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Solutions of the Laplace equation in spherical polar coordinates
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions
Solid_harmonics
Special mathematical functions defined on the surface of a sphere
fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal
Spherical_harmonics
Mathematical series
R_{L}^{M}} are irregular and regular solid harmonics, respectively. The translation of the regular solid harmonic gives a finite expansion, R L M ( r A
Multipole_expansion
Second-order partial differential equation
} where the fℓm are constants and the factors rℓ Yℓm are known as solid harmonics. Such an expansion is valid in the ball r < R = 1 lim sup ℓ → ∞ | f
Laplace's_equation
Laser science process
the high-order harmonics of the generation beam (above the fifth harmonic). Due to the coherent nature of the process, high-harmonics generation is a
High_harmonic_generation
Arrangement that creates a quadrupole field of some sort
order, e.g., 2 ℓ − 1 {\displaystyle 2^{\ell -1}} . Multipole expansion Solid harmonics Axial multipole moments Cylindrical multipole moments Spherical multipole
Quadrupole
Differential operator in mathematics
log r . {\displaystyle R(r)=A+B\log r.} These give the classical solid harmonics. In particular, if f ( x ) = F ( r ) {\displaystyle f(x)=F(r)} is radial
Laplace_operator
Atomic model
often partially replaced by cubic harmonics for a number of reasons. These harmonics are usually named tesseral harmonics in the field of condensed matter
Cubic_harmonic
Measure of positive and negative charges
moment Dynamic toroidal dipole Multipole expansion Multipole moments Solid harmonics Axial multipole moments Cylindrical multipole moments Spherical multipole
Electric_dipole_moment
output will only contain even harmonics. This is because the currents which would generate the fundamental and the odd harmonics in this circuit are canceled
Radio_transmitter_design
Nonlinear optical process
happen: the signal at higher harmonics will be very low, and requires very intense lasers to be generated. To generate high harmonics (like n = 30 {\displaystyle
Harmonic_generation
Electromagnetic phenomenon
Monopole Solid harmonics Axial multipole moments Cylindrical multipole moments Spherical multipole moments Laplace expansion Molecular solid Magnetic
Dipole
Polyhedra in which all vertices are the same
Archimedean solids appeared in Albrecht Dürer's Underweysung der Messung, copied from the Pacioli's work. By around 1620, Johannes Kepler in his Harmonices Mundi
Archimedean_solid
The quasi-harmonic approximation is a phonon-based model of solid-state physics used to describe volume-dependent thermal effects, such as the thermal
Quasi-harmonic_approximation
Nonlinear optical process
inversion symmetry is broken, allowing for SHG and other even order harmonics to occur. For a colloidal system of microparticles at relatively low concentrations
Second-harmonic_generation
}^{m}} is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics, 1 ‖ r − r ′ ‖ = ∑ ℓ = 0 ∞ ∑
Laplace_expansion_(potential)
Coefficients in a series expansion of a potential
multipole moments are defined as the complex conjugate of irregular solid harmonics I ℓ m = d e f q ( r ′ ) ℓ + 1 4 π 2 ℓ + 1 Y ℓ m ∗ ( θ ′ , ϕ ′ )
Spherical_multipole_moments
Device that changes direct current (DC) to alternating current (AC)
different harmonics. For the first one, through Fourier Analysis, the magnitude of harmonics would be 4/(pi*k) (k is the order of harmonics). So the majority
Power_inverter
Electrical string musical instrument
pedal Electric pipa Electromagnetic induction Electronic tuner Guitar harmonics Guitar synthesizer Guitar amplifier Keytar List of guitars List of guitarists
Electric_guitar
American Christian rock record label
Fold Zandura Ghost Ship The Glorious Unseen Hangnail Hawk Nelson iLL Harmonics KJ-52 Dustin Kensrue Joy Electric The O.C. Supertones Phillip LaRue Jadon
Tooth_&_Nail_Records
Any of 4 regular star polyhedra
dodecahedron, for the first time treating it as a surface rather than a solid. He noticed that by extending the edges or faces of the convex dodecahedron
Kepler–Poinsot_polyhedron
Characteristic quality of sounds from vacuum tube amplifiers
MOSFETs) produce a monotonically decaying harmonic distortion spectrum. Even-order harmonics and odd-order harmonics are both natural number multiples of the
Tube_sound
Archimedean solid with 26 faces
Archimedean solid, and its dual is a Catalan solid, the deltoidal icositetrahedron. The elongated square gyrobicupola, the 37th Johnson solid, is a polyhedron
Rhombicuboctahedron
Bowed string instrument
the pitch of the harmonic, or by diamond-shaped note heads. There are two types of harmonics: natural harmonics and artificial harmonics (also known as
Violin
required in gas harmonics. However to generate high harmonics from plasma plumes, we require another laser pulse focused onto the surface of a solid target to
Resonant high harmonic generation from laser ablated plasma plumes
Resonant_high_harmonic_generation_from_laser_ablated_plasma_plumes
Functions such that f(–x) equals f(x) or –f(x)
of harmonics produced depend on the response function f: When the response function is even, the resulting signal will consist of only even harmonics of
