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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
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
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
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
}^{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)
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
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
Electromagnetic phenomenon
Monopole Solid harmonics Axial multipole moments Cylindrical multipole moments Spherical multipole moments Laplace expansion Molecular solid Magnetic
Dipole
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
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
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
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
American Christian rock record label
Ghost Ship Aaron Gillespie The Glorious Unseen Hangnail Hawk Nelson iLL Harmonics KJ-52 Dustin Kensrue Joy Electric The O.C. Supertones Phillip LaRue Jadon
Tooth_&_Nail_Records
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
Nonlinear optical process
"Detection of order–disorder transition in organic solids by using temperature resolved second harmonic generation (TR-SHG)". Journal of Molecular Structure
Second-harmonic_generation
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
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
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
Pattern of oscillating motion in a system
called its harmonic overtones. He assigned to one of the oscillators the frequency of the fundamental vibration of the whole block of solid. He assigned
Normal_mode
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
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 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
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
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
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)
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)
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
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
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
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)
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
expansion Spherical multipole moments Cylindrical multipole moments Solid harmonics Laplace expansion Eyges, Leonard (2012-06-11). The Classical Electromagnetic
Axial_multipole_moments
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)
Quasiparticle of mechanical vibrations
G.; Ipatova, I. (1971). Theory of lattice dynamics in the harmonic approximation. Solid State Physics. Vol. Supplement 3 (Second ed.). New York: Academic
Phonon
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
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
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
Electronic amplifier for musical instruments
produces even-order harmonics, which are generally perceived as musical and pleasing. As the volume or gain is increased, it produces harmonic distortion. The
Guitar_amplifier
Force needed to pull a spring grows linearly with distance
Series and parallel springs Spring system Simple harmonic motion of a mass on a spring Sine wave Solid mechanics Spring pendulum The anagram was given
Hooke's_law
Function describing an electron in an atom
orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. Letting ψn,ℓ,m denote a complex orbital with quantum
Atomic_orbital
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
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
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
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
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
In the Greek world from 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
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
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
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 guitarist (born 1981)
learning from any guitarist. She expressed interest in how Gates "uses harmonics and creates the darker sounds" that let him "make things sound creepy"
Synyster_Gates
Number between two given numbers
Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode Dispersion Average absolute deviation
Heronian_mean
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
Geopotential descriptions used by the US DoD
d/o 5540 with a different spheroidal harmonic construction followed by conversion back into spherical harmonics. It was made by authors affiliated with
Earth_Gravitational_Model
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
Ground vibration associated with volcanic activity
also generate distinct iceberg harmonic tremor signals that propagate to large distances as ocean acoustic and solid Earth seismic wavefields. The source
Harmonic_tremor
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
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
Geometric figure which has infinite surface area but finite volume
Another approach is to treat the solid as a stack of disks with diminishing radii. The sum of the radii produces a harmonic series that goes to infinity.
Gabriel's_horn
0+ (M1+) RoboSport Robot Battle Robot Rover Rock'N'Roll Dice Roller Rock Solid Richard White 1998 Puzzle ? 7.0 - 7.6 - 9.0 Rocket Golf Rogue RollerCoaster
List_of_Mac_games
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
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
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
Capacity of a material to conduct heat
be a second-rank tensor in the case of anisotropic materials. Consider a solid material placed between two environments of different temperatures. Let
Thermal conductivity and resistivity
Thermal_conductivity_and_resistivity
Group of rotations in 3 dimensions
Hilbert space L2(𝐒2)=span{Ymℓ,ℓ∈ℕ+,−ℓ≤m≤ℓ}, where Ymℓ are spherical harmonics. Its elements are square integrable complex-valued functions on the sphere
3D_rotation_group
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
Technique in attosecond physics
technique involves the interference of two-photon interband transitions in solids. It is especially suited for diagnostics on the temporal structure of XUV
Reconstruction of attosecond beating by interference of two-photon transitions
Reconstruction_of_attosecond_beating_by_interference_of_two-photon_transitions
Hydrodynamic distortions of tides
dissipates energy from the principal tide to its higher harmonics. For the sake of consistency, higher harmonics having a frequency that is an even or odd multiple
Nonlinear_tides
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
Spectroscopic technique
electronic structure of a molecule, as discussed in the Theory section. In solid-state physics, Raman spectroscopy is used to characterize materials, measure
Raman_spectroscopy
Brass instrument
adjustment, compensating for imperfections in the tuning of different harmonics. The fifth partial is rather flat on most trombones and usually requires
Trombone
Kepler–Poinsot polyhedron
stellated dodecahedra can be seen in antiquity, as in Johannes Kepler's Harmonices Mundi, and a floor mosaic in Paolo Uccello's St Mark's Basilica c. 1430
Small_stellated_dodecahedron
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
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
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
Change in sea level due to gravity
fundamental cycle. These multiples are called harmonics of the fundamental frequency, and the process is termed harmonic analysis. If the basis set of sinusoidal
Tide
Solid with six equal square faces
A cube is a three-dimensional solid object in geometry. It has eight vertices and twelve straight edges of the same length, so that these edges form six
Cube
Disruption to reception of television signals
power controller using the variable phase angle method will generate harmonics of the mains supply, while the spark at a contact will be a very wide
Television_interference
Experimental acoustical apparatus
physicist August Kundt for the measurement of the speed of sound in a gas or a solid rod. The experiment is still taught today due to its ability to demonstrate
Kundt's_tube
Audio effect
or swoosh or jet plane sweeping effect as shifting sum-and-difference harmonics are created, analogous to the use of a variable notch filter. The term
Flanging
Keyboard instrument
content of these harmonics. Different instruments have different harmonic content for the same pitch. A real string vibrates at harmonics that are not perfect
Piano
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
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
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