Harmonics are the allowed standing-wave frequencies of a vibrating system. The first harmonic is the fundamental frequency, and higher harmonics are whole-number multiples of it.
A string or air column can vibrate in several allowed patterns, each with its own frequency.
Showing a random 20 of 50 problems.
Example 1
medium
The 4th and 5th harmonics of a string are 280 Hz and 350 Hz. Verify and find the fundamental.
Example 2
challenge
Two open organ pipes have lengths L1 and L2=L1/2. The longer pipe's 4th harmonic equals the shorter pipe's which harmonic?
Example 3
medium
A closed pipe has fundamental 170 Hz. What is its next allowed harmonic? (Odd harmonics only.)Closed pipe: f₁ = 170 Hz; find the next harmonic
Example 4
hard
An open pipe and a closed pipe have the same length. The open pipe's fundamental is 400 Hz. State both pipes' first three modal frequencies.
Example 5
medium
A closed pipe's 7th harmonic is 770 Hz. State its fundamental and check that 7 is an allowed mode.
Example 6
easy
The 3rd harmonic of an open pipe is 660 Hz. What is its fundamental? (fn=nf1.)3rd harmonic of an open pipe (f₃ = 660 Hz)
Example 7
easy
On a string of length L, the wavelength of the nth harmonic is ___.
Example 8
challenge
A pipe of length 0.85 m has sound speed 340 m/s and is closed at one end. Find the harmonic numbers and frequencies of all allowed modes below 1000 Hz.
Example 9
easy
True or false: every harmonic on a string-fixed-at-both-ends is a standing wave.
Example 10
medium
A string plays its 2nd harmonic at 400 Hz. The string is shortened to half its length with no tension change. Find the new 2nd-harmonic frequency.
Example 11
medium
A string's fundamental is 110 Hz. What is the frequency of its 6th harmonic, and is it an octave-related note?String: f₁ = 110 Hz, find the 6th harmonic
Example 12
easy
A string's fundamental frequency is 150 Hz. Using fn=nf1, find the 2nd harmonic.Fundamental (f₁ = 150 Hz) vs 2nd harmonic
Example 13
medium
An open pipe of length 0.5 m has wave speed 340 m/s. Find its fundamental frequency. (f1=2Lv.)Open pipe fundamental, L = 0.5 m, v = 340 m/s
Example 14
easy
A guitar string fundamental is 200 Hz. List the 1st, 2nd, and 3rd harmonics.
Example 15
challenge
A string of linear density μ=0.01 kg/m is stretched to tension T=100 N over length 0.5 m. Find the frequency of the 3rd harmonic.
Example 16
medium
An open pipe and a closed pipe have the same length 0.4 m (v = 340 m/s). Find the ratio of their fundamental frequencies.Open vs closed pipe fundamental, L = 0.4 m, v = 340 m/s
Example 17
medium
A string fixed at both ends has wavelengths λ1=2 m, λ2=1 m, λ3=?.
Example 18
hard
A pipe open at both ends has consecutive harmonics at 360 Hz and 480 Hz. Find its fundamental.
Example 19
medium
A string fixed at both ends, length 1 m, has wave speed 240 m/s. Find its 2nd harmonic frequency. (fn=2Lnv.)String harmonics: L = 1 m, v = 240 m/s; find f₂
Example 20
hard
A string of length 50 cm vibrates with three loops (3 antinodes). Wave speed is 240 m/s. Find the frequency.