17-5 Sources of Musical Sound#

Prompts

For a pipe open at both ends, where are the nodes and antinodes? Sketch the fundamental and first two harmonics. What is \(\lambda\) for the fundamental?
For a pipe closed at one end and open at the other, where are the node and antinode? Why are only odd harmonics (\(n = 1, 3, 5, \ldots\)) possible?
Write the resonant frequencies for (a) open-open and (b) open-closed pipes in terms of \(L\), \(v\), and \(n\).
A flute (open-open) and a clarinet (open-closed) are the same length. Which has the lower fundamental frequency? By what factor?
Two instruments play the same note (same fundamental). Why do they sound different? What is timbre?

Lecture Notes#

Overview#

Musical instruments produce sound through standing waves—resonance when the wavelength matches the geometry.
Pipes (flute, organ, clarinet): standing waves in air columns. Strings (guitar, piano): standing waves on stretched strings (section 16-7).
Two pipe types: open at both ends (all harmonics) and open at one end, closed at the other (odd harmonics only).
Timbre (tone color) comes from the mix of harmonics—different instruments playing the same note sound different.

Standing waves in pipes: boundary conditions#

Sound waves reflect at pipe ends. The boundary condition determines whether the end is a node (zero displacement) or antinode (maximum displacement):

End type	Displacement	Analogy
Closed	Node	Fixed end of string
Open	Antinode	Free end of string

At a closed end, air cannot move longitudinally → node.
At an open end, air moves freely → antinode (slightly beyond the physical end).

Pipe open at both ends#

Antinodes at both ends. The simplest pattern has one node in the middle:

(131)#\[ \lambda = \frac{2L}{n}, \quad f = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots \]

Fundamental (\(n = 1\)): \(\lambda = 2L\), one node at center.
All harmonics present: \(n = 1, 2, 3, \ldots\) (odd and even).

Distance between nodes

For standing waves, the distance between adjacent nodes is \(\lambda/2\). So a pipe of length \(L\) fits \(n\) half-wavelengths when \(\lambda = 2L/n\).

Poll: Open-open pipe fundamental

An open-open pipe of length 1 m has \(v = 343\) m/s. What is the first harmonic frequency?

(A) 172 Hz
(B) 343 Hz
(C) 686 Hz
(D) 1029 Hz
(E) None of these

Pipe closed at one end (open-closed)#

Antinode at open end, node at closed end. The simplest pattern has \(\lambda = 4L\) (quarter-wavelength fits in the pipe):

(132)#\[ \lambda = \frac{4L}{n}, \quad f = \frac{nv}{4L}, \quad n = 1, 3, 5, \ldots \]

Fundamental (\(n = 1\)): \(\lambda = 4L\).
Only odd harmonics: \(n = 1, 3, 5, \ldots\)—even \(n\) would require an antinode at the closed end, which is impossible.

Pipe type	Fundamental \(\lambda\)	Resonant frequencies
Open-open	\(2L\)	\(f = nv/(2L)\), \(n = 1, 2, 3, \ldots\)
Open-closed	\(4L\)	\(f = nv/(4L)\), \(n = 1, 3, 5, \ldots\)

Poll: Closed-closed pipe fundamental

A pipe closed at both ends, length 1 m, \(v = 343\) m/s. What is the first harmonic?

(A) 172 Hz
(B) 343 Hz
(C) 686 Hz
(D) 1029 Hz
(E) None of these

Poll: Open-closed pipe from harmonic sequence

An open-closed pipe has adjacent harmonics 428.75 Hz, 600.25 Hz, 771.75 Hz. What is the first harmonic?

(A) 86 Hz
(B) 172 Hz
(C) 257 Hz
(D) 343 Hz
(E) None of these

Musical instruments#

Strings: guitar, piano, violin—standing waves on strings (section 16-7).
Air columns: flute (open-open), clarinet (open-closed), pipe organ, saxophone.
Shorter instrument → higher fundamental frequency.

Flute vs clarinet

Same length \(L\): flute (open-open) has \(f_1 = v/(2L)\); clarinet (open-closed) has \(f_1 = v/(4L)\). The clarinet’s fundamental is half the flute’s—it sounds an octave lower.

Timbre#

Real sounds are a superposition of the fundamental and higher harmonics. Two instruments playing the same note have the same fundamental frequency but different harmonic content (relative amplitudes of \(n = 1, 2, 3, \ldots\)). That mix is timbre—why a flute and oboe sound different even on the same pitch.

Summary#

Closed end → node; open end → antinode.
Open-open: \(f = nv/(2L)\), \(n = 1, 2, 3, \ldots\); all harmonics.
Open-closed: \(f = nv/(4L)\), \(n = 1, 3, 5, \ldots\); odd harmonics only.
Timbre: different harmonic mix → different tone color for the same pitch.