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78-829: A pattern is an original object used to make copies, or a set of repeating objects in a decorative design and in other disciplines. Pattern , patterns , or patterning may also refer to: Pattern Any of the senses may directly observe patterns. Conversely, abstract patterns in science , mathematics , or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic , rarely exactly repeating, and often involve fractals . Natural patterns include spirals , meanders , waves , foams , tilings , cracks , and those created by symmetries of rotation and reflection . Patterns have an underlying mathematical structure; indeed, mathematics can be seen as

156-401: A node . Halfway between two nodes there is an antinode , where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time. A soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in

234-470: A standing wave . Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut , where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing

312-428: A transmission medium . The propagation and reflection of plane waves—e.g. Pressure waves ( P wave ) or Shear waves (SH or SV-waves) are phenomena that were first characterized within the field of classical seismology, and are now considered fundamental concepts in modern seismic tomography . The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding

390-634: A vacuum and through some dielectric media (at wavelengths where they are considered transparent ). Electromagnetic waves, as determined by their frequencies (or wavelengths ), have more specific designations including radio waves , infrared radiation , terahertz waves , visible light , ultraviolet radiation , X-rays and gamma rays . Other types of waves include gravitational waves , which are disturbances in spacetime that propagate according to general relativity ; heat diffusion waves ; plasma waves that combine mechanical deformations and electromagnetic fields; reaction–diffusion waves , such as in

468-405: A wave is a propagating dynamic disturbance (change from equilibrium ) of one or more quantities . Periodic waves oscillate repeatedly about an equilibrium (resting) value at some frequency . When the entire waveform moves in one direction, it is said to be a travelling wave ; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave . In

546-538: A chosen effect on the viewer. Nature provides examples of many kinds of pattern, including symmetries , trees and other structures with a fractal dimension, spirals , meanders , waves , foams , tilings , cracks and stripes. Symmetry is widespread in living things. Animals that move usually have bilateral or mirror symmetry as this favours movement. Plants often have radial or rotational symmetry , as do many flowers, as well as animals which are largely static as adults, such as sea anemones . Fivefold symmetry

624-413: A container of gas by a function F ( x , t ) {\displaystyle F(x,t)} that gives the pressure at a point x {\displaystyle x} and time t {\displaystyle t} within that container. If the gas was initially at uniform temperature and composition, the evolution of F {\displaystyle F} is constrained by

702-618: A family of waves by a function F ( A , B , … ; x , t ) {\displaystyle F(A,B,\ldots ;x,t)} that depends on certain parameters A , B , … {\displaystyle A,B,\ldots } , besides x {\displaystyle x} and t {\displaystyle t} . Then one can obtain different waves – that is, different functions of x {\displaystyle x} and t {\displaystyle t} – by choosing different values for those parameters. For example,

780-402: A family of waves is to give a mathematical equation that, instead of explicitly giving the value of F ( x , t ) {\displaystyle F(x,t)} , only constrains how those values can change with time. Then the family of waves in question consists of all functions F {\displaystyle F} that satisfy those constraints – that is, all solutions of

858-537: A homogeneous isotropic non-conducting solid. Note that this equation differs from that of heat flow only in that the left-hand side is ∂ 2 F / ∂ t 2 {\displaystyle \partial ^{2}F/\partial t^{2}} , the second derivative of F {\displaystyle F} with respect to time, rather than the first derivative ∂ F / ∂ t {\displaystyle \partial F/\partial t} . Yet this small change makes

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936-442: A huge difference on the set of solutions F {\displaystyle F} . This differential equation is called "the" wave equation in mathematics, even though it describes only one very special kind of waves. Consider a traveling transverse wave (which may be a pulse ) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling This wave can then be described by

1014-510: A medium – air or water, making it oscillate as they pass by. Wind waves are surface waves that create the chaotic patterns of the sea. As they pass over sand, such waves create patterns of ripples; similarly, as the wind passes over sand, it creates patterns of dunes . Foams obey Plateau's laws , which require films to be smooth and continuous, and to have a constant average curvature . Foam and bubble patterns occur widely in nature, for example in radiolarians , sponge spicules , and

1092-411: A roughly pyramidal form, where elements of the pattern repeat in a fractal -like way at different sizes. Mathematics is sometimes called the "Science of Pattern", in the sense of rules that can be applied wherever needed. For example, any sequence of numbers that may be modeled by a mathematical function can be considered a pattern. Mathematics can be taught as a collection of patterns. Gravity

1170-593: A standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. There are two types of waves that are most commonly studied in classical physics : mechanical waves and electromagnetic waves . In a mechanical wave, stress and strain fields oscillate about a mechanical equilibrium. A mechanical wave is a local deformation (strain) in some physical medium that propagates from particle to particle by creating local stresses that cause strain in neighboring particles too. For example, sound waves are variations of

1248-410: A sum of sine waves of various frequencies, relative phases, and magnitudes. A plane wave is a kind of wave whose value varies only in one spatial direction. That is, its value is constant on a plane that is perpendicular to that direction. Plane waves can be specified by a vector of unit length n ^ {\displaystyle {\hat {n}}} indicating the direction that

1326-635: A wave may be constant (in which case the wave is a c.w. or continuous wave ), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form: u ( x , t ) = A ( x , t ) sin ⁡ ( k x − ω t + ϕ ) , {\displaystyle u(x,t)=A(x,t)\sin \left(kx-\omega t+\phi \right),} where A ( x ,   t ) {\displaystyle A(x,\ t)}

1404-435: A wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as: v p = ω k , {\displaystyle v_{\