Standing Waves — Flute Second Harmonic (Worked Example)

Medium

Problem

A flute (open pipe) is 0.60 m long. What is the frequency of its second harmonic? (v = 343 m/s)

Given

L = 0.60 m
v = 343 m/s
n = 2

Find

The second harmonic f₂.

Principle

Higher harmonics are integer multiples: f_n = nv/(2L) with n = 2.

Formula

f_n = \frac{nv}{2L}

f_n = nv/(2L)

Solution

Substitute: $f_2 = 2 \times 343/(2 \times 0.60)$
Numerator: $686$ ; denominator: $1.20$
Divide: $f_2 = 572\ \mathrm{Hz}$

572 Hz

Unit check

(m/s) ÷ m = Hz, times the dimensionless harmonic number — a frequency.

What this means physically

572 Hz — exactly twice the 286 Hz fundamental. Overblowing to harmonics is how flutes (and all wind instruments) reach high notes without extra holes.

Common mistake

Calling n = 2 the 'first harmonic'. n = 1 is the first harmonic (fundamental); n = 2 is the second.