---
title: "Derive or solve wave equation"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:derive-or-solve-wave-equation"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/derive-or-solve-wave-equation
site: BCT Engineering
---

# Derive or solve wave equation

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 4 marks
- Video: [Solution of the Wave Equation by the Method of Separation of Variables](https://www.youtube.com/watch?v=g9_i-rUWNls) — Ananga Das
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/derive-or-solve-wave-equation
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Solve the wave equation u_tt = c²u_xx under boundary conditions u(0,t) = 0, u(π,t) = 0 and initial conditions ∂u/∂t = 0 at t = 0 and u(x, 0) = x for 0 < x < π.

## Short answer

Separating variables gives u(x,t)= sum b_n sin(nx) cos(nct). Using x = sum b_n sin(nx), b_n = 2(-1)^(n+1)/n. Hence u(x,t)= 2 sum_{n=1}^∞ [(-1)^(n+1)/n] sin(nx) cos(nct).

## Answer

Separate variables in u_tt = c² u_xx with u = X(x)T(t): X″/X = T″/(c²T) = −p².

```text
X″ + p²X = 0     →  X = A cos px + B sin px
T″ + c²p²T = 0   →  T = C cos cpt + D sin cpt
```

**Boundary conditions.** u(0, t) = 0 gives A = 0. u(π, t) = 0 gives B sin pπ = 0, so p = n (an integer).

u = sin nx (C cos nct + D sin nct)

**Initial velocity.** ∂u/∂t = 0 at t = 0. The t-derivative is sin nx · nc(−C sin nct + D cos nct); at t = 0 it gives D = 0.

So u = Σ bₙ sin nx cos nct.

**Initial displacement.** u(x, 0) = x on (0, π):

```text
x = Σ bₙ sin nx
bₙ = (2/π)∫₀^π x sin nx dx = (2/π)[−x cos nx/n + sin nx/n²]₀^π
   = −2 cos nπ / n = 2(−1)ⁿ⁺¹/n
```

**Answer:**

u(x, t) = 2 Σ (n=1 to ∞) [(−1)ⁿ⁺¹/n] sin nx cos nct

= 2( sin x cos ct − sin 2x cos 2ct / 2 + sin 3x cos 3ct / 3 − … )

(Note that the initial shape x jumps to 0 at x = π, so the series converges to x only for 0 ≤ x < π.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Fixed ends give:
   A. B = 0
   B. p = 0
   C. A = 0 and p = n
   D. A = B
   Answer: C. X(0) = 0, X(π) = 0. (Key point: Boundary)

2. u_t = 0 at t = 0 gives:
   A. C = 0
   B. D = C
   C. n = 0
   D. D = 0, leaving cos nct
   Answer: D. Derivative at t=0. (Key point: Velocity condition)

3. bn for u(x,0) = x is:
   A. 2(−1)^(n+1)/n
   B. 2(−1)^n/n
   C. 4/(n^2)
   D. 0
   Answer: A. Sine series of x. (Key point: bn)

4. u(x,t) =
   A. 2Σ [(−1)^(n+1)/n] sin nx sin nct
   B. 2Σ [(−1)^(n+1)/n] sin nx cos nct
   C. Σ sin nx cos nct
   D. 2Σ cos nx cos nct
   Answer: B. Product solution. (Key point: Answer)

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