. *Fix:* ek right-mover wahan fixed value rakhta hai jahan $x-ct=$const ho, toh jaise $t\uparrow$ hota hai, $x$ bhi $\uparrow$ hona chahiye — yahi $f(x-ct)$ hai. Yaad rakho: **minus = +x direction mein move karta hai.** > [!mistake] Integral pe $\dfrac1{2c}$ ki jagah $\dfrac1{2}$ likhna. > *Kyun sahi lagta hai:* $f$ pe $\tfrac12$ ka mirror lagta hai. *Fix:* yeh eq. (2) mein $c$ se divide karne se aaya tha. Dimensionally, $g$ ke units velocity hain; $\int g\,ds$ ka (velocity·length) hai, aur $c$ (velocity) se divide karne pe length = displacement milta hai. $c$ **zaroor** hona chahiye. --- ## #flashcards/maths D'Alembert kaunsa PDE solve karta hai? ::: Infinite line pe 1D wave equation $u_{tt}=c^2u_{xx}$. Wave operator ko factor karo. ::: $(\partial_t-c\partial_x)(\partial_t+c\partial_x)$. Kaunse characteristic coordinates ise linearise karte hain? ::: $\xi=x-ct,\ \eta=x+ct$, jo ise $u_{\xi\eta}=0$ mein reduce karte hain. Wave equation ka general solution kya hai? ::: $u=F(x-ct)+G(x+ct)$ (right- + left-moving waves). D'Alembert ka formula batao. ::: $u(x,t)=\tfrac12[f(x-ct)+f(x+ct)]+\tfrac1{2c}\int_{x-ct}^{x+ct}g(s)\,ds$. Integral pe $\tfrac1{2c}$ kyun hai ($\tfrac12$ nahi)? ::: Yeh velocity condition ko $c$ se divide karne se aata hai; dimensionally displacement restore karta hai. $(x_0,t_0)$ ka domain of dependence kya hai? ::: Initial line pe interval $[x_0-ct_0,\ x_0+ct_0]$. Finite $c$ ka physical matlab kya hai? ::: Disturbances speed $c$ pe travel karti hain; $[x-ct,x+ct]$ ke bahar ka data $u(x,t)$ ko influence nahi kar sakta. Agar $g=0$ ho toh initial bump ka kya hota hai? ::: Yeh do half-height copies mein split ho jaata hai jo speed $c$ pe opposite directions mein move karte hain. --- > [!recall]- Feynman: 12-saal ke bachche ko samjhao > Socho tumne ek lamba skipping rope flick kiya. Jo bump tumne banaya woh wahan nahi rukta — woh **do** chhote bumps mein split ho jaata hai: ek right ki taraf bhaagta hai, ek left ki taraf, dono same speed se. Agar tumne rope ko sirf ek push diya (koi starting shape nahi, bas ek dhakka), toh woh push do travelling edges ke beech ek phailte hue flat region mein spread out ho jaata hai. D'Alembert ka formula bas yeh recipe hai: "apni starting shape lo, use aadha karo, halves ko opposite directions mein bhejo; apna starting push lo aur use us stretch pe phailao jahan wave pahunch sakti hai." Bas yahi poora gaana hai. > [!mnemonic] **"Shape ke dono halves alag jaate hain; speed push ko slice karti hai."** > $\tfrac12$ splitting shape ke liye, aur $\tfrac1{2c}$ (yahan $c$ = "slice") velocity integral ke liye $x\pm ct$ pe. --- ## Connections - [[Method of Characteristics]] — same $\xi,\eta$ lines, doosre hyperbolic PDEs tak generalise hota hai. - [[Transport Equation]] — first-order factors $\partial_t\pm c\partial_x$. - [[Separation of Variables — Wave Equation]] — bounded strings pe alternative (Fourier) route. - [[Domain of Dependence and Influence]] — causality / finite speed. - [[Classification of Second-Order PDEs]] — wave equation hyperbolic ka prototype hai. - [[Heat Equation]] — contrast: infinite propagation speed, koi characteristics nahi. ## 🖼️ Concept Map ```mermaid flowchart TD WE[Wave equation u_tt=c2 u_xx] -->|initial data| IC[f shape and g velocity] WE -->|factor operator| FAC[Difference of squares] FAC -->|yields| TRANS[Two transport equations] TRANS -->|constant along| CHAR[Characteristic lines x-ct and x+ct] CHAR -->|define coords| XI[xi=x-ct, eta=x+ct] XI -->|chain rule| SIMP[Equation becomes u_xi_eta=0] SIMP -->|integrate twice| GEN[u=F of xi plus G of eta] GEN -->|right-mover| RIGHT[F x-ct shifts right] GEN -->|left-mover| LEFT[G x+ct shifts left] IC -->|split into| GEN ```