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Standing Waves
AP Physics 1
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Standing Waves

Standing waves are periodic waves that are reflected and
inverted so that they interfere with one another.

The result of the interference is that specific points along the
medium appear to be standing still while other points
vibrated back and forth.

NODES: locations of total destructive interference

ANTINODES: locations of constructive interference
.
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Pitch

Pitch is our brain’s interpretation of frequency.

440 Hz is typically concert pitch and is the A above middle C.
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Harmonics

Such patterns are only created within the medium at specific
frequencies of vibration. These frequencies are known as
harmonic frequencies or merely harmonics. At any
frequency other than a harmonic frequency, the interference
of reflected and incident waves results in a disturbance of the
medium that is irregular and non-repeating

n is called the harmonic number

n = 1 is the fundamental frequency

n= 2 is the 2nd harmonic

n = 3 is the 3rd harmonic, etc.
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Standing Waves for Vibrating
Strings

This is just one possible
example. There are a variety
of patterns by which the guitar
string could naturally vibrate.
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Standing Waves for a Vibrating
String

Each pattern is associated with
one of the natural frequencies
of the guitar strings.. Each
standing wave pattern is
referred to as a harmonic of
the instrument (in this case,
the guitar string). The three
diagrams at the right
represent the standing wave
patterns for the first, second,
and third harmonics of a guitar
string.
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For guitar strings, each pattern is characterized by
some basic traits:

There is an alternating pattern of nodes and antinodes.

There are either a half-number or a whole number of waves
within the pattern established on the string.

Nodal positions (points of no displacement) are established
at the ends of the string where the string is clamped down in
a fixed position.

One pattern is related to the next pattern by the addition (or
subtraction) of one or more nodes (and antinodes).
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Nodes at each end of string

First, consider a guitar string vibrating at its natural
frequency or harmonic frequency. Because the ends of the
string are attached and fixed in place to the guitar's structure
(the bridge at one end and the frets at the other), the ends of
the string are unable to move. Subsequently, these ends
become nodes - points of no displacement. In between these
two nodes at the end of the string, there must be at least one
antinode.
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Vibrating String: Fundamental
Frequency

The most fundamental harmonic
for a guitar string is the
harmonic associated with a
standing wave having only one
antinode positioned between
the two nodes on the end of the
string.

This would be the harmonic
with the longest wavelength and
the lowest frequency.

The lowest frequency produced
by any particular instrument is
known as the fundamental
frequency. The fundamental
frequency is also called the first
harmonic of the instrument.
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Vibrating String: 2nd Harmonic

The second harmonic of a guitar string is
produced by adding one more node
between the ends of the guitar string. And
of course, if a node is added to the pattern,
then an antinode must be added as well in
order to maintain an alternating pattern of
nodes and antinodes.

In order to create a regular and repeating
pattern, that node must be located midway
between the ends of the guitar string

. This additional node gives the second
harmonic a total of three nodes and two
antinodes. The standing wave pattern for
the second harmonic is shown at the right.
A careful investigation of the pattern
reveals that there is exactly one full wave
within the length of the guitar string. For
this reason, the length of the string is equal
to the length of the wave.
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Vibrating String: 3rd Harmonic

The third harmonic of a guitar string is produced
by adding two nodes between the ends of the
guitar string. And of course, if two nodes are
added to the pattern, then two antinodes must be
added as well in order to maintain an alternating
pattern of nodes and antinodes.

In order to create a regular and repeating pattern
for this harmonic, the two additional nodes must
be evenly spaced between the ends of the guitar
string. This places them at the one-third mark and
the two-thirds mark along the string. These
additional nodes give the third harmonic a total of
four nodes and three antinodes.

The standing wave pattern for the third harmonic
is shown at the right. A careful investigation of the
pattern reveals that there is more than one full
wave within the length of the guitar string. In fact,
there are three-halves of a wave within the length
of the guitar string. For this reason, the length of
the string is equal to three-halves the length of the
wave.
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Vibrating Strings: Pattern
Harmonic
#
1
2
3
4
5
# Waves in
String
# Nodes
# Antinodes
WavelengthLength
Relationship
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Vibrating String Example Problem

Consider an 80-cm long guitar string that has a fundamental
frequency (1st harmonic) of 400 Hz.

What is the wavelength of the wave pattern produced?

What is the speed of this standing wave?

Find the wavelength, speed and frequency for the 2nd and 3rd
harmonics.
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Vibrating String: Frequencies

the individual frequencies in the set of natural frequencies
produced by a guitar string are related to each other by
whole number ratios.

The frequency of the second harmonic is two times the
frequency of the first harmonic. The frequency of the third
harmonic is three times the frequency of the first harmonic.
The frequency of the nth harmonic (where n represents the
harmonic # of any of the harmonics) is n times the frequency
of the first harmonic. In equation form, this can be written as
 fn
= n • f1
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Summary of Guitar Strings
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Open-End Air Columns

Many musical instruments consist of an air column enclosed
inside of a hollow metal tube. Though the metal tube may be
more than a meter in length, it is often curved upon itself one
or more times in order to conserve space. If the end of the
tube is uncovered such that the air at the end of the tube can
freely vibrate when the sound wave reaches it, then the end
is referred to as an open end. If both ends of the tube are
uncovered or open, the musical instrument is said to contain
an open-end air column.

