Standing Waves — Fundamental on a String (Worked Example)

Medium

Problem

A 1.2 m guitar string is fixed at both ends. Waves travel along it at 240 m/s. What are the wavelength and frequency of the fundamental (first harmonic) standing wave?

Given

L = 1.2 m
v = 240 m/s
both ends fixed
fundamental mode (n = 1)

Find

Wavelength λ₁ and frequency f₁ of the fundamental.

Principle

Fixed ends must be nodes. The longest wave that fits has a single antinode in the middle, so the string holds half a wavelength.

Formula

\lambda_1 = 2L, \qquad f_1 = \frac{v}{\lambda_1}

λ₁ = 2L; f₁ = v/λ₁

Solution

Fundamental wavelength: $\lambda_1 = 2L = 2 \times 1.2 = 2.4\ \mathrm{m}$
Frequency: $f_1 = v/\lambda_1 = 240/2.4$
Calculate: $f_1 = 100\ \mathrm{Hz}$

λ₁ = 2.4 m; f₁ = 100 Hz

Unit check

(m/s)/m = 1/s = Hz. Consistent.

What this means physically

100 Hz is about the pitch of a low male voice — and near the open-string region of a guitar. Shortening the string (fretting) raises this frequency, which is how notes are made.

Common mistake

Using λ = L instead of 2L. One full wavelength needs two antinodes; the fundamental fits only half a wave between the fixed ends.