Solving wave equation — separation of variables

4.7.12 · Maths › Partial Differential Equations

Woh problem jo hum solve kar rahe hain

KYA chahiye humein: ek function $u (x, t)$ jo PDE + 2 BCs + 2 ICs satisfy kare. KYUN separation kaam karta hai: PDE linear & homogeneous hai, aur BCs homogeneous hain ( $= 0$ ), isliye solutions ka sum bhi solution hota hai — yahi superposition ka engine hai.

Step-by-step derivation SCRATCH SE

Step 1 — Separation ansatz banao

Assume karo $u (x, t) = X (x) T (t) .$ PDE mein plug karo. Kyunki $u_{tt} = X T^{''}$ aur $u_{xx} = X^{''} T$ : $X T^{''} = c^{2} X^{''} T .$

Yeh step kyun? Yeh 2-variable function ki derivatives ko 1-variable functions ki derivatives mein convert karta hai.

Step 2 — Variables ko separate karo

Dono sides ko $c^{2} X T$ se divide karo: $\frac{T ^{''}}{c ^{2} T} = \frac{X ^{''}}{X} .$

Yeh step kyun? Left side sirf $t$ pe depend karta hai, right side sirf $x$ pe. Do cheezein jo saare $x, t$ ke liye equal hon, tabhi possible hai jab dono same constant ke equal hon. Ise $- λ$ kahte hain (minus isliye choose kiya — hum dekhenge ki humein oscillation chahiye): $\frac{X ^{''}}{X} = \frac{T ^{''}}{c ^{2} T} = - λ .$

Isse do ODEs milti hain: $X^{''} + λ X = 0, T^{''} + c^{2} λ T = 0.$

Step 3 — Spatial ODE pe BCs apply karo (eigenvalue problem)

BCs $u (0, t) = u (L, t) = 0$ saare $t$ ke liye hold karni chahiye, aur $T (t) \neq \equiv 0$ , isliye $X (0) = 0, X (L) = 0.$

Hum $X^{''} + λ X = 0$ solve karte hain aur $λ$ ke teen cases check karte hain (har ek ko steel-man karo!):

$λ < 0$ (maano $λ = - μ^{2}$ ): $X = A e^{μx} + B e^{- μx}$ . BCs force karte hain $A = B = 0$ . Sirf trivial solution. Rejected.
$λ = 0$ : $X = A x + B$ . BCs force karte hain $A = B = 0$ . Trivial. Rejected.
$λ > 0$ (maano $λ = k^{2}$ ): $X = A cos k x + B sin k x$ .

$λ > 0$ case pe BCs apply karo:

$X (0) = A = 0 \Rightarrow A = 0$ .
$X (L) = B sin k L = 0$ . Nontrivial $B \neq = 0$ ke liye: $sin k L = 0 \Rightarrow k L = nπ$ .

Toh eigenvalues aur eigenfunctions hain $λ_{n} = (\frac{nπ}{L})^{2}, X_{n} (x) = sin \frac{nπ x}{L}, n = 1, 2, 3, \dots$

Step 4 — Har mode ke liye time ODE solve karo

$λ_{n} = (nπ / L)^{2}$ ke saath: $T_{n}^{''} + c^{2} λ_{n} T_{n} = 0 \Rightarrow T_{n}^{''} + ω_{n}^{2} T_{n} = 0, ω_{n} = \frac{nπ c}{L} .$ Yeh simple harmonic motion hai: $T_{n} (t) = a_{n} cos ω_{n} t + b_{n} sin ω_{n} t .$

Yeh step kyun? $ω_{n}$ string ki natural frequencies hain — woh harmonics jo tumhe guitar pe sunai deti hain.

Step 5 — Superpose karo (general solution banao)

Sum kyun? Har $u_{n} = X_{n} T_{n}$ (linear, homogeneous) PDE + BCs solve karta hai. Superposition se, koi bhi sum bhi karta hai. Hum $a_{n}, b_{n}$ ki freedom use karte hain ICs fit karne ke liye.

Step 6 — Coefficients find karne ke liye ICs use karo (Fourier!)

