Physics:Quantum Node - Revision history

WikiHarold: Remove imported red links from Quantum page

2026-05-23T23:46:30Z

Remove imported red links from Quantum page

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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	~~{{Other uses\|Node (disambiguation)}}~~

	A '''node''' is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an '''antinode''', a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes.<ref name="Stanford">{{cite book		A '''node''' is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an '''antinode''', a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes.<ref name="Stanford">{{cite book
	\| last1 = Stanford		\| last1 = Stanford
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	Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection:		Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection:

	*<u>Fixed boundary</u>: Examples of this type of boundary are the attachment point of a guitar string, the closed end of an open pipe like an organ pipe, or a woodwind pipe, the periphery of a drumhead, a transmission line with the end short circuited, or the mirrors at the ends of a laser cavity. In this type, the amplitude of the wave is forced to zero at the boundary, so there is a node at the boundary, and the other nodes occur at multiples of half a wavelength from it:~~{{block indent \| text = 0,  λ/2,  λ,  3λ/2,  2λ, ..., nλ/2}}~~		*<u>Fixed boundary</u>: Examples of this type of boundary are the attachment point of a guitar string, the closed end of an open pipe like an organ pipe, or a woodwind pipe, the periphery of a drumhead, a transmission line with the end short circuited, or the mirrors at the ends of a laser cavity. In this type, the amplitude of the wave is forced to zero at the boundary, so there is a node at the boundary, and the other nodes occur at multiples of half a wavelength from it:
	*<u>Free boundary</u>: Examples of this type are an open-ended organ or woodwind pipe, the ends of the vibrating resonator bars in a xylophone, glockenspiel or tuning fork, the ends of an antenna, or a transmission line with an open end. In this type the derivative (slope) of the wave's amplitude (in sound waves the pressure, in electromagnetic waves, the current) is forced to zero at the boundary. So there is an amplitude maximum (antinode) at the boundary, the first node occurs a quarter wavelength from the end, and the other nodes are at half wavelength intervals from there: ~~{{block indent \| text = λ/4,  3λ/4,  5λ/4,  7λ/4, ..., (2n+1)λ/4}}~~		*<u>Free boundary</u>: Examples of this type are an open-ended organ or woodwind pipe, the ends of the vibrating resonator bars in a xylophone, glockenspiel or tuning fork, the ends of an antenna, or a transmission line with an open end. In this type the derivative (slope) of the wave's amplitude (in sound waves the pressure, in electromagnetic waves, the current) is forced to zero at the boundary. So there is an amplitude maximum (antinode) at the boundary, the first node occurs a quarter wavelength from the end, and the other nodes are at half wavelength intervals from there:

	== Examples ==		== Examples ==
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	==References==		==References==
	<references />		<references />

	~~{{Acoustics}}~~
	~~{{Strings (music)}}~~

	{{DEFAULTSORT:Node (Physics)}}		{{DEFAULTSORT:Node (Physics)}}
	~~[[Category:Concepts in physics]]~~
	~~[[Category:Sound]]~~
	~~[[Category:Waves]]~~

	{{Sourceattribution\|Node (physics)}}		{{Sourceattribution\|Node (physics)}}

WikiHarold: Clean Quantum page image and red links

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Clean Quantum page image and red links

← Older revision		Revision as of 23:33, 23 May 2026
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	{{Other uses\|Node (disambiguation)}}		{{Other uses\|Node (disambiguation)}}

