Start with the recap, study the fully worked examples, then use the practice problems to
check your understanding of Harmonics.
This page combines explanation, solved examples, and follow-up practice so you can move
from recognition to confident problem-solving in Physics.
Concept Recap
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.
Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Harmonics asks what oscillates, what travels, and which wave quantity is being measured.
Common stuck point:Students often know a formula related to harmonics but skip the recognition step: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition? That leads to a correct-looking substitution attached to the wrong physical model.
Sense of Study hint:Ask: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition?
Worked Examples
Example 1
medium
An open pipe has length 0.85 m and air speed 340 m/s. Find its first three harmonic frequencies.
Answer
f1=200,f2=400,f3=600 Hz
First step
1
Open pipe: f1=v/(2L)=340/(2×0.85)=200 Hz.
See the full worked solution + why-it-works coaching
Setup·Key insight·Why it works·Common pitfall·Connection
A pipe open at both ends produces harmonics at 256 Hz, 512 Hz, 768 Hz, ... Now one end is closed without changing length. List the lowest three frequencies it can play (speed of sound unchanged).
Example 3
hard
A string of length 50 cm vibrates with three loops (3 antinodes). Wave speed is 240 m/s. Find the frequency.
Example 4
hard
On a string of length L, a finger lightly touches the midpoint while plucking. Which harmonic is enhanced and which is suppressed?
Example 5
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.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A string's fundamental frequency is 150 Hz. Using fn=nf1, find the 2nd harmonic.Fundamental (f₁ = 150 Hz) vs 2nd harmonic
Example 2
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 3
easy
Is the 1st harmonic the same as the fundamental frequency?
Example 4
easy
A closed pipe (one end closed) allows only odd harmonics. Which of 2, 3, 4 is allowed?
Example 5
easy
For an open pipe with fundamental 100 Hz, list the first three harmonic frequencies.Harmonic series: f₁ = 100, f₂ = 200, f₃ = 300 Hz
Example 6
easy
The 4th harmonic of a string is 480 Hz. What is the 2nd harmonic? (fn=nf1.)
Example 7
easy
Higher harmonics of a string have higher or lower frequencies than the fundamental?
Example 8
easy
An open pipe's fundamental is 256 Hz. What is its 3rd harmonic?Open pipe: f₁ = 256 Hz, find f₃
Example 9
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 10
medium
A closed pipe (one end closed) of length 0.25 m has sound speed 340 m/s. Find its fundamental frequency. (f1=4Lv.)
Example 11
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 12
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 13
medium
The first overtone of an open pipe is 400 Hz. What is the fundamental? (First overtone = 2nd harmonic for open pipes.)1st overtone (f₂ = 400 Hz) of an open pipe
Example 14
medium
A closed pipe's 3rd harmonic is 450 Hz. What is the 5th harmonic? (Odd harmonics, fn=nf1.)Closed pipe: f₃ = 450 Hz, find f₅
Example 15
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 16
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 17
medium
A closed pipe of length 0.5 m has sound speed 340 m/s. Find its 3rd harmonic frequency. (Closed pipe: fn=4Lnv, odd n.)Closed pipe: L = 0.5 m, v = 340 m/s; find f₃
Example 18
challenge
An open pipe of length L and a closed pipe of length L/2 are compared (same v). Find the ratio of their fundamental frequencies.
Example 19
challenge
A guitar string sounds 196 Hz fundamental. Pressing a fret shortens it so the new fundamental is the 4th harmonic of the open string's pitch... actually the new fundamental is 392 Hz. By what factor was the effective length shortened? (f1∝1/L.)
Example 20
challenge
An open pipe is 1 m long with v = 340 m/s. How many harmonics lie at or below 1000 Hz?
Example 21
easy
A string's fundamental is 120 Hz. State its 5th harmonic.
Example 22
easy
A guitar string fundamental is 200 Hz. List the 1st, 2nd, and 3rd harmonics.
Example 23
easy
An open pipe has 6th harmonic at 1200 Hz. Find its fundamental.
Example 24
easy
A closed pipe has fundamental 100 Hz. State its 3rd and 5th harmonic frequencies.
Example 25
easy
If the fundamental of a string is f1 and you double the tension, by what factor does f1 change? (v∝T.)
Example 26
easy
On a string of length L, the wavelength of the nth harmonic is ___.
Example 27
medium
A closed pipe has length 0.5 m, speed 340 m/s. Find its fundamental and first two allowed harmonics.
Example 28
medium
A string of length 0.6 m, wave speed 180 m/s, plays its 3rd harmonic. Find its frequency.
Example 29
medium
The 4th and 5th harmonics of a string are 280 Hz and 350 Hz. Verify and find the fundamental.
Example 30
medium
A closed pipe's 7th harmonic is 770 Hz. State its fundamental and check that 7 is an allowed mode.
Example 31
medium
A string fixed at both ends has wavelengths λ1=2 m, λ2=1 m, λ3=?.
Example 32
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 33
medium
How many nodes (including the ends) does the 4th harmonic on a fixed-fixed string have?
Example 34
medium
Sound speed in air at 20°C is about 343 m/s. Find the fundamental of a 1.0 m open pipe.
Example 35
medium
A 0.4 m closed pipe in air with v=340 m/s plays its 3rd harmonic. Find that frequency.
Example 36
hard
Two strings have the same length but tensions in ratio 9:1. The looser string plays fundamental 100 Hz. Find the tighter string's fundamental.
Example 37
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 38
hard
A guitar string is 65 cm long with wave speed 520 m/s. A finger frets it at the 12th-fret position, halving the vibrating length to 32.5 cm. Find the new fundamental.
Example 39
hard
A pipe open at both ends has consecutive harmonics at 360 Hz and 480 Hz. Find its fundamental.
Example 40
hard
A closed pipe has consecutive observed harmonics at 270 Hz and 450 Hz. Find its fundamental.
Example 41
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 42
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?