Wave equation (hyperbolic) 1D — derivation

4.7.10 · Maths › Partial Differential Equations

HUM KYA derive kar rahe hain?

Hyperbolic KYU? Ise $u_{tt} - c^{2} u_{xx} = 0$ likhne par, second-order coefficients hain $A = 1$ ( $u_{tt}$ ke liye), $B = 0$ , $C = - c^{2}$ . Discriminant $B^{2} - 4 A C = 0 - 4 (1) (- c^{2}) = 4 c^{2} > 0$ , isliye yeh hyperbolic hai — iske do real families of characteristics hain aur yeh travelling waves support karta hai.

KAISE: First principles se Derivation

Hum string ko in assumptions ke saath model karte hain (inhe clearly state karo — yahan physics rehti hai):

Bilkul flexible string (bending stiffness nahi).
Small displacements aur small slopes: $∣ u_{x} ∣ ≪ 1$ .
Tension $T$ magnitude mein string ke saath constant hai.
Motion purely transverse (vertical) hai, koi horizontal motion nahi.
Linear density $ρ$ constant; gravity tension ke comparison mein negligible.

Figure — Wave equation (hyperbolic) 1D — derivation

Step 1 — Ek tiny element alag karo

$x$ aur $x + Δ x$ ke beech string ka ek piece lo. Tension dono siron pe tangentially khinchti hai. Maan lo $θ (x)$ woh angle hai jo string horizontal ke saath banati hai.

Yeh step kyun? Forces string ke saath act karte hain, isliye hum inhe local slope use karke resolve karte hain.

Step 2 — Horizontal balance (koi horizontal motion nahi)

Horizontal components cancel hone chahiye: $T cos θ (x + Δ x) - T cos θ (x) = 0.$ Small slopes ke liye, $cos θ \approx 1$ , toh horizontal tension essentially har jagah $T$ hai — assumption 3 ke saath consistent.

Yeh step kyun? Yeh confirm karta hai ki $T$ ko horizontally constant treat kiya ja sakta hai, jo humein aage chahiye.

Step 3 — Vertical balance = Newton's 2nd law

Vertical force tension ke vertical components ka difference hai: $F_{net} = T sin θ (x + Δ x) - T sin θ (x) .$ Element ka mass $ρ Δ x$ hai aur uska vertical acceleration $u_{tt}$ hai. Toh $F = ma$ deta hai $T [sin θ (x + Δ x) - sin θ (x)] = ρ Δ x \frac{\partial ^{2} u}{\partial t ^{2}} .$

Yeh step kyun? Yeh literally $ma$ hai: net upward pull equals mass times upward acceleration.

Step 4 — Small-angle: slope = derivative

Small angles ke liye, $sin θ \approx tan θ = \frac{\partial u}{\partial x}$ (slope!). Substitute karo: $T [u_{x} (x + Δ x, t) - u_{x} (x, t)] = ρ Δ x u_{tt} .$

Yeh step kyun? Hum geometric angles ko $u$ ke derivatives se replace kar rahe hain — physics ko calculus mein badal rahe hain.

Step 5 — $Δ x$ se divide karo aur ushe shrink karo

$T \frac{u _{x} ( x + Δ x , t ) - u _{x} ( x , t )}{Δ x} = ρ u_{tt} .$ Jab $Δ x \to 0$ , left fraction bilkul definition hai $\frac{\partial}{\partial x} (u_{x}) = u_{xx}$ ki: $T u_{xx} = ρ u_{tt} .$

Yeh step kyun? Slope ka difference-quotient exactly second derivative hai — curvature.

Step 6 — Constant ka naam rakho

$ρ$ se divide karo aur $c^{2} = T / ρ$ set karo: $u_{tt} = c^{2} u_{xx}$

d'Alembert: travelling-wave solution

Yeh kyun include kiya? Yeh dikhata hai ki equation sach mein speed $c$ par chalti waves produce karti hai.

