---
title: "Vibrating string deflection"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:vibrating-string-deflection"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/vibrating-string-deflection
site: BCT Engineering
---

# Vibrating string deflection

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Kartik (new course), 4 marks
- Video: [Problems on vibrating string| wave equation | KTU | Partial Differential equations](https://www.youtube.com/watch?v=-Rrm2y4flBo) — Prijimol V B
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/vibrating-string-deflection
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

A tightly stretched string fixed at two ends x = 0 and x = π is initially at rest in equilibrium position. If it is set in vibration by giving to each of its points a velocity ∂u/∂t = 0.03 sin x − 0.04 sin 3x, find the deflection u(x, t) at any point of the string at any time.

## Short answer

With u(x,0)=0, solution is u(x,t)= Σ C_n sin(nct) sin(nx). Matching u_t(x,0)=0.03 sin x − 0.04 sin 3x gives C1=0.03/c, C3=−0.04/(3c). Deflection: u(x,t) = (0.03/c) sin(ct) sin x − (0.04/(3c)) sin(3ct) sin 3x.

## Answer

Let the string vibrate according to the wave equation u_tt = c² u_xx, with

- u(0, t) = u(π, t) = 0 (fixed ends),
- u(x, 0) = 0 (initially in equilibrium position),
- u_t(x, 0) = 0.03 sin x − 0.04 sin 3x.

**General solution.** Separation of variables with the end conditions gives p = n:

u = Σ (Bₙ cos nct + Cₙ sin nct) sin nx

**Initial displacement.** u(x, 0) = 0 gives Σ Bₙ sin nx = 0, so every Bₙ = 0.

**Initial velocity.** Differentiating with respect to t and putting t = 0:

```text
u_t(x, 0) = Σ nc Cₙ sin nx = 0.03 sin x − 0.04 sin 3x
```

Comparing coefficients of sin x and sin 3x (all others are zero):

- n = 1: c C₁ = 0.03, so C₁ = 0.03/c
- n = 3: 3c C₃ = −0.04, so C₃ = −0.04/(3c)

**Answer:**

u(x, t) = (0.03/c) sin ct sin x − (0.04/3c) sin 3ct sin 3x

Here c² = T/ρ, where c is the wave speed on the string (with c = 1 this is 0.03 sin t sin x − 0.0133 sin 3t sin 3x).

## Check yourself

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

1. u(x,0) = 0 gives:
   A. all Cn = 0
   B. all Bn = 0
   C. B1 only
   D. nothing
   Answer: B. Σ Bn sin nx = 0. (Key point: Zero displacement)

2. u_t(x,0) = Σ nc Cn sin nx; compare coefficients:
   A. C1 = 0.03
   B. c C1 = −0.04
   C. n = 1: cC1 = 0.03
   D. C1 = 0.03c
   Answer: C. Coefficient of sin x. (Key point: Velocity)

3. C3 =
   A. −0.04/c
   B. 0.04/(3c)
   C. −0.04c/3
   D. −0.04/(3c)
   Answer: D. 3c C3 = −0.04. (Key point: C3)

4. u =
   A. (0.03/c) sin ct sin x − (0.04/3c) sin 3ct sin 3x
   B. 0.03 cos ct sin x − 0.04 cos 3ct sin 3x
   C. (0.03/c) cos ct sin x
   D. 0.03 sin x − 0.04 sin 3x
   Answer: A. Sine in time since displacement is zero. (Key point: Answer)

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