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Special type of functions in mathematics
In mathematics, prolate spheroidal wave functions (PSWFs) are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions
Prolate spheroidal wave function
Prolate_spheroidal_wave_function
Solutions of the Helmholtz equation
called oblate spheroidal wave functions if oblate spheroidal coordinates are used and prolate spheroidal wave functions if prolate spheroidal coordinates
Spheroidal_wave_function
Special type of functions in mathematics
product of a radial spheroidal wave function R m n ( − i c , i ξ ) {\displaystyle R_{mn}(-ic,i\xi )} and an angular spheroidal wave function S m n ( − i c
Oblate spheroidal wave function
Oblate_spheroidal_wave_function
Math equation
In mathematics, the spheroidal wave equation is given by ( 1 − t 2 ) d 2 y d t 2 − 2 ( b + 1 ) t d y d t + ( c − 4 q t 2 ) y = 0 {\displaystyle (1-t^{2}){\frac
Spheroidal_wave_equation
Function used in signal processing
approximate closely the prolate spheroidal wave functions of order zero. Kaiser, James F. (Nov 1964). "A family of window functions having nearly ideal properties"
Window_function
Special mathematical functions defined on the surface of a sphere
2026-05-02. "DLMF: §14.30 Spherical and Spheroidal Harmonics". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology
Spherical_harmonics
Mathematical function
"Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions". Bell
Slepian_function
Solutions of Lamé's equation
corresponding calculations for Mathieu functions, and oblate spheroidal wave functions and prolate spheroidal wave functions). With the following boundary conditions
Lamé_function
Austrian-American mathematician (1927–2026)
Dolciani Award in 2020. Slepian, D.; Pollak, H. O. (1961). "Prolate spheroidal wave functions, Fourier analysis and uncertainty. I". The Bell System Technical
Henry_O._Pollak
Quartic potential in quantum mechanics
corresponding results for Mathieu functions, Lamé functions, prolate spheroidal wave functions, oblate spheroidal wave functions and others). In field theory
Double-well_potential
Surface formed by rotating an ellipse
dome Equatorial bulge Great ellipse Lentoid Oblate spheroidal coordinates Oval Prolate spheroidal coordinates Torge, Wolfgang (2001). Geodesy (3rd ed
Spheroid
Special function occurring in problems possessing elliptic symmetry
periodic differential equations, as for Lamé functions and prolate and oblate spheroidal wave functions. The asymptotic nature of the asymptotic solutions
Mathieu_function
American mathematician (1923–2007)
"Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty -- I" (PDF). BSTJ. Retrieved June 15, 2012. "Prolate Spheroidal Wave Functions, Fourier
David_Slepian
American engineer (1913–1973)
Stratton JA, Morse PM, Chu LJ, Hunter RA, Elliptic Cylinder and Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients, Wiley-MIT
Lan_Jen_Chu
Solitons in Euclidean spacetime
cosine potential (cf. Mathieu function) or other periodic potentials (cf. e.g. Lamé function and spheroidal wave function) and irrespective of whether
Instanton
Spectral density estimation technique
the Slepian functions or discrete prolate spheroidal sequences, after David Slepian (also known in literature as discrete prolate spheroidal sequences or
Multitaper
Topics referred to by the same term
failure of the performance of a task Posterior spinal fusion Prolate spheroidal wave function Pounds per square foot, a measure of pressure; see also Pounds
PSF
German theoretical physicist
mechanics, Meixner polynomials, Meixner–Pollaczek polynomials, and spheroidal wave functions. Meixner began his studies in theoretical physics with Arnold
Josef_Meixner
American university president (1901–1994)
Stratton JA, Morse PM, Chu LJ, Hunter RA, Elliptic Cylinder and Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients, Wiley-MIT
Julius_Adams_Stratton
Shape of a spinning body of self-gravitating fluid
A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity
Maclaurin_spheroid
1964 mathematical reference work edited by M. Abramowitz and I. Stegun
