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Vibrations in Strings and Pipes –
Learning Outcomes Describe stationary waves in strings and pipes,
particularly the relationship between frequency and
length.
Use string and wind instruments.
Discuss string and woodwind sections in orchestras.
HL: Describe harmonics in strings and pipes.
HL: Solve problems about harmonics in strings and pipes.
1
Harmonics In addition to the fundamental frequency, recall that
instruments often produce overtones.
Overtones are frequencies above the fundamental
frequency that instruments produce.
In physics, we discuss the similar idea of harmonics –
integer multiples of the fundamental frequency.
The 1st harmonic is the fundamental frequency, f.
The 2nd harmonic is 2f.
The 3rd harmonic is 3f, etc.
2
Stationary Waves in Pipes Three rules for stationary waves in pipes:
Closed ends have a node,
Open ends have an anti-node,
There must be a node between two adjacent anti-
nodes and an anti-node between two adjacent nodes.
We discuss two types of pipes, those with two open
ends, and those with one open and one closed end.
Examples of open-end pipes are flutes, piccolos.
Examples of closed-end pipes are panpipes, clarinet.
3
Open PipesOpen pipes have anti-nodes at both ends.
The fundamental frequency of an open pipe has two
anti-nodes at the ends with one node between them.
4
e.g. if the above pipe is 1 m long, what is the
wavelength of the standing wave in the pipe?
e.g. if the speed of sound in air is 340 m s-1, what is the
fundamental frequency / 1st harmonic of the pipe?
HL: Harmonics in Open Pipes Recalling the rules for open pipes, what does the 2nd
harmonic look like?
5
What does the 3rd harmonic look like?
HL: Open Pipes e.g. draw the first three harmonics produced in an open
pipe of length 0.75 m.
Calculate the frequency of each and compare them to
the fundamental frequency.
e.g. In the open pipe instrument below, the note A4 is
produced with the illustrated fingering. If A4 is at 440 Hz,
what is the distance between the mouthpiece and the
open hole?
6
Closed PipesClosed pipes have an anti-node at the open end and a
node at the closed end.
The fundamental frequency of a closed pipe has one
node and one anti-node:
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e.g. if the pipe above is 1 m long, what is the
wavelength of the standing wave in the pipe?
e.g. if the speed of sound in air is 340 m s-1, what is the
fundamental frequency / 1st harmonic of the pipe?
HL: Harmonics in Closed Pipes Recalling the rules for closed pipes, what does next
harmonic look like?
8
What about the harmonic after that?
HL: Closed Pipes e.g. Draw the first three harmonics that appear in a
closed pipe of length 0.90 m.
Calculate the frequency of each and compare them to
the fundamental frequency.
9
Open Pipe Closed Pipe
Harmonic Overtone Harmonic Overtone
1st - 1st -
2nd 1st 3rd 1st
3rd 2nd 5th 2nd
4th 3rd 7th 3rd
5th 4th 9th 4th
Stationary Waves in Strings Two rules for stationary waves in strings:
Nodes at both ends,
Anti-node between adjacent nodes and node between
adjacent anti-nodes.
The fundamental frequency has a node at both ends
and one anti-node between them:
10
If the string above is 1 m long, what is the wavelength of
the wave produced within the string?
HL: Harmonics in Strings Recalling the rules for stretched strings, what does the
second harmonic look like?
11
What does the third harmonic look like?
Strings Strings work slightly differently to pipes because the
sound is produced in a different medium (pipes produce
sound in air) which travels into air.
When waves travel between media, they refract, which
changes their speed and wavelength. We can’t use just
wavelength and the speed of sound in air to determine
frequency as we did with pipes.
When waves refract, their frequency is preserved, so the
frequency of a wave within the string is the same as the
frequency after it enters air. We need the frequency
produced in the string.
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HL: Strings The fundamental frequency of a stretched string is
affected by three factors:
inversely proportional to length: 𝑓 ∝1
𝑙
proportional to root of tension: 𝑓 ∝ 𝑇
inversely proportional to root of mass per unit length
(aka linear density): 𝑓 ∝1
𝜇
Combine these with a constant of proportionality:
𝑓 =1
2𝑙
𝑇
𝜇
13
𝜇 is a Greek letter pronounced “mew”
HL: Strings e.g. A wire of length 3 m and mass 0.6 kg is stretched
between two points so that the tension in the wire is 200
N. Calculate its fundamental frequency.
e.g. When the tension in a stretched string is 40 N, its
fundamental frequency is 260 Hz. Find its fundamental
frequency if its tension is increased to 160 N.
14
OrchestrasOrchestras are made up of many different instruments.
Brass and woodwind sections use air columns in pipes to
create their sound. Trumpets, trombones and various
horns in the brass section use cylindrical pipes. Flutes and
clarinets in the woodwind section use them also.
The string section uses stretched strings to create music.
Violins and cellos vary length to create different notes.
Harps and pianos use separate strings for each note.
15
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