---
title: "Lateral vibration of a stretched string"
subject: "Theory of Mechanism and Machine II"
chapter: 10
key: "semester_7/theory_of_mechanism_and_machine_ii:string-wave-equation-lateral-vibration"
source: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/string-wave-equation-lateral-vibration
site: BCT Engineering
---

# Lateral vibration of a stretched string

Theory of Mechanism and Machine II · Chapter 10: Vibration of Continuous Systems

- Asked in: year not given, 6 marks
- Video: [Modes on a String](https://www.youtube.com/watch?v=cnH2ltfW48U) — geoff martin
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/string-wave-equation-lateral-vibration
- Whole chapter: https://www.bctengineer.com/md/semester_7/theory_of_mechanism_and_machine_ii/chapter-10

## Question

Derive the wave equation for the lateral vibration of a taut string and obtain its natural frequencies and mode shapes for a string fixed at both ends. Find the first three natural frequencies of a string 0.65 m long, with tension 70 N and mass per unit length 0.0006 kg/m.

## Short answer

Wave equation ∂²w/∂t² = c² ∂²w/∂x² with c = √(P/ρ) follows from transverse force balance. Fixed ends give f_n = (n/(2l))c. For l=0.65 m, P=70 N, ρ=0.0006 kg/m: f₁=262.7 Hz, f₂=525.5 Hz, f₃=788.2 Hz.

## Answer

#### Wave equation
A string of length $l$, tension $P$ (constant) and mass per unit length $\rho$ vibrates with small transverse displacement $w(x,t)$.

```text
  P  ^ (theta at x)        ^ P (theta at x+dx)
      \                   /
  ----- \_______________/ -----
        x          x+dx
```
For an element $dx$, the net transverse force from the tension is

$$
P\left[\frac{\partial w}{\partial x}\right]_{x+dx} - P\left[\frac{\partial w}{\partial x}\right]_x = P\frac{\partial^2 w}{\partial x^2}dx
$$

By Newton's law, this equals $\rho\,dx\,\dfrac{\partial^2 w}{\partial t^2}$. Hence

$$
\frac{\partial^2 w}{\partial t^2} = c^2\frac{\partial^2 w}{\partial x^2}, \qquad c = \sqrt{\frac{P}{\rho}}
$$

$c$ is the wave speed.

#### Solution
Let $w(x,t) = W(x)\,T(t)$. Separating variables:

$$
\frac{W''}{W} = \frac{\ddot T}{c^2T} = -\frac{\omega^2}{c^2} \;\Rightarrow\; W = A\cos\frac{\omega x}{c} + B\sin\frac{\omega x}{c}
$$

Boundary conditions $W(0) = 0 \Rightarrow A = 0$ and $W(l) = 0 \Rightarrow \sin\dfrac{\omega l}{c} = 0$ (frequency equation):

$$
\omega_n = \frac{n\pi c}{l}, \qquad f_n = \frac{n}{2l}\sqrt{\frac{P}{\rho}}, \qquad n = 1, 2, 3,\ldots
$$

Mode shape: $W_n(x) = B_n\sin\dfrac{n\pi x}{l}$. The $n$th mode has $n-1$ nodes between the supports.

#### Numerical values
$$
c = \sqrt{\frac{70}{0.0006}} = 341.6\ \text{m/s}
$$

$$
f_1 = \frac{c}{2l} = \frac{341.6}{2 \times 0.65} = 262.7\ \text{Hz}, \quad f_2 = 2f_1 = 525.5\ \text{Hz}, \quad f_3 = 3f_1 = 788.2\ \text{Hz}
$$

**Answer:** $f_1 = 262.7$ Hz, $f_2 = 525.5$ Hz, $f_3 = 788.2$ Hz.

## Check yourself

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

1. Newton's law on a string element gives the wave equation:
   A. w_tt = c^2 w_xx with c = sqrt(P/rho)
   B. w_tt = c w_x with c = P/rho
   C. w_xx = c^2 w_t
   D. w_tt = c^2 w_xx with c = sqrt(rho/P)
   Answer: A. Net tension force P w_xx dx = rho dx w_tt, so c^2 = P/rho (wave speed). (Key point: Wave equation)

2. For a string fixed at both ends, the frequency equation is:
   A. cos(w l/c) = 0
   B. sin(w l/c) = 0
   C. tan(w l/c) = 1
   D. sinh(w l/c) = 0
   Answer: B. W(0) = 0 kills the cosine; W(l) = 0 needs sin(w l/c) = 0. (Key point: Boundary conditions)

3. The natural frequencies of the fixed-fixed string are:
   A. f_n = (2n-1) c/4l
   B. f_n = n^2 c/2l
   C. f_n = c/(2 n l)
   D. f_n = n c/2l
   Answer: D. w_n = n pi c/l gives f_n = (n/2l) sqrt(P/rho), integer multiples of the fundamental. (Key point: Natural frequencies)

4. For l = 0.65 m, P = 70 N, rho = 0.0006 kg/m, the fundamental frequency is about:
   A. 525 Hz
   B. 131 Hz
   C. 341.6 Hz
   D. 262.7 Hz
   Answer: D. c = sqrt(70/0.0006) = 341.6 m/s, f1 = c/2l = 262.7 Hz; f2 and f3 are 2 and 3 times that. (Key point: Numerical result)

## Related in Theory of Mechanism and Machine II

- [Three-mass chain frequencies and modes](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Athree-dof-eigenvalues-mode-shapes) — Theory of Mechanism and Machine II, chapter 8
- [Lateral vibration of a simply supported beam](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Abeam-lateral-vibration-simply-supported) — Theory of Mechanism and Machine II, chapter 10
- [Natural frequencies of a two-mass system](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Atwo-dof-natural-frequencies-mode-shapes) — Theory of Mechanism and Machine II, chapter 7
- [Rod longitudinal and shaft torsional frequencies](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Alongitudinal-rod-torsional-shaft-frequencies) — Theory of Mechanism and Machine II, chapter 10
- [Matrix iteration for the first mode](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Amatrix-iteration-fundamental-frequency) — Theory of Mechanism and Machine II, chapter 9
- [Free damped vibration: three cases](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Adamped-free-vibration-cases-derivation) — Theory of Mechanism and Machine II, chapter 6

## Related in other subjects

- [Three moment equation: derive and apply to fixed beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Athree-moment-derive-fixed-beam-udl) — Theory of Structures II, chapter 2
- [Euler's formula: both ends fixed](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-fixed) — Strength of Materials, chapter 8, asked 2×
- [Euler's formula both ends fixed, with limitations](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-fixed-limitation) — Strength of Materials, chapter 8
- [Derive 1D wave equation and solve](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Awave-eq-derive-and-solve) — Applied Mathematics, chapter 3, asked 5×