	A '''node''' is a point along a ~~[[Physics:Standing wave\|~~standing wave]] where the wave has minimum ~~[[Company:Amplitude\|~~amplitude]]. For instance, in a vibrating ~~[[Engineering:Guitar\|~~guitar]] string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an '''antinode''', a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes.<ref name="Stanford">{{cite book		A '''node''' is a point along a standing wave where the wave has minimum amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an '''antinode''', a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes.<ref name="Stanford">{{cite book
	\| last1 = Stanford		\| last1 = Stanford
	\| first1 = A. L.		\| first1 = A. L.
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	== Explanation ==		== Explanation ==
	[[File:Node.svg\|thumb\|Pattern of two waves' interference (from up to down). The point represents the node.]]		[[File:Node.svg\|thumb\|Pattern of two waves' interference (from up to down). The point represents the node.]]
	~~[[Physics:~~Standing ~~wave\|Standing wave]]s~~ result when two sinusoidal wave trains of the same ~~[[Physics:Frequency\|~~frequency]] are moving in opposite directions in the same space and interfere with each other.<ref>{{cite book		Standing waves result when two sinusoidal wave trains of the same frequency are moving in opposite directions in the same space and interfere with each other.<ref>{{cite book
	\| last = Feynman		\| last = Feynman
	\| first = Richard P.		\| first = Richard P.
	\|author2=Robert Leighton \|author3=Matthew Sands		\|author2=Robert Leighton \|author3=Matthew Sands
	\| title = ~~[[Physics:The Feynman Lectures on Physics\|~~The Feynman Lectures on Physics, Vol.1]]		\| title = The Feynman Lectures on Physics, Vol.1
	\| publisher = Addison-Wesley		\| publisher = Addison-Wesley
	\| year = 1963		\| year = 1963
Line 50:		Line 50:
	\| doi =		\| doi =
	\| id =		\| id =
	\| isbn = 0-201-02011-4}}</ref> They occur when waves are reflected at a boundary, such as ~~[[Physics:Sound wave\|~~sound ~~wave]]s~~ reflected from a wall or electromagnetic waves reflected from the end of a ~~[[Engineering:Transmission line\|~~transmission line]], and particularly when waves are confined in a ~~[[Physics:Resonator\|~~resonator]] at ~~[[Resonance\|~~resonance]], bouncing back and forth between two boundaries, such as in an organ pipe or guitar string.		\| isbn = 0-201-02011-4}}</ref> They occur when waves are reflected at a boundary, such as sound waves reflected from a wall or electromagnetic waves reflected from the end of a transmission line, and particularly when waves are confined in a resonator at resonance, bouncing back and forth between two boundaries, such as in an organ pipe or guitar string.

	In a standing wave the '''nodes''' are a series of locations at equally spaced intervals where the wave ~~[[Company:Amplitude\|~~amplitude]] (motion) is zero (see animation above). At these points the two waves add with opposite ~~[[Physics:Phase (waves)\|~~phase]] and cancel each other out. They occur at intervals of half a ~~[[Wavelength\|~~wavelength]] (λ/2). Midway between each pair of nodes are locations where the amplitude is maximum. These are called the '''antinodes'''. At these points the two waves add with the same phase and reinforce each other.		In a standing wave the '''nodes''' are a series of locations at equally spaced intervals where the wave amplitude (motion) is zero (see animation above). At these points the two waves add with opposite phase and cancel each other out. They occur at intervals of half a wavelength (λ/2). Midway between each pair of nodes are locations where the amplitude is maximum. These are called the '''antinodes'''. At these points the two waves add with the same phase and reinforce each other.

	In cases where the two opposite wave trains are not the same amplitude, they do not cancel perfectly, so the amplitude of the standing wave at the nodes is not zero but merely a minimum. This occurs when the reflection at the boundary is imperfect. This is indicated by a finite ~~[[Engineering:Standing wave ratio\|~~standing wave ratio]] (SWR), the ratio of the amplitude of the wave at the antinode to the amplitude at the node.		In cases where the two opposite wave trains are not the same amplitude, they do not cancel perfectly, so the amplitude of the standing wave at the nodes is not zero but merely a minimum. This occurs when the reflection at the boundary is imperfect. This is indicated by a finite standing wave ratio (SWR), the ratio of the amplitude of the wave at the antinode to the amplitude at the node.

	In ~~[[Resonance\|~~resonance]] of a two dimensional surface or membrane, such as a drumhead or vibrating metal plate, the nodes become nodal lines, lines on the surface where the surface is motionless, dividing the surface into separate regions vibrating with opposite phase. These can be made visible by sprinkling sand on the surface, and the intricate patterns of lines resulting are called Chladni figures.		In resonance of a two dimensional surface or membrane, such as a drumhead or vibrating metal plate, the nodes become nodal lines, lines on the surface where the surface is motionless, dividing the surface into separate regions vibrating with opposite phase. These can be made visible by sprinkling sand on the surface, and the intricate patterns of lines resulting are called Chladni figures.