Variables ko characteristics $ξ = x - c t$ , $η = x + c t$ mein badlo. Chain rule se: $u_{x} = u_{ξ} + u_{η}, u_{xx} = u_{ξ ξ} + 2 u_{ξ η} + u_{η η},$ $u_{t} = - c u_{ξ} + c u_{η}, u_{tt} = c^{2} (u_{ξ ξ} - 2 u_{ξ η} + u_{η η}) .$ $u_{tt} - c^{2} u_{xx} = 0$ mein plug karo: $c^{2} (u_{ξ ξ} - 2 u_{ξ η} + u_{η η}) - c^{2} (u_{ξ ξ} + 2 u_{ξ η} + u_{η η}) = 0 \Rightarrow - 4 c^{2} u_{ξ η} = 0.$ Toh $u_{ξ η} = 0$ , do baar integrate karo: $u = F (ξ) + G (η) = right move karta hai F (x - c t) + left move karta hai G (x + c t) .$

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachhe ko explain karo

Socho ek skipping rope jo do doston ne tight pakdi hai. Agar aap ek end flick karo, ek bump doosri taraf run karta hai. Kyun? Kyunki rope ka har chota bit apne saath wale bits se khincha ja raha hai. Agar ek bit kisi dip mein baitha hai (smile ki tarah curved), uske neighbours use upar khinchte hain; agar hump pe hai, toh neeche khinchte hain. Toh har bit hamesha "straight" ki taraf khicha jata rehta hai, aur jab woh overshoot karta hai, bump aage badhta jaata hai. Wave equation bas is baat ka maths hai ki "jitna zyada curved ek bit hai, utna hi zyada use straight push kiya jaata hai." Rope ko tighter khicho aur bump tezi se bhaagta hai; bhaari rope use aur slow kar deti hai.

Active-recall flashcards

1D wave equation kya hai?

u_{tt} = c^{2} u_{xx}

jahan

c = T / ρ

Derivation ke neeche physical law kya hai?

Newton's second law

F = ma

ek small string element pe (vertical balance).

u_{xx}

physically kya represent karta hai?

String ki curvature; yeh vertical acceleration ka sign/size set karta hai.

sin θ

u_{x}

se kyun replace kiya jaata hai?

Small-angle approximation:

sin θ \approx tan θ = u_{x} =

slope.

Equation ko hyperbolic kyun kaha jaata hai?

Discriminant

B^{2} - 4 A C = 4 c^{2} > 0

for

u_{tt} - c^{2} u_{xx} = 0

Characteristic variables kya hain?

ξ = x - c t

aur

η = x + c t

General (d'Alembert) solution form?

u = F (x - c t) + G (x + c t)

— right- aur left-moving waves.

d'Alembert formula

u (x, 0) = f

u_{t} (x, 0) = g

ke saath?

u = \frac{1}{2} [f (x - c t) + f (x + c t)] + \frac{1}{2 c} \int_{x - c t}^{x + c t} g d s

c = T / ρ

ke units?

(N) / (kg/m) =

m/s, ek speed.

Kya wave speed amplitude pe depend karta hai (linear model mein)?

Nahi — sirf medium pe (

T, ρ

Derivation ki key assumptions kya hain?

Small slopes, constant tension, flexible string, transverse motion, constant

ρ

Connections

Heat equation (parabolic) 1D — derivation — same "neighbours influence" idea lekin $u_{t}$ ke saath (first order in time) → diffusive, oscillatory nahi.
Laplace equation (elliptic) — steady state ( $u_{tt} = 0$ ) limit.
Classification of second-order PDEs — jahan discriminant test rehta hai.
Method of characteristics — $x \pm c t$ trick ko generalise karta hai.
Separation of variables for the wave equation — bounded-string Fourier-series solution.
d'Alembert solution — travelling-wave form ka full treatment.
Newton's second law — poori derivation ka physical seed.