Weierstrass Elliptic and Related Functions Parabolic Cylinder Functions Mathieu Functions Spheroidal Wave Functions Orthogonal Polynomials Bernoulli and
Abramowitz_and_Stegun
Indian American academic (born 1954)
Vaidyanathan, "MIMO radar space-time adaptive processing using prolate spheroidal wave functions", IEEE Transactions on Signal Processing, Feb, 2008. https://ieeexplore
P._P._Vaidyanathan
Pattern of oscillating motion in a system
Rayleigh waves. Toroidal modes only involve SH waves (like Love waves) and do not exist in fluid outer core. Radial modes are just a subset of spheroidal modes
Normal_mode
Refractive property of materials
as a wave with field components in transverse polarization (perpendicular to the direction of the wave vector). A mathematical description of wave propagation
Birefringence
Aspect of crystal optics
the index ellipsoid; but the principal axes of the spheroidal index ellipsoid and the spheroidal sheet of the index surface are interchanged. In the
Index_ellipsoid
Belgian geophysicist
and to the theory of spatiospectral localization via prolate spheroidal wave functions. Simons is involved in the design of mobile marine instrumentation
Frederik_J._Simons
Component of blood aiding in coagulation
(thrómbos) 'clot' and κύτος (kútos) 'cell') are a part of blood whose function (along with the coagulation factors) is to react to bleeding from blood
Platelet
German physicist
Asymptotic expansions of Mathieu functions, spheroidal wave functions, Lamé functions and ellipsoidal wave functions and their eigenvalues. Asymptotic
Harald_J._W._Mueller-Kirsten
History of research by Augustin-Jean Fresnel
assumption that the secondary waves are spherical for the ordinary refraction (which satisfies Snell's law) and spheroidal for the extraordinary refraction
Fresnel's_physical_optics
Range of physical processes in physics
related to wave–particle duality. Scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering
Scattering
Set of imaging methods for determining soft-tissue hardness
"tissue" stiffness. It functions by vibrating the skin with a motor to create a passing distortion in the tissue (a shear wave), and imaging the motion
Elastography
Molecular ion
correlation). The wave equation (a partial differential equation) separates into two coupled ordinary differential equations when using prolate spheroidal coordinates
Dihydrogen_cation
Problem in physics and astronomy
fixed centers. It is well known that the Schrödinger wave equation is separable in prolate spheroidal coordinates and can be decoupled into two ordinary
Euler's_three-body_problem
Light rays follow quickest paths
known as the principle of least time, is the link between ray optics and wave optics. Fermat's principle states that the path taken by a ray between two
Fermat's_principle
PMID 24785032. Rainwater J (August 1950). "Nuclear Energy Level Argument for a Spheroidal Nuclear Model". Physical Review. 79 (3): 432–434. Bibcode:1950PhRv...79
Shape_of_the_atomic_nucleus
French optical physicist (1788–1827)
physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, fully supplanting Newton's corpuscular theory, from the
Augustin-Jean_Fresnel
French mathematician and physicist (1781–1840)
on the theory of waves (Mém. ft. l'acad., 1825). In his work on heat conduction, Joseph Fourier maintained that the arbitrary function may be represented
Siméon_Denis_Poisson
Surface of a fluid that is subject to zero parallel shear stress
liquid is disturbed, waves are produced on the surface. These waves are not elastic waves due to any elastic force; they are gravity waves caused by the force
Free_surface
doi:10.1080/14786443109461714. Havelock, T. H. (1931). "The wave resistance of a spheroid. Proceedings of the Royal Society of London. Series A, Containing
Thomas_Henry_Havelock
Set of coordinates where the coordinate hypersurfaces all meet at right angles
general but still analytical orthogonal coordinate system is Similar Oblate Spheroidal (SOS) system, in which the transformation from the Cartesian coordinates
Orthogonal_coordinates
Visible regularity of form found in the natural world