	In ~~[[Engineering:Transmission line\|~~transmission ~~line]]s~~ a ~~[[Physics:Voltage\|~~voltage]] node is a ~~[[Physics:Electric~~ current~~\|current]]~~ antinode, and a voltage antinode is a current node.		In transmission lines a voltage node is a current antinode, and a voltage antinode is a current node.

	Nodes are the points of zero displacement, not the points where two constituent waves intersect.		Nodes are the points of zero displacement, not the points where two constituent waves intersect.
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	Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection:		Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection:

	*<u>Fixed boundary</u>: Examples of this type of boundary are the attachment point of a ~~[[Engineering:Guitar\|~~guitar]] string, the closed end of an open pipe like an organ pipe, or a woodwind pipe, the periphery of a drumhead, a ~~[[Engineering:Transmission line\|~~transmission line]] with the end ~~[[Physics:Short circuit\|~~short ~~circuit]]ed~~, or the mirrors at the ends of a laser cavity. In this type, the amplitude of the wave is forced to zero at the boundary, so there is a node at the boundary, and the other nodes occur at multiples of half a wavelength from it:{{block indent \| text = 0,  λ/2,  λ,  3λ/2,  2λ, ..., nλ/2}}		*<u>Fixed boundary</u>: Examples of this type of boundary are the attachment point of a guitar string, the closed end of an open pipe like an organ pipe, or a woodwind pipe, the periphery of a drumhead, a transmission line with the end short circuited, or the mirrors at the ends of a laser cavity. In this type, the amplitude of the wave is forced to zero at the boundary, so there is a node at the boundary, and the other nodes occur at multiples of half a wavelength from it:{{block indent \| text = 0,  λ/2,  λ,  3λ/2,  2λ, ..., nλ/2}}
	*<u>Free boundary</u>: Examples of this type are an open-ended organ or woodwind pipe, the ends of the vibrating resonator bars in a xylophone, glockenspiel or ~~[[Physics:Tuning fork\|~~tuning fork]], the ends of an ~~[[Engineering:Antenna (radio)\|~~antenna]], or a transmission line with an open end. In this type the derivative (slope) of the wave's amplitude (in ~~[[Physics:Sound wave\|~~sound ~~wave]]s~~ the pressure, in electromagnetic waves, the ~~[[Physics:Electric current\|~~current]]) is forced to zero at the boundary. So there is an amplitude maximum (antinode) at the boundary, the first node occurs a quarter wavelength from the end, and the other nodes are at half wavelength intervals from there: {{block indent \| text = λ/4,  3λ/4,  5λ/4,  7λ/4, ..., (2n+1)λ/4}}		*<u>Free boundary</u>: Examples of this type are an open-ended organ or woodwind pipe, the ends of the vibrating resonator bars in a xylophone, glockenspiel or tuning fork, the ends of an antenna, or a transmission line with an open end. In this type the derivative (slope) of the wave's amplitude (in sound waves the pressure, in electromagnetic waves, the current) is forced to zero at the boundary. So there is an amplitude maximum (antinode) at the boundary, the first node occurs a quarter wavelength from the end, and the other nodes are at half wavelength intervals from there: {{block indent \| text = λ/4,  3λ/4,  5λ/4,  7λ/4, ..., (2n+1)λ/4}}

	== Examples ==		== Examples ==
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	The number of nodes in a specified length is directly proportional to the frequency of the wave.		The number of nodes in a specified length is directly proportional to the frequency of the wave.