Even_and_odd_functions
Audio signal processing technique
frequencies, the harmonics are derived from a purer frequency band resulting in clearer highs. Exciters are also used to synthesize harmonics of low frequency
Exciter_(effect)
Model of a crystalline solid
The Einstein solid is a model of a crystalline solid that contains a large number of independent three-dimensional quantum harmonic oscillators of the
Einstein_solid
Type of laser
A diode-pumped solid-state laser (DPSSL) is a solid-state laser made by pumping a solid gain medium, for example, a ruby or a neodymium-doped YAG crystal
Diode-pumped solid-state laser
Diode-pumped_solid-state_laser
Method in physics
non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low-temperature dependence of the heat capacity of solids, which is proportional
Debye_model
Type of electronic audio manipulation
power in higher harmonics. As clipping increases, a tone input progressively begins to resemble a square wave which has odd number harmonics. This is generally
Distortion_(music)
Archimedean solid with 62 faces
the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon
Rhombicosidodecahedron
British audio equipment manufacturer
Solid State Logic Ltd. (SSL) is a British company based in Begbroke, Oxfordshire, England that designs and markets audio mixing consoles, signal processors
Solid_State_Logic
Pattern of oscillating motion in a system
also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of hν. The spectrum of waveforms can be described
Normal_mode
Solid conductive electrolyte
polymer electrolytic capacitor, is an electrolytic capacitor (e-cap) with a solid conductive polymer electrolyte. There are four different types: Polymer
Polymer_capacitor
Hungarian mathematician (1895–1985)
1090/s0002-9947-1936-1501884-1. MR 1501884. Szegő, G. (1940). "On the gradient of solid harmonic polynomials". Trans. Amer. Math. Soc. 47: 51–65. doi:10.1090/s0002-9947-1940-0000847-6
Gábor_Szegő
expansion Spherical multipole moments Cylindrical multipole moments Solid harmonics Laplace expansion Eyges, Leonard (2012-06-11). The Classical Electromagnetic
Axial_multipole_moments
physics Solid State Communications Solid angle Solid harmonics Solid hydrogen Solid mechanics Solid solution Solid state dye lasers Solidus (chemistry)
Index_of_physics_articles_(S)
Form of waveform distortion
where third harmonics are contextual outliers in a Fourier Transform. In the case of an expected sine wave, the presence of odd harmonics will often suggest
Clipping_(audio)
Ratio of active power to apparent power
2013-11-26 "Harmonic Effects" (PDF), Harmonics and IEEE 519, CA: EnergyLogix Solutions Sankaran, C. (1999), "Transformers", Effects of Harmonics on Power
Power_factor
Classification of power amplifier
square waves consist of odd harmonics only. In a class-D amplifier, the output filter blocks all harmonics; i.e., the harmonics see an open load. So even
Power_amplifier_classes
Mathematical objects more general than vectors
{n}}+1)\tau +i(\mathbf {r} {\hat {\mathbf {A} }})} . Tensor Spherical harmonics Operator (physics) Laplace-Runge-Lenz vector Angular momentum operator
Harmonic_tensors
Quasiparticle of mechanical vibrations
arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized
Phonon
Crystal used as a lasing medium for solid-state lasers
mainly via their second and third harmonics, are widely used to excite dye lasers either in the liquid or solid-state. They are also used as pump sources
Nd:YAG_laser
American fantasy television series
website's critics consensus reads, "Avatar: The Last Airbender serves as a solid live-action entry point into the beloved franchise, although it only sporadically
Avatar: The Last Airbender (2024 TV series)
Avatar:_The_Last_Airbender_(2024_TV_series)
Type of adjustable-speed drive
are partially cancelled by three-phase diode bridge harmonics because their 5th and 7th harmonics are in counterphase. However, when the proportion of
Variable-frequency_drive
Details of violin playing technique
pitch of the harmonic. There are two types of harmonics: natural and artificial (also known as "false harmonics"). Artificial harmonics are more advanced
Violin_technique
Oscillator that produces a nonsinusoidal repetitive waveform
waveform. This contrasts with the other type of electronic oscillator, the harmonic or linear oscillator, which uses an amplifier with feedback to excite resonant
Relaxation_oscillator
Quantum number denoting orbital angular momentum
liquids and solids. The quantum number ℓ plays an important role here via the connection to the angular dependence of the spherical harmonics for the different
Azimuthal_quantum_number
Greco-Roman astronomer and geographer (c. 100–170)
conjecture. Ptolemy's Harmonics (Greek: Ἁρμονικόν) is a work in three books on music theory and the mathematics behind musical scales. Harmonics begins with a
Ptolemy
Ancient philosophical concept
cosmological model, and on the grounds that "excessive noises ... shatter the solid bodies even of inanimate things", and therefore any sounds made by the planets
Musica_universalis
American bassist (born 1964)
exclusively. Trujillo is known for playing "massive chords" and "chord-based harmonics" on the bass. Trujillo uses the slap bass technique, seen mostly in his