crown-shaped splash pattern formed when a drop falls into a pond, and both the spheroidal shape and rings of a planet like Saturn. Symmetry has a variety of causes
Patterns_in_nature
Field of radiation
An Airy beam is a propagation invariant wave whose main intensity lobe propagates along a curved parabolic trajectory while being resilient to perturbations
Airy_beam
Hypothetical invisible cosmic material
telescopes have set limits on the annihilation of dark matter in dwarf spheroidal galaxies and in clusters of galaxies. The PAMELA experiment (launched
Dark_matter
Loudspeaker using an acoustic horn
describe horns with low acoustic loading, such as conic, quadratic, oblate spheroidal or elliptic cylindrical horns. These are designed more to control the
Horn_loudspeaker
American theoretical physicist
His thesis Ph.D. thesis is entitled Variational calculation of atomic wave functions. As a postdoc he worked at the University of Pennsylvania from 1932
John_P._Vinti
Length of a line segment
Vincenty's formulae also known as "Vincent distance" for distance on a spheroid. Euclidean distance is the distance in Euclidean space. Both concepts are
Euclidean_distance
Approximation in mathematics
Mathieu equation (best example), Lamé and ellipsoidal wave equations, oblate and prolate spheroidal wave equations, and equations with anharmonic potentials
Method of matched asymptotic expansions
Method_of_matched_asymptotic_expansions
English polymath (1642–1727)
law) of the speed of sound in air, inferred the oblateness of Earth's spheroidal figure, accounted for the precession of the equinoxes as a result of the
Isaac_Newton
Exact solution for the Einstein field equations
coordinates r , θ , ϕ {\displaystyle r,\theta ,\phi } are standard oblate spheroidal coordinates, which are equivalent to the cartesian coordinates where r
Kerr_metric
Low-Density Gases
C.; Di Staso, G.; Toschi, F. (2024). "Modeling drag coefficients of spheroidal particles in rarefied flow conditions". European Journal of Mechanics
Rarefied_gas_dynamics
Subfield of astronomy
structure on large scales. Gravitational-wave astronomy – study of astrophysical sources of gravitational waves and the information they carry. High-energy
Outline_of_astrophysics
Project to measure the expansion of the universe
2019, the team also discovered a sixth star cluster in the Fornax Dwarf Spheroidal Galaxy and a tidally Disrupted Ultra-Faint Dwarf Galaxy. The signature
Dark_Energy_Survey
Process by which dust, particulates, etc. scatter light
transmitted, and scattered field are expanded into spherical vector wave functions. Scattering from any spherical particles with arbitrary size parameter
Light_scattering_by_particles
System of stars and interstellar matter
galaxies are classified as either irregular or dwarf elliptical/dwarf spheroidal galaxies. A study of 27 Milky Way neighbors found that in all dwarf galaxies
Galaxy
DC Comics equipment
be deployed by throwing or breaking open these small hardened-gelatin spheroid capsules. Often used to quickly provide cover for Batman's stealthy exits
Batman's_utility_belt
Large self-illuminated object in space
A star is a luminous spheroid of plasma held together by self-gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked
Star
American mathematician (1929–1991)
Wisconsin–Madison. His Ph.D. thesis Asymptotic Forms for the Whitaker Functions of Large Complex Order m was supervised by Rudolf Ernest Langer. In the
Nicholas_D._Kazarinoff
Gene editing method
new phage DNA into the CRISPR array, which is how they fight off the next wave of attacking phage. In 2012, the research team led by professor Jennifer
CRISPR_gene_editing
Explosion of a vessel containing liquid above and beyond boiling point
sequence. Fireballs can rise to significant heights above ground. They are spheroidal when developed and rise from the ground in a mushroom shape. The diameter
Boiling liquid expanding vapor explosion
Boiling_liquid_expanding_vapor_explosion
Type of lens
Johannes; Osten, Wolfgang (20 July 2017). "Measuring aspheres quickly: tilted wave interferometry". Optical Engineering. 56 (11) 111713. doi:10.1117/1.OE.56
Aspheric_lens
French polymath (1749–1827)