	Occasionally on a guitar, violin, or other stringed instrument, nodes are used to create ~~[[Physics:Harmonic\|harmonic]]s~~. When the finger is placed on top of the string at a certain point, but does not push the string all the way down to the fretboard, a third node is created (in addition to the bridge and nut) and a harmonic is sounded. During normal play when the frets are used, the harmonics are always present, although they are quieter. With the artificial node method, the ~~[[Physics:Overtone\|~~overtone]] is louder and the ~~[[Physics:Fundamental frequency\|~~fundamental]] tone is quieter. If the finger is placed at the midpoint of the string, the first overtone is heard, which is an octave above the fundamental note which would be played, had the harmonic not been sounded. When two additional nodes divide the string into thirds, this creates an octave and a perfect fifth (twelfth). When three additional nodes divide the string into quarters, this creates a double octave. When four additional nodes divide the string into fifths, this creates a double-octave and a major third (17th). The octave, major third and perfect fifth are the three notes present in a major chord.		Occasionally on a guitar, violin, or other stringed instrument, nodes are used to create harmonics. When the finger is placed on top of the string at a certain point, but does not push the string all the way down to the fretboard, a third node is created (in addition to the bridge and nut) and a harmonic is sounded. During normal play when the frets are used, the harmonics are always present, although they are quieter. With the artificial node method, the overtone is louder and the fundamental tone is quieter. If the finger is placed at the midpoint of the string, the first overtone is heard, which is an octave above the fundamental note which would be played, had the harmonic not been sounded. When two additional nodes divide the string into thirds, this creates an octave and a perfect fifth (twelfth). When three additional nodes divide the string into quarters, this creates a double octave. When four additional nodes divide the string into fifths, this creates a double-octave and a major third (17th). The octave, major third and perfect fifth are the three notes present in a major chord.

	The characteristic sound that allows the listener to identify a particular instrument is largely due to the relative magnitude of the harmonics created by the instrument.		The characteristic sound that allows the listener to identify a particular instrument is largely due to the relative magnitude of the harmonics created by the instrument.
Line 86:		Line 86:
	In two dimensional standing waves, nodes are curves (often straight lines or circles when displayed on simple geometries.) For example, sand collects along the nodes of a vibrating [[Biography:Ernst Chladni\|Chladni plate]] to indicate regions where the plate is not moving.<ref>Comer, J. R., et al. [https://aapt.scitation.org/doi/pdf/10.1119/1.1758222 "Chladni plates revisited."] American journal of physics 72.10 (2004): 1345-1346.</ref>		In two dimensional standing waves, nodes are curves (often straight lines or circles when displayed on simple geometries.) For example, sand collects along the nodes of a vibrating [[Biography:Ernst Chladni\|Chladni plate]] to indicate regions where the plate is not moving.<ref>Comer, J. R., et al. [https://aapt.scitation.org/doi/pdf/10.1119/1.1758222 "Chladni plates revisited."] American journal of physics 72.10 (2004): 1345-1346.</ref>

	In chemistry, [[Physics:Quantum mechanics\|quantum mechanical]] waves, or "~~[[Physics:Atomic orbital\|~~orbitals]]", are used to describe the wave-like properties of electrons. Many of these quantum waves have nodes and antinodes as well. The number and position of these nodes and antinodes give rise to many of the properties of an atom or ~~[[Chemistry:Covalent bond\|~~covalent bond]]. Atomic orbitals are classified according to the number of radial and angular nodes. A radial node for the hydrogen atom is a sphere that occurs where the wavefunction for an atomic orbital is equal to zero, while the angular node is		In chemistry, [[Physics:Quantum mechanics\|quantum mechanical]] waves, or "orbitals", are used to describe the wave-like properties of electrons. Many of these quantum waves have nodes and antinodes as well. The number and position of these nodes and antinodes give rise to many of the properties of an atom or covalent bond. Atomic orbitals are classified according to the number of radial and angular nodes. A radial node for the hydrogen atom is a sphere that occurs where the wavefunction for an atomic orbital is equal to zero, while the angular node is
	a flat plane.<ref>Supplemental modules (physical and Theoretical Chemistry). Chemistry LibreTexts. (2020, December 13). Retrieved September 13, 2022, from https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)</ref>		a flat plane.<ref>Supplemental modules (physical and Theoretical Chemistry). Chemistry LibreTexts. (2020, December 13). Retrieved September 13, 2022, from https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)</ref>

