Class 11 > Unit # 12:Acoustics > Modes of Vibration in a Stretched String

Talha's Physics Academy

Modes of Vibrations in a Stretched String - Unit #12 Acoustics

Video Lecture: Modes of Vibrations in a Stretched String

Watch the complete lecture demonstrating musical string variations, standing wave harmonics, and mathematical frequency derivations:

Modes of Vibrations in a Stretched String

When you pluck strings of different thicknesses on a guitar, you notice a distinct change in pitch—thicker strings produce lower, heavier notes, while thinner strings produce higher, sharper tones. This behavior is governed by the modes of vibration in a stretched string.

1. Fundamental Mode of Vibration (First Harmonic)

When a string of length $L$ is fixed at both ends and vibrates in a single loop, it forms the fundamental mode (or first harmonic). The length of the string corresponds to half of the wavelength:

$L = \frac{\lambda_1}{2} \implies \lambda_1 = 2L$

Using the wave speed equation $v = f \lambda$, the fundamental frequency $f_1$ is:

$f_1 = \frac{v}{2L}$

2. Speed of Waves in a Stretched String

The velocity $v$ of waves traveling along a stretched string depends on the tension ($T$) applied to the string and its linear mass density ($\mu = \frac{m}{L}$, mass per unit length):

$v = \sqrt{\frac{T}{\mu}}$

Substituting this into the fundamental frequency equation gives:

$f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$

3. Second Harmonic (First Overtone)

When the string vibrates in two segments (two loops), a node appears in the center. The length of the string equals one full wavelength:

$L = \lambda_2 \implies f_2 = \frac{v}{L} = 2\left(\frac{v}{2L}\right) = 2f_1$

4. Third Harmonic (Second Overtone)

When the string vibrates in three loops, the length equals three half-wavelengths:

$L = \frac{3\lambda_3}{2} \implies f_3 = 3\left(\frac{v}{2L}\right) = 3f_1$

5. General Equation for $n$ Loops (Harmonics)

Extending this pattern to $n$ loops, the frequency of the $n$-th harmonic is given by:

$f_n = n f_1 = \frac{n}{2L}\sqrt{\frac{T}{\mu}} \quad (\text{where } n = 1, 2, 3, \dots)$

© 2026 Talha's Physics Academy. All rights reserved.