magnitude and direction. A potential function is a scalar function that defines how the vectors will behave. A scalar function is computationally and conceptually
Pierre-Simon_Laplace
Barred spiral galaxy in the Local Group
radio results as approximately 8×1011 M☉. In 2006, the Andromeda Galaxy's spheroid was determined to have a higher stellar density than that of the Milky
Andromeda_Galaxy
Change in sea level due to gravity
stations measure water level over time. Gauges ignore variations caused by waves with periods shorter than minutes. These data are compared to the reference
Tide
Motion of Earth's rotational axis relative to its crust
is a standing wave of the form: (3) p = p0Θ1 −3(θ) cos[2πνA(t − t0)] cos(λ − λ0) with p0 a pressure amplitude, Θ1 −3 a Hough function describing the
Polar_motion
seminal explorations of artificial materials for manipulating electromagnetic waves at the end of the 19th century. Hence, the history of metamaterials is essentially
History_of_metamaterials
World War II Allied nuclear weapons program
trinitite (radioactive glass) in the desert 250 feet (76 m) wide. The shock wave was felt over 100 miles (160 km) away, and the mushroom cloud reached 7.5
Manhattan_Project
Nanotechnology simulation of human organ function
testing. Although multiple publications claim to have translated organ functions onto this interface, the development of these microfluidic applications
Organ-on-a-chip
World Geodetic System (WGS) A standard geographic coordinate system, spheroidal reference ellipsoid (for raw altitude data), and geoid (which defines
Glossary of geography terms (N–Z)
Glossary_of_geography_terms_(N–Z)
Type of programmed cell death
glutathione state, nicotinamide adenine dinucleotide phosphate (NADP) function, lipid peroxidation, and iron homeostasis. In 1989, work by the groups
Ferroptosis
Precipitation in the form of ice crystal flakes
depth and seasonal persistence of snowpack in snow-prone areas. Mountain waves have also been found to help enhance precipitation amounts downwind of mountain
Snow
Team field sport
but they may also function as decoys or as blockers during running plays. Tight ends line up next to the offensive line and function both as receivers
American_football
Ancient Greek storage container
was still fresh clay. Some common themes are spirals, meanders, and waves. Waves in relief give the appearance of slack rope; whether they were intended
Pithos
Cloud of debris and smoke from a large explosion
cooled by energy radiation, turning from a sphere to a violently rotating spheroidal vortex. A Rayleigh–Taylor instability is formed as the cool air underneath
Mushroom_cloud
Massive fermion wave equation in Kerr spacetime
Chandrasekhar–Page equations describe the wave function of the spin-1/2 massive particles, that resulted by seeking a separable solution to the Dirac
Chandrasekhar–Page_equations
seismic waves produced by the earthquake are usually most noticeable. equal-area projection equator 1. An imaginary line dividing a spheroid such as
Glossary of geography terms (A–M)
Glossary_of_geography_terms_(A–M)
Ocean shape without winds and tides
parallel to the geoid. Earth's gravitational field is not uniform. An oblate spheroid is typically used as the idealized Earth, but even if the Earth were spherical
Geoid
Historical development of physics
wave (soliton) in the Union Canal near Edinburgh, Scotland, and used a water tank to study the dependence of solitary water wave velocities on wave amplitude
History_of_physics
Particle motion in a non-uniform electric field due to dipole-field interactions
nanoparticles and nanowires. Furthermore, a study of the change in DEP force as a function of frequency can allow the electrical (or electrophysiological in the case
Dielectrophoresis
Biomedical engineering discipline
structural properties for proper functioning. The term has also been applied to efforts to perform specific biochemical functions using cells within an artificially
Tissue_engineering
Galaxy in the constellation Virgo
galaxy is a strong source of multi-wavelength radiation, particularly radio waves. It has an isophotal diameter of 40.55 kiloparsecs (132,000 light-years)
Messier_87