	~~[[Chemistry:~~Molecular ~~orbital\|Molecular orbital]]s~~ are classified according to bonding character. Molecular orbitals with an antinode between nuclei are very stable, and are known as "bonding orbitals" which strengthen the bond. In contrast, molecular orbitals with a node between nuclei will not be stable due to electrostatic repulsion and are known as "anti-bonding orbitals" which weaken the bond. Another such [[Physics:Quantum mechanics\|quantum mechanical]] concept is the ~~[[Physics:Particle in a box\|~~particle in a box]] where the number of nodes of the wavefunction can help determine the quantum energy state—zero nodes corresponds to the ground state, one node corresponds to the 1st excited state, etc. In general,<ref>[[Biography:Albert Messiah\|Albert Messiah]], 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. [https://archive.org/details/QuantumMechanicsVolumeI online] Ch 3 §12</ref> ''If one arranges the eigenstates in the order of increasing energies, <math>\epsilon_1,\epsilon_2, \epsilon_3,...</math>, the eigenfunctions likewise fall in the order of increasing number of nodes; the ''n''th eigenfunction has ''n−1'' nodes, between each of which the following eigenfunctions have at least one node''.		Molecular orbitals are classified according to bonding character. Molecular orbitals with an antinode between nuclei are very stable, and are known as "bonding orbitals" which strengthen the bond. In contrast, molecular orbitals with a node between nuclei will not be stable due to electrostatic repulsion and are known as "anti-bonding orbitals" which weaken the bond. Another such [[Physics:Quantum mechanics\|quantum mechanical]] concept is the particle in a box where the number of nodes of the wavefunction can help determine the quantum energy state—zero nodes corresponds to the ground state, one node corresponds to the 1st excited state, etc. In general,<ref>[[Biography:Albert Messiah\|Albert Messiah]], 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. [https://archive.org/details/QuantumMechanicsVolumeI online] Ch 3 §12</ref> ''If one arranges the eigenstates in the order of increasing energies, <math>\epsilon_1,\epsilon_2, \epsilon_3,...</math>, the eigenfunctions likewise fall in the order of increasing number of nodes; the ''n''th eigenfunction has ''n−1'' nodes, between each of which the following eigenfunctions have at least one node''.

	==References==		==References==

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 {{Short description|Point with minimum wave amplitude}}
 {{Other uses|Node (disambiguation)}}
 A '''node''' is a point along a [[Physics:Standing wave|standing wave]] where the wave has minimum [[Company:Amplitude|amplitude]].  For instance, in a vibrating [[Engineering:Guitar|guitar]] string, the ends of the string are nodes.  By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an '''antinode''', a point where the amplitude of the standing wave is at maximum.  These occur midway between the nodes.<ref name="Stanford">{{cite book
@@ Line 19: / Line 28: @@
   | isbn   = 978-1483220291
   }}</ref>
 == Explanation ==

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WikiHarold moved page Physics:Node to Physics:Quantum Node without leaving a redirect

← Older revision	Revision as of 16:33, 4 May 2026
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imported>WikiHarold: WikiHarold moved page Physics:Node to Physics:Quantum Node without leaving a redirect

2026-05-04T16:33:34Z

WikiHarold moved page Physics:Node to Physics:Quantum Node without leaving a redirect

New page

{{Short description|Point with minimum wave amplitude}}
{{Other uses|Node (disambiguation)}}
[[File:Standing wave.gif|thumb|300px|right|A standing wave. The red dots are the wave nodes]]

A '''node''' is a point along a [[Physics:Standing wave|standing wave]] where the wave has minimum [[Company:Amplitude|amplitude]]. For instance, in a vibrating [[Engineering:Guitar|guitar]] string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the note played. The opposite of a node is an '''antinode''', a point where the amplitude of the standing wave is at maximum. These occur midway between the nodes.<ref name="Stanford">{{cite book
| last1 = Stanford
| first1 = A. L.
| last2 = Tanner
| first2 = J. M.
| title = Physics for Students of Science and Engineering
| publisher = Academic Press
| date = 2014
| location =
| pages = 561
| language =
| url = https://books.google.com/books?id=XONbAwAAQBAJ&dq=antinode+%22midway+between+nodes%22&pg=PA561
| doi =
| id =
| isbn = 978-1483220291
}}</ref>