A variety of instruments operate on the basis of open-end air
columns; examples include the flute and the recorder. Even
some organ pipes serve as open-end air columns.
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Standing Wave Patterns in Air
Columns

In the case of stringed instruments (discussed earlier), standing wave
patterns were drawn to depict the amount of movement of the string at
various locations along its length. Such patterns show nodes - points of
no displacement or movement - at the two fixed ends of the string.

In the case of air columns, a closed end in a column of air is analogous
to the fixed end on a vibrating string. That is, at the closed end of an air
column, air is not free to undergo movement and thus is forced into
assuming the nodal positions of the standing wave pattern.

Conversely, air is free to undergo its back-and-forth longitudinal motion
at the open end of an air column; and as such, the standing wave
patterns will depict antinodes at the open ends of air columns.

So the basis for drawing the standing wave patterns for air columns is
that vibrational antinodes will be present at any open end and
vibrational nodes will be present at any closed end.
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Open-End Air Column:
1st harmonic

the pattern for the fundamental
frequency (the lowest frequency
and longest wavelength pattern)
will have antinodes at the two
open ends and a single node in
between.

The distance between antinodes
on a standing wave pattern is
equivalent to one-half of a
wavelength. A careful analysis of
the diagram above shows that
adjacent antinodes are
positioned at the two ends of the
air column. Thus, the length of
the air column is equal to onehalf of the wavelength for the
first harmonic.
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Open-End Air Column:
2nd harmonic

total of three antinodes and
two nodes.

there is one full wave in the
length of the air column. One
full wave is twice the number
of waves that were present in
the first harmonic. For this
reason, the frequency of the
second harmonic is two times
the frequency of the first
harmonic.
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Open-End Air Column:
3rd harmonic

a total of four antinodes and
three nodes.

there are one and one-half
waves present in the length of
the air column. One and onehalf waves is three times the
number of waves that were
present in the first harmonic.
For this reason, the frequency
of the third harmonic is three
times the frequency of the first
harmonic.
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Open-End Air Columns: Pattern
Harmonic
#
1
2
3
4
5
# Waves in
String
# Nodes
# Antinodes
WavelengthLength
Relationship
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Closed-End Air Columns

An instrument consisting of a closed-end column typically
contains a metal tube in which one of the ends is covered and
not open to the surrounding air. Some pipe organs and the
air column within the bottle of a soda-bottle orchestra are
examples of closed-end instruments.

Some instruments that operate as open-end air columns can
be transformed into closed-end air columns by covering the
end opposite the mouthpiece with a mute. As we will see the
presence of the closed end on such an air column will affect
the actual frequencies that the instrument can produce.
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Standing Wave Patterns in Air
Columns (again)

In the case of stringed instruments (discussed earlier), standing wave
patterns were drawn to depict the amount of movement of the string at
various locations along its length. Such patterns show nodes - points of
no displacement or movement - at the two fixed ends of the string.

In the case of air columns, a closed end in a column of air is analogous
to the fixed end on a vibrating string. That is, at the closed end of an air
column, air is not free to undergo movement and thus is forced into
assuming the nodal positions of the standing wave pattern.

Conversely, air is free to undergo its back-and-forth longitudinal motion
at the open end of an air column; and as such, the standing wave
patterns will depict antinodes at the open ends of air columns.

So the basis for drawing the standing wave patterns for air columns is
that vibrational antinodes will be present at any open end and
vibrational nodes will be present at any closed end.
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Closed-End Air Column:
1st harmonic

the pattern for the fundamental
frequency (the lowest frequency
and longest wavelength pattern)
will have a node at the closed
end and an antinode at the open
end.

The distance between an
antinode and a node must be
equivalent to one-fourth of a
wavelength. A careful analysis of
the diagram above shows that a
node and an adjacent antinode
are positioned at the two ends of
the air column. Thus, the length
of the air column is equal to onefourth of the wavelength for the
first harmonic.
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Closed-End Air Column:
3rd harmonic

. Just as for all the instruments, the
next harmonic for a closed-end air
column is the harmonic that has one
more node. And just as for all the
instruments, the addition of an extra
node also means that an extra
antinode must also be added to the
pattern. This would result in a total of
two vibrational antinodes and one
vibrational node

Observe in the pattern that there is
three-fourths of a full wave in the
length of the air column. That is three
times the number of waves in the
first harmonic. Since, the frequency
of this harmonic is three times the
frequency of the first harmonic, this
is called the third harmonic.
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What happened to the 2nd
harmonic?

what happened to the second harmonic?

Unlike the other instrument types, there is no second
harmonic for a closed-end air column. The next frequency
above the fundamental frequency is the third harmonic (three
times the frequency of the fundamental). In fact, a closed-end
instrument does not possess any even-numbered harmonics.
Only odd-numbered harmonics are produced, where the
frequency of each harmonic is some odd-numbered multiple
of the frequency of the first harmonic.
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Closed-End Air Column:
5th harmonic

the standing wave pattern for
the fifth harmonic of a closedend air column is produced by
adding another node to the
pattern. This would result in a
total of three anti-nodes and
three nodes.

Observe in the pattern that there
are one and one-fourth waves
present in the length of the air
column. That is five times the
number of waves in the first
harmonic. For this reason, the
frequency of the fifth harmonic
is five times the frequency of the
first harmonic.
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Closed-End Air Columns: Pattern
Harmonic
#
1
2
3
4
5
# Waves in
String
# Nodes
# Antinodes
WavelengthLength
Relationship