slightly flatter at the poles while bulging at the Equator: an oblate spheroid. There are consequently slight deviations in the magnitude of gravity across
Gravity_of_Earth
Optical coating that reduces reflection
thickness equal to a quarter of some design wavelength is called a "quarter-wave layer". The most common type of optical glass is crown glass, which has an
Anti-reflective_coating
Ratio of energy extracted from sunlight in solar cells
Elvira; Martins, Rodrigo (1 August 2016). "Design of optimized wave-optical spheroidal nanostructures for photonic-enhanced solar cells". Nano Energy
Solar-cell_efficiency
Plane curve
\end{aligned}}} where atan2 is the 2-argument arctangent function. Using trigonometric functions, a parametric representation of the standard ellipse x
Ellipse
Method of approximation in general relativity
relativistic two-body problem, which includes the emission of gravitational waves. In general, the perturbed metric can be written as d s 2 = a 2 ( τ ) [
Post-Newtonian_expansion
Secret laboratory established by the Manhattan Project
Trinitite (radioactive glass) in the desert 250 feet (76 m) wide. The shock wave was felt over 100 miles (160 km) away, and the mushroom cloud reached 7.5
Project_Y
Apparent force in a rotating reference frame
for the propagation of many types of waves in the ocean and atmosphere, including Rossby waves and Kelvin waves. It is also instrumental in the so-called
Coriolis_force
in South Africa form – contain some of the oldest microfossils mostly spheroidal and carbonaceous alga-like bodies. c. 3,200–2,600 Ma – Assembly of the
Timeline_of_Earth
Curve from a cone intersecting a plane
the Poisson equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic. Eccentricity classifications include: Möbius transformations
Conic_section
Deterioration of rocks and minerals through exposure to the elements
Formation of authigenic clay minerals Soil production function Space weathering – Type of weathering Spheroidal weathering – Form of chemical weathering that
Weathering
gravitational-wave astronomy A branch of observational astronomy which analyzes minute distortions in the curvature of spacetime known as gravitational waves to
Glossary_of_astronomy
Type of inertial force
material is often explained in terms of centrifugal force. The oblate spheroid shape reflects, following Clairaut's theorem, the balance between containment
Centrifugal_force
Emergence of modern science (1572-1687)
principles. His prediction that the Earth should be shaped as an oblate spheroid was later vindicated by other scientists. His laws of motion were to be
Scientific_Revolution
Method in molecular biology to make pores in cell membranes
"Analytical description of transmembrane voltage induced by electric fields on spheroidal cells". Biophysical Journal. 79 (2): 670–679. Bibcode:2000BpJ....79..670K
Electroporation
Granite reef in False Bay, South Africa
has since then been exposed by prolonged erosion. The characteristic spheroidal shapes of granite boulders are a result of preferential weathering along
Whittle_Rock
Change of rotational axis in an astronomical body
The value in the direction of y (sinδ cosδ (−cosα)) for the Sun is a sine wave varying from zero at the four equinoxes and solstices to ±0.19364 (slightly
Axial_precession
Bioluminescent, marine dinoflagellate
a tentacle.[citation needed] Noctiluca scintillans is a single-celled spheroid organism, ranging from 400 to 1500 μm in length. It moves with the current
Noctiluca
shock wave propagation within the explosive mass, this requires the pit to be a prolate spheroid, that is, roughly egg shaped. The shock wave first reaches
Nuclear_weapon_design
Subfield of cosmology
mass of the Sun. The black hole mass is tied to the host galaxy bulge or spheroid mass. It is now recognized that the black hole mass scaling relation is
Galaxy formation and evolution
Galaxy_formation_and_evolution
Order of reptiles (fossil)
surprise schools of fish, the head arriving before the sight or pressure wave of the trunk could alert them. "Plesiosauromorphs" hunted visually, as shown
Plesiosaur
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