Robert_Trujillo
American rock band
harmonies and distinctive lead guitar techniques, such as delays and natural harmonics. Throughout several lineup changes, Burchell has been the band's only
Saosin
Musical term
percussion Anharmonicity Dissonance Pseudo-octave Subharmonic How harmonic are harmonics? by Joe Wolfe, accessed 29 June 2008 The Indian Musical Drums by
Inharmonicity
1619 book by Johannes Kepler
describe the formation of Platonic solids in terms of basic triangles. The book features illustrations of solids and tiling patterns, some of which are
Harmonice_Mundi
Form of distortion that limits a signal in processing
Hard clipping results in many high-frequency harmonics; soft clipping results in fewer higher-order harmonics and intermodulation distortion components.
Clipping_(signal_processing)
Semi-analytic method of computational electromagnetism
Fourier-space method so devices and fields are represented as a sum of spatial harmonics. The method is based on Floquet's theorem that the solutions of periodic
Rigorous coupled-wave analysis
Rigorous_coupled-wave_analysis
the measurements to a set of spherical harmonics. This was first done by Carl Friedrich Gauss. Spherical harmonics are functions that oscillate over the
Earth's_magnetic_field
Baseband analog video signal format
a modulated color signal that consists mainly of harmonic frequencies that fall between the harmonics in the baseband luma signal, rather than both being
Composite_video
Flat-sided three-dimensional shape
vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly
Polyhedron
Mathematics of Ancient Greece and the Mediterranean, 5th BC to 6th AD
Works in mathematical harmonics in the Hellenistic period include the Sectio Canonis, attributed to Euclid, and Ptolemy's Harmonics. The study of optics
Ancient_Greek_mathematics
Type of substation
complex harmonic filters are necessary because there are odd numbered harmonics of the orders 6n + 1 and 6n - 1 produced on the AC side and even harmonics of
HVDC_converter_station
Sustained ground vibration associated with underground movement of magma or volcanic gas
also generate distinct iceberg harmonic tremor signals that propagate to large distances as ocean acoustic and solid Earth seismic wavefields. The source
Harmonic_tremor
Deviation of a physical system from being a harmonic oscillator
molecule or a solid vibrate about their equilibrium positions. When these vibrations have small amplitudes they can be described by harmonic oscillators
Anharmonicity
Technology of power electronics
complex harmonics with the higher-order harmonics being filtered by the machine inductance. Causing the machine current to have fewer harmonics, while
Power_electronics
Radiation emitted by charged particles deflected by magnetic field
fundamental frequency as the particle's orbit, and harmonics at higher integral factors. Harmonics are the result of imperfections in the actual emission
Cyclotron_radiation
Mathematical function
the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes, one obtains 2 W ( q ) = ℏ 2 q 2 6 M
Debye_function
1987 album by David Murray and Randy Weston
slightly known) and Butch Morris' 'Clever Beggar.' Weston provides a solid harmonic and rhythmic foundation for Murray's thick-toned but sometimes screeching
The_Healers_(album)
Function describing an electron in an atom
orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. Letting ψ n , ℓ , m {\displaystyle \psi _{n,\ell
Atomic_orbital
Vibrational energy transfer in Earth or other planetary body
nSlm, where l is the angular order number (or spherical harmonic degree, see Spherical harmonics for more details). The number m is the azimuthal order
Seismic_wave
Electronic amplifier for musical instruments
signal clipped, losing high and low frequencies but gaining compression, harmonics, and a "musical type of distortion". While an unintended technical shortcoming
Guitar_amplifier
French mathematician and physicist (1768–1830)
of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations
Joseph_Fourier
Group of rotations in 3 dimensions
\ell \right\},} where Y m ℓ {\displaystyle Y_{m}^{\ell }} are spherical harmonics. Its elements are square integrable complex-valued functions on the sphere
3D_rotation_group
Audible vibration that travels via pressure waves in matter
means there are non-linear propagation effects, such as the production of harmonics and mixed tones not present in the original sound (see parametric array)
Sound
Method for reconstructing a harmonic function in a domain
shell theorem is an example. Consider a uniform mass distribution within a solid ball B {\displaystyle B} in R 3 {\displaystyle \mathbb {R} ^{3}} . The balayage
Balayage
Device designed to reduce vibrations in structures
A tuned mass damper (TMD), also known as a harmonic absorber or seismic damper, is a device mounted in structures to reduce mechanical vibrations, consisting
Tuned_mass_damper
Stochastic differential equation
velocity distribution (blue) and position distributions (orange) in a harmonic potential ( U = 1 2 k x 2 {\displaystyle U={\tfrac {1}{2}}kx^{2}} ) is
Langevin_equation
Ocean shape without winds and tides
mantle. Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM2020
Geoid