== Explanation ==
[[File:Node.svg|thumb|Pattern of two waves' interference (from up to down). The point represents the node.]]
[[Physics:Standing wave|Standing wave]]s result when two sinusoidal wave trains of the same [[Physics:Frequency|frequency]] are moving in opposite directions in the same space and interfere with each other.<ref>{{cite book
| last = Feynman
| first = Richard P.
|author2=Robert Leighton |author3=Matthew Sands
| title = [[Physics:The Feynman Lectures on Physics|The Feynman Lectures on Physics, Vol.1]]
| publisher = Addison-Wesley
| year = 1963
| location = USA
| pages = ch.49
| url =
| doi =
| id =
| isbn = 0-201-02011-4}}</ref> They occur when waves are reflected at a boundary, such as [[Physics:Sound wave|sound wave]]s reflected from a wall or electromagnetic waves reflected from the end of a [[Engineering:Transmission line|transmission line]], and particularly when waves are confined in a [[Physics:Resonator|resonator]] at [[Resonance|resonance]], bouncing back and forth between two boundaries, such as in an organ pipe or guitar string.

In a standing wave the '''nodes''' are a series of locations at equally spaced intervals where the wave [[Company:Amplitude|amplitude]] (motion) is zero (see animation above). At these points the two waves add with opposite [[Physics:Phase (waves)|phase]] and cancel each other out. They occur at intervals of half a [[Wavelength|wavelength]] (λ/2). Midway between each pair of nodes are locations where the amplitude is maximum. These are called the '''antinodes'''. At these points the two waves add with the same phase and reinforce each other.

In cases where the two opposite wave trains are not the same amplitude, they do not cancel perfectly, so the amplitude of the standing wave at the nodes is not zero but merely a minimum. This occurs when the reflection at the boundary is imperfect. This is indicated by a finite [[Engineering:Standing wave ratio|standing wave ratio]] (SWR), the ratio of the amplitude of the wave at the antinode to the amplitude at the node.

In [[Resonance|resonance]] of a two dimensional surface or membrane, such as a drumhead or vibrating metal plate, the nodes become nodal lines, lines on the surface where the surface is motionless, dividing the surface into separate regions vibrating with opposite phase. These can be made visible by sprinkling sand on the surface, and the intricate patterns of lines resulting are called Chladni figures.

In [[Engineering:Transmission line|transmission line]]s a [[Physics:Voltage|voltage]] node is a [[Physics:Electric current|current]] antinode, and a voltage antinode is a current node.

Nodes are the points of zero displacement, not the points where two constituent waves intersect.

== Boundary conditions ==
Where the nodes occur in relation to the boundary reflecting the waves depends on the end conditions or boundary conditions. Although there are many types of end conditions, the ends of resonators are usually one of two types that cause total reflection:

*<u>Fixed boundary</u>: Examples of this type of boundary are the attachment point of a [[Engineering:Guitar|guitar]] string, the closed end of an open pipe like an organ pipe, or a woodwind pipe, the periphery of a drumhead, a [[Engineering:Transmission line|transmission line]] with the end [[Physics:Short circuit|short circuit]]ed, or the mirrors at the ends of a laser cavity. In this type, the amplitude of the wave is forced to zero at the boundary, so there is a node at the boundary, and the other nodes occur at multiples of half a wavelength from it:{{block indent | text = 0,  λ/2,  λ,  3λ/2,  2λ, ..., nλ/2}}
*<u>Free boundary</u>: Examples of this type are an open-ended organ or woodwind pipe, the ends of the vibrating resonator bars in a xylophone, glockenspiel or [[Physics:Tuning fork|tuning fork]], the ends of an [[Engineering:Antenna (radio)|antenna]], or a transmission line with an open end. In this type the derivative (slope) of the wave's amplitude (in [[Physics:Sound wave|sound wave]]s the pressure, in electromagnetic waves, the [[Physics:Electric current|current]]) is forced to zero at the boundary. So there is an amplitude maximum (antinode) at the boundary, the first node occurs a quarter wavelength from the end, and the other nodes are at half wavelength intervals from there: {{block indent | text = λ/4,  3λ/4,  5λ/4,  7λ/4, ..., (2n+1)λ/4}}

== Examples ==

=== Sound ===
A sound wave consists of alternating cycles of compression and expansion of the wave medium. During compression, the molecules of the medium are forced together, resulting in the increased pressure and density. During expansion the molecules are forced apart, resulting in the decreased pressure and density.