Position IC $u (x, 0) = f (x)$ : $t = 0$ set karo ( $cos 0 = 1, sin 0 = 0$ ): $f (x) = \sum_{n = 1}^{\infty} a_{n} sin \frac{nπ x}{L} .$ Yeh ek Fourier sine series hai, toh orthogonality $\int_{0}^{L} sin \frac{mπ x}{L} sin \frac{nπ x}{L} d x = \frac{L}{2} δ_{mn}$ se: $a_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{nπ x}{L} d x$

Velocity IC $u_{t} (x, 0) = g (x)$ : $u$ ko $t$ mein differentiate karo, $t = 0$ set karo ( $\frac{d}{d t} cos ω_{n} t ∣_{0} = 0$ , $\frac{d}{d t} sin ω_{n} t ∣_{0} = ω_{n}$ ): $g (x) = \sum_{n = 1}^{\infty} b_{n} ω_{n} sin \frac{nπ x}{L} \Rightarrow b_{n} ω_{n} = \frac{2}{L} \int_{0}^{L} g (x) sin \frac{nπ x}{L} d x .$ Toh $b_{n} = \frac{2}{nπ c} \int_{0}^{L} g (x) sin \frac{nπ x}{L} d x (using ω_{n} = \frac{nπ c}{L}) .$

Figure — Solving wave equation — separation of variables

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek guitar string pluck karo. Woh sirf khaas "shapes" mein jiggle kar sakti hai — ek bada hump, do humps, teen humps — kyunki uske ends chipke hue hain aur move nahi kar sakte. Har shape ka apna musical note hota hai (tezi se wiggle = zyada ucha note). Jab tum pluck karte ho, tum kisi shape se start karte ho; string secretly in saare khaas humps ko theek us amount mein add kar rahi hai taaki woh shape bane, aur phir har hump apni speed se hamesha upar-neeche bounces karta hai. Hamara math bas yeh figure out karta hai starting shape mein kitna har hump hai (yahi "Fourier" part hai) aur kitni tezi se har ek bounces karta hai (yahi frequency hai).

Active-recall flashcards

Separation of variables ke liye PDE ka linear hona aur BCs ka homogeneous hona kyun zaroori hai?

Taaki superposition hold kare — product solutions ka sum bhi solutions hain aur zero BCs satisfy karte hain, jisse hum ICs fit kar sakein.

X^{''} = - λ X

mein

X (0) = X (L) = 0

ke saath,

λ \leq 0

kyun reject hote hain?

Woh sirf exponential/linear

X

dete hain, jo do alag points pe zero nahi ho sakti bina identically zero (trivial solution) hue.

Clamped string ke eigenvalues aur eigenfunctions kya hain?

λ_{n} = (nπ / L)^{2}

X_{n} (x) = sin (nπ x / L)

n = 1, 2, \dots

Spatial part sine kyun hai, cosine kyun nahi?

u (0, t) = 0

cosine ko kill karta hai (

cos 0 = 1

), sirf

sin

bachta hai.

Natural frequencies

ω_{n}

kya hain?

ω_{n} = nπ c / L

, isliye time part hai

a_{n} cos ω_{n} t + b_{n} sin ω_{n} t

Initial shape

f (x)

a_{n}

ka formula?

a_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{nπ x}{L} d x

Initial velocity

g (x)

b_{n}

ka formula, aur extra factor kyun?

b_{n} = \frac{2}{nπ c} \int_{0}^{L} g (x) sin \frac{nπ x}{L} d x

;

T

ko time mein differentiate karne se

ω_{n} = nπ c / L

neeche aata hai, isliye usse divide karo.

Agar

u_{t} (x, 0) = 0

ho, toh saare

b_{n}

ka kya hota hai?

Woh saare zero ho jaate hain; sirf

cos ω_{n} t

terms bachte hain.

Clamped wave equation ka general solution?

u = \sum_{n} (a_{n} cos \frac{nπ c t}{L} + b_{n} sin \frac{nπ c t}{L}) sin \frac{nπ x}{L}

Coefficients extract karne ke liye use hone wala orthogonality relation?

\int_{0}^{L} sin \frac{mπ x}{L} sin \frac{nπ x}{L} d x = \frac{L}{2} δ_{mn}

Connections

Fourier sine series — $a_{n}$ , $b_{n}$ kaise compute hote hain; wahi orthogonality engine.
Sturm–Liouville theory — spatial problem $X^{''} + λ X = 0$ ek special eigenvalue problem hai; real eigenvalues aur orthogonal eigenfunctions guarantee karta hai.
Heat equation — separation of variables — wahi spatial ODE, lekin time ODE first order hai ( $T^{'} = - c^{2} λ T$ ) → oscillation ki jagah decay.
d'Alembert solution of the wave equation — alternative method ( $u = F (x - c t) + G (x + c t)$ ); modes ko sum karne ke equivalent hai.
Standing waves and harmonics — $ω_{n} = nπ c / L$ , overtones, timbre ki physics.
Superposition principle — modes ko sum karna legal kyun hai.