Structures of the Earth's mantle
at the D″ layer beneath Africa and the Pacific. The global spherical harmonics of the D″ layer are uniform throughout most of the mantle but anomalies
Large low-shear-velocity provinces
Large_low-shear-velocity_provinces
Electric power transmission system
damping at lower-order, noncharacteristic harmonics such as 3rd or 5th harmonics. The task of designing AC harmonic filters for HVDC converter stations is
High-voltage_direct_current
Number between two given numbers
Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode Dispersion Average absolute deviation
Heronian_mean
Musical instrument
another in external appearance, and include instruments made from rectangular solid blocks of wood. Some may be small enough to be played on the lap; others
Lap_steel_guitar
Type of electronic circuit
push–pull circuits must cancel even order harmonics, like 2f, 4f, 6f and therefore promote odd order harmonics, like f, 3f, 5f when driven into the nonlinear
Push–pull_output
Bowed string instrument
harmonic series on a particular string. Artificial harmonics (also called false harmonics or stopped harmonics), in which the player depresses the string fully
Cello
Wave that remains in a constant position
allowing harmonics to be identified. Nodes occur at fixed ends and anti-nodes at open ends. If fixed at only one end, only odd-numbered harmonics are available
Standing_wave
Solid with 12 equal pentagonal faces
attribution with nature by Plato, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids, one of them is a regular dodecahedron. In his Mysterium
Regular_dodecahedron
Amplifies the signal from a turntable
Tube phono stage - produces even order harmonic distortion Solid state phono stage - produces odd order harmonic distortion Tube phono stages were the
Phono_stage
Empirical thermodynamic law
of vibrations in a crystalline solid lattice can be modeled as an Einstein solid, i.e. by considering N quantum harmonic oscillator potentials along each
Dulong–Petit_law
Chemical compound
crystal for ultraviolet applications and generates the fourth and fifth harmonics of the Nd:YAG fundamental laser wavelength (1064 nm). Mori, Y.; Kuroda
Caesium_lithium_borate
1972 song by Yes
followed by musical and lyrical structure which sounds similar to "The Solid Time of Change", except this time with exclusively major chords. Rick Wakeman's
Close_to_the_Edge_(song)
Branch of engineering
generally depend upon the work they do. For example, quantum mechanics and solid state physics might be relevant to an engineer working on VLSI (the design
Electrical_engineering
Physics theorem
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete
Virial_theorem
Device that emits light via optical amplification
nonlinearity of the oscillating cloud would produce both spatial and temporal harmonics, so nuclear transitions of higher multipolarity could also be driven at
Laser
Electrical device that converts AC to DC
Non-linear loads like rectifiers produce current harmonics of the source frequency on the AC side and voltage harmonics of the source frequency on the DC side,
Rectifier
Polyhedron resembling a soccerball
complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book, Harmonices Mundi. Chamfered dodecahedron
Truncated_icosahedron
Bowed string instrument
bows the note. Bowed harmonics are used in contemporary music for their "glassy" sound. Both natural harmonics and artificial harmonics, where the thumb stops
Double_bass
Branch of mechanics concerned with balance of forces in nonmoving systems
major figures in the development of hydrostatics. Cremona diagram Dynamics Solid mechanics Physics portal Lindberg, David C. (1992). The Beginnings of Western
Statics
Sequence of notes that results from inverting the intervals of the overtone series
software and electronics. Subharmonics can be contrasted with harmonics. While harmonics can "occur in any linear system", there are "only fairly restricted
Undertone_series
Irreducible representation of the rotation group SO
elements with second index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon
Wigner_D-matrix
Device or system that exhibits resonance
frequencies of resonators, called normal modes, are equally spaced multiples (harmonics) of a lowest frequency called the fundamental frequency. The above analysis
Resonator
SOLID HARMONICS
SOLID HARMONICS
Boy/Male
Arabic, Muslim
Solid
Boy/Male
Muslim/Islamic
Firm Solid
Boy/Male
Indian
Firm, Solid
Boy/Male
Indian, Sanskrit
Strong; Solid
Boy/Male
Muslim/Islamic
Solid
Boy/Male
Hindu, Indian
Solid Redemption
Boy/Male
Muslim
Firm, Solid
Girl/Female
Indian
Strong, Solid
Boy/Male
Muslim
Solid
Boy/Male
Muslim
Solid structure
Girl/Female
Indian
Firm, Solid, Determined
Girl/Female
Muslim
Strong, Solid
Boy/Male
Arabic, Muslim, Parsi
Firm; Solid
Surname or Lastname
Spanish and Asturian-Leonese (SolÃs)
Spanish and Asturian-Leonese (SolÃs) : habitational name from SolÃs in Asturies or a similarly named place elsewhere.English : from a medieval personal name bestowed on a child born after the death of a sibling, from Middle English solace ‘comfort’, ‘consolation’. The word also came to have the sense ‘delight’, ‘amusement’, and in some cases the surname may have arisen from a nickname for a playful or entertaining person.
