1.6.19 · D5Oscillations & Waves
Question bank — Harmonics and overtones — on strings and in pipes
Before the questions, three anchors you will lean on constantly. Read them once slowly.
True or false — justify
Each answer must say why, not just "true/false".
T or F: A string fixed at both ends can vibrate at any frequency you drive it at.
False. Only frequencies whose half-wavelength fits a whole number of times survive; other frequencies destructively interfere with their own reflection and die out.
T or F: In a pipe open at both ends, every integer harmonic is present.
True. Open–open means antinode at each end, antinode-to-antinode spacing is , so works for all integers — nothing forbids any of them.
T or F: A closed pipe of length has the same fundamental as an open pipe of length .
False. Closed needs only a quarter wave (), so versus for open; the closed wavelength is doubled, hence its is half the open pipe's .
T or F: The "first overtone" and the "second harmonic" are always the same frequency.
False. They coincide for strings and open pipes, but in a closed pipe the second harmonic doesn't exist, so the first overtone is the third harmonic .
T or F: Doubling the length of a fixed–fixed string halves every harmonic frequency.
True. ; if (tension and density unchanged) is fixed, then , so doubling halves all of them together.
T or F: A closed pipe's harmonics are spaced apart, just like an open pipe's.
False. Only odd multiples survive (), so consecutive allowed frequencies differ by , not .
T or F: The clean formulas and are exactly right for real pipes.
False. The antinode sits slightly outside an open end, so the effective length is a bit longer ( per open end), which lowers the true frequencies below the ideal ones.
T or F: Raising the tension in a guitar string raises all its harmonics.
True. rises with tension, and scales with , so every harmonic goes up by the same factor.
T or F: A "node" and an "overtone" describe the same thing.
False. A node is a place on the standing wave with zero motion; an overtone is a frequency above the fundamental. One is spatial, the other is a pitch.
Spot the error
Each line states a piece of (flawed) reasoning; the reveal names the mistake and repairs it.
"A closed pipe sounds an octave lower, so every one of its harmonics is half the matching open-pipe harmonic."
The octave-lower fact applies to the fundamental formula ( vs ). But the closed pipe only keeps odd harmonics, so its higher modes don't line up half-for-half with the open pipe's at all.
"An open pipe and a closed pipe of the same length must share their third harmonic, since both allow ."
They allow of their own different . Open gives ; closed gives — different frequencies, so they do not coincide.
"Between any node and any antinode the distance is ."
No — node to nearest antinode is . It is node to node (or antinode to antinode) that spans .
"The 5th harmonic of a closed pipe is the 5th allowed mode."
The 5th harmonic is , which is only the third allowed mode of a closed pipe (allowed list: ). Harmonic number and overtone/mode index differ here.
"Since , making the pipe longer makes the wave travel faster."
is set by the medium (speed of sound), not by geometry. A longer pipe changes the allowed and hence , but stays the same.
"A free end of a string must be a node because the end is a boundary."
A free end is an antinode — it is free to move maximally. Only a fixed clamped end is forced to be a node.
"Adding mass per length to a string raises its pitch."
The opposite: falls as grows, so drops. That's why thicker bass strings sound lower.
Why questions
Why must a fixed end of a string be a node?
The clamp physically cannot move, so the displacement there is forced to zero at all times — that is exactly the definition of a node.
Why can only whole numbers of half-wavelengths fit on a fixed–fixed string?
Each end must be a node, and nodes are spaced apart; fitting nodes at both ends with the wave in between requires an integer count of these segments.
Why does a closed pipe skip even harmonics?
It needs a node at the closed end and an antinode at the open end. That node–antinode geometry only fits when is an odd number of quarter-wavelengths, which forces frequencies to be odd multiples of .
Why do a string and an open pipe share the exact same frequency formula?
Both have matching boundary conditions at the two ends (node–node for the string, antinode–antinode for the pipe), and both boundary types repeat every , so the fitting condition is identical.
Why does a longer wavelength mean a lower pitch?
Because with fixed by the medium: a bigger in the denominator gives fewer oscillations per second, i.e. a lower frequency.
Why does the end correction lower the observed frequency?
It makes the effective length longer, which lengthens the fitting wavelength, and a longer gives a smaller .
Why is "harmonic" a safer word than "overtone" when comparing systems?
"-th harmonic" always means the fixed number , whereas "-th overtone" just means the -th allowed mode above , and which modes are allowed changes between open and closed pipes.
Edge cases
What happens if a string is driven at a frequency exactly between two harmonics?
No stable standing wave forms — the reflected wave does not reinforce the incoming one, so the pattern interferes with itself and dies away; only the true harmonics resonate.
Is there a "zeroth harmonic" ()?
No. gives , meaning no vibration at all (an infinite wavelength), so it is not a real oscillation — the ladder starts at .
What is the largest possible wavelength that fits a fixed–fixed string of length ?
, from the fundamental . Any longer wavelength cannot place nodes at both ends, so is the ceiling.
For a closed pipe, what is the longest wavelength allowed?
, from — this is the fundamental and gives the deepest note the pipe can sound.
If you double a closed pipe's length, what happens to its fundamental?
It halves, since ; the pipe sounds an octave lower.
What frequency does a standing wave have at a node itself?
A node has zero displacement at all times, but it still belongs to the same oscillation — the frequency of the mode is a property of the whole pattern, not of a single point; the node just never moves.
As the harmonic number grows very large on a string, what happens to the spacing between nodes?
The wavelength shrinks, so nodes crowd closer together ( apart) — very high harmonics have many tightly packed nodes.
Recall One-line survival kit
Ask which modes are allowed before naming any overtone; keep (node–node) and (node–antinode) at your fingertips; remember belongs to the medium, geometry only picks .
Connections
- Standing waves and superposition — why unfit frequencies self-destruct.
- Reflection of waves at boundaries — why fixed vs free ends behave oppositely.
- Speed of a wave on a string — the origin of used in the tension traps.
- Speed of sound in gases — what sets for every pipe question here.
- Beats and resonance — how a driven column locks onto exactly these frequencies.
- Timbre and Fourier synthesis — why the set of allowed overtones shapes an instrument's sound.