The number of nodes in a specified length is directly proportional to the frequency of the wave.

Occasionally on a guitar, violin, or other stringed instrument, nodes are used to create [[Physics:Harmonic|harmonic]]s. When the finger is placed on top of the string at a certain point, but does not push the string all the way down to the fretboard, a third node is created (in addition to the bridge and nut) and a harmonic is sounded. During normal play when the frets are used, the harmonics are always present, although they are quieter. With the artificial node method, the [[Physics:Overtone|overtone]] is louder and the [[Physics:Fundamental frequency|fundamental]] tone is quieter. If the finger is placed at the midpoint of the string, the first overtone is heard, which is an octave above the fundamental note which would be played, had the harmonic not been sounded. When two additional nodes divide the string into thirds, this creates an octave and a perfect fifth (twelfth). When three additional nodes divide the string into quarters, this creates a double octave. When four additional nodes divide the string into fifths, this creates a double-octave and a major third (17th). The octave, major third and perfect fifth are the three notes present in a major chord.

The characteristic sound that allows the listener to identify a particular instrument is largely due to the relative magnitude of the harmonics created by the instrument.

[[File:Bowing chladni plate.png|thumb|left|Sand highlights nodes on a Chladni plate.]]

===Waves in two or three dimensions===
[[File:HAtomOrbitals.png|thumb|Radial and angular nodes on hydrogen wavefunctions.]]

In two dimensional standing waves, nodes are curves (often straight lines or circles when displayed on simple geometries.) For example, sand collects along the nodes of a vibrating [[Biography:Ernst Chladni|Chladni plate]] to indicate regions where the plate is not moving.<ref>Comer, J. R., et al. [https://aapt.scitation.org/doi/pdf/10.1119/1.1758222 "Chladni plates revisited."] American journal of physics 72.10 (2004): 1345-1346.</ref>

In chemistry, [[Physics:Quantum mechanics|quantum mechanical]] waves, or "[[Physics:Atomic orbital|orbitals]]", are used to describe the wave-like properties of electrons. Many of these quantum waves have nodes and antinodes as well. The number and position of these nodes and antinodes give rise to many of the properties of an atom or [[Chemistry:Covalent bond|covalent bond]]. Atomic orbitals are classified according to the number of radial and angular nodes. A radial node for the hydrogen atom is a sphere that occurs where the wavefunction for an atomic orbital is equal to zero, while the angular node is
a flat plane.<ref>Supplemental modules (physical and Theoretical Chemistry). Chemistry LibreTexts. (2020, December 13). Retrieved September 13, 2022, from https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)</ref>

[[Chemistry:Molecular orbital|Molecular orbital]]s are classified according to bonding character. Molecular orbitals with an antinode between nuclei are very stable, and are known as "bonding orbitals" which strengthen the bond. In contrast, molecular orbitals with a node between nuclei will not be stable due to electrostatic repulsion and are known as "anti-bonding orbitals" which weaken the bond. Another such [[Physics:Quantum mechanics|quantum mechanical]] concept is the [[Physics:Particle in a box|particle in a box]] where the number of nodes of the wavefunction can help determine the quantum energy state—zero nodes corresponds to the ground state, one node corresponds to the 1st excited state, etc. In general,<ref>[[Biography:Albert Messiah|Albert Messiah]], 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. [https://archive.org/details/QuantumMechanicsVolumeI online] Ch 3 §12</ref> ''If one arranges the eigenstates in the order of increasing energies, <math>\epsilon_1,\epsilon_2, \epsilon_3,...</math>, the eigenfunctions likewise fall in the order of increasing number of nodes; the ''n''th eigenfunction has ''n−1'' nodes, between each of which the following eigenfunctions have at least one node''.

==References==
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