Boy/Male
Indian
Solid
Girl/Female
Arabic, British, Danish, English, Muslim
Solid; Strong
Girl/Female
Biblical
Nativity, generation.
Biblical
nativity; generation;begetter;
Boy/Male
Indian
Solid structure
Boy/Male
Tamil
Solid redemption
SOLID HARMONICS
SOLID HARMONICS
Girl/Female
Indian
Energy, Enthusiasm
Boy/Male
Assamese, Hindu, Indian, Marathi, Sanskrit, Traditional
Fame
Girl/Female
German
Pure; Little and Womanly; Female Version of Charles or Carl
Girl/Female
Latin
A flower name.
Male
Celtic
, sea circle.
Girl/Female
American, Australian, Chinese, Christian, Danish, French, German, Hebrew
Supplanter; He who Supplants; God May Protect
Boy/Male
Biblical
Brother of the right hand.
Surname or Lastname
English (northern)
English (northern) : habitational name from places in Lancashire (in the parish of Middleton) and West Yorkshire (part of Halifax) called Siddal, from Old English sīd ‘wide’ + halh ‘nook’, ‘recess’, and possibly also from Siddle in East Harsley, North Yorkshire.
Girl/Female
Hindu
Tender, Beautiful, Delicate
Girl/Female
Hindu
Pure, Clean, Spotless, Without blemish
SOLID HARMONICS
SOLID HARMONICS
SOLID HARMONICS
SOLID HARMONICS
SOLID HARMONICS
superl.
Solid; nourishing; as, strong meat.
a.
United; without division; unanimous; as, the delegation is solid for a candidate.
v. i.
To become solid; to harden.
a.
Of a fleshy, uniform, undivided substance, as a bulb or root; not spongy or hollow within, as a stem.
a.
Partially solid.
a.
Sound; not weakly; as, a solid constitution of body.
a.
Firm; compact; strong; stable; unyielding; as, a solid pier; a solid pile; a solid wall.
a.
Impenetrable; resisting or excluding any other material particle or atom from any given portion of space; -- applied to the supposed ultimate particles of matter.
n.
A magnitude which has length, breadth, and thickness; a part of space bounded on all sides.
a.
Having the constituent parts so compact, or so firmly adhering, as to resist the impression or penetration of other bodies; having a fixed form; hard; firm; compact; -- opposed to fluid and liquid or to plastic, like clay, or to incompact, like sand.
n.
A substance that is held in a fixed form by cohesion among its particles; a substance not fluid.
a.
Not hollow; full of matter; as, a solid globe or cone, as distinguished from a hollow one; not spongy; dense; hence, sometimes, heavy.
a.
Solid; gross; opaque.
a.
Not having the lines separated by leads; not open.
a.
Fig.: Worthy of credit, trust, or esteem; substantial, as opposed to frivolous or fallacious; weighty; firm; strong; valid; just; genuine.
n.
The art of delineating the forms of solid bodies on a plane; a branch of solid geometry which shows the construction of all solids which are regularly defined.
n.
Solid coal on the side of a gallery; solid ore in a vein.
a.
Enduring; solid; firm; substantial.
a.
Having all the geometrical dimensions; cubic; as, a solid foot contains 1,728 solid inches.
a.
Applied to a compound word whose parts are closely united and form an unbroken word; -- opposed to hyphened.