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.

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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.

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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.

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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.

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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?

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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?

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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?

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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.

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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:

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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?

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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

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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.

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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.

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