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Vibrations Of A Circular Drum
A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental frequency. There exist infinitely many ways in which a membrane can vibrate, each depending on the shape of the membrane at some initial time, and the transverse velocity of each point on the membrane at that time. The vibrations of the membrane are given by the solutions of the two-dimensional wave equation with Dirichlet boundary conditions which represent the constraint of the frame. It can be sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drum Vibration Mode12
The drum is a member of the percussion group of musical instruments. In the Hornbostel–Sachs classification system, it is a membranophone. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a shell and struck, either directly with the player's hands, or with a percussion mallet, to produce sound. There is usually a resonant head on the underside of the drum. Other techniques have been used to cause drums to make sound, such as the thumb roll. Drums are the world's oldest and most ubiquitous musical instruments, and the basic design has remained virtually unchanged for thousands of years. Drums may be played individually, with the player using a single drum, and some drums such as the djembe are almost always played in this way. Others are normally played in a set of two or more, all played by one player, such as bongo drums and timpani. A number of different drums together with cymbals form the basic modern drum kit. Many d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eardrum
In the anatomy of humans and various other tetrapods, the eardrum, also called the tympanic membrane or myringa, is a thin, cone-shaped membrane that separates the external ear from the middle ear. Its function is to transmit changes in pressure of sound from the air to the ossicles inside the middle ear, and thence to the oval window in the fluid-filled cochlea. The ear thereby converts and amplifies vibration in the air to vibration in cochlear fluid. The malleus bone bridges the gap between the eardrum and the other ossicles. Rupture or perforation of the eardrum can lead to conductive hearing loss. Collapse or retraction of the eardrum can cause conductive hearing loss or cholesteatoma. Structure Orientation and relations The tympanic membrane is oriented obliquely in the anteroposterior, mediolateral, and superoinferior planes. Consequently, its superoposterior end lies lateral to its anteroinferior end. Anatomically, it relates superiorly to the middle crani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Orbital
In quantum mechanics, an atomic orbital () is a Function (mathematics), function describing the location and Matter wave, wave-like behavior of an electron in an atom. This function describes an electron's Charge density, charge distribution around the Atomic nucleus, atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus. Each orbital in an atom is characterized by a set of values of three quantum numbers , , and , which respectively correspond to electron's energy, its angular momentum, orbital angular momentum, and its orbital angular momentum projected along a chosen axis (magnetic quantum number). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of and orbitals, and are often labeled using associated Spherical harmonics#Harmonic polynomial representation, harmonic polynomials (e.g., ''xy'', ) which describe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cymatics
Cymatics (from ) is a subset of modal vibrational phenomena. The term was coined by Swiss physician Hans Jenny (1904–1972). Typically the surface of a plate, diaphragm, or membrane is vibrated, and regions of maximum and minimum displacement are made visible in a thin coating of particles, paste, or liquid. Different patterns emerge in the excitatory medium depending on the geometry of the plate and the driving frequency. The apparatus employed can be simple, such as the Chinese spouting bowl, in which copper handles are rubbed and cause the copper bottom elements to vibrate. Other examples include the Chladni plate and the so-called cymascope. History On July 8, 1680, Robert Hooke was able to see the nodal patterns associated with the modes of vibration of glass plates. Hooke ran a bow along the edge of a glass plate covered with flour, and saw the nodal patterns emerge. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chladni Patterns
Ernst Florens Friedrich Chladni (, , ; 30 November 1756 – 3 April 1827) was a German physicist and musician. His most important work, for which he is sometimes labeled the father of acoustics, included research on vibrating plates and the calculation of the speed of sound for different gases. He also undertook pioneering work in the study of meteorites and is regarded by some as the father of meteoritics. Early life Although Chladni was born in Wittenberg in Saxony, his family originated from Kremnica, then part of the Kingdom of Hungary and today a mining town in central Slovakia. Chladni has therefore been identified as German, Hungarian and Slovak. Chladni came from an educated family of academics and learned men. Chladni's great-grandfather, the Lutheran clergyman Georg Chladni (1637–1692), had left Kremnica in 1673 during the Counter Reformation. Chladni's grandfather, Martin Chladni (1669–1725), was also a Lutheran theologian and, in 1710, became profes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Vibration
A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. For an homogenous string, the motion is given by the wave equation. Wave The velocity of propagation of a wave in a string (v) is proportional to the square root of the force of tension of the string (T) and inversely proportional to the square root of the linear density (\mu) of the string: v = \sqrt. This relationship was discovered by Vincenzo Galilei in the late 1500s. Derivation Source: Let \Delta x be the length of a piece of string, m its mass, and \mu its linear density. If angles \alpha and \beta are small, then the horizontal components of tension on either side can both be approximated by a constant T, for which the net horizontal force ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel's Differential Equation
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. #Spherical Bessel functions, Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotational Symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational symmetry is the number of distinct Orientation (geometry), orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids. Formal treatment Formally the rotational symmetry is symmetry with respect to some or all rotations in -dimensional Euclidean space. Rotations are Euclidean group#Direct and indirect isometries, direct isometries, i.e., Isometry, isometries preserving Orientation (mathematics), orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of (see Euclidean g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress Resultants
Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells. The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions. As a consequence the three traction components that vary from point to point in a cross-section can be replaced with a set of resultant forces and resultant moments. These are the stress resultants (also called '' membrane forces'', ''shear forces'', and ''bending moment'') that may be used to determine the detailed stress state in the structural element. A three-dimensional problem can then be reduced to a one-dimensional problem (for beams) or a two-dimensional problem (for plates and shells). Stress resultants are defined as integrals of stress over the thickness of a structural element. The integrals are weighted by integer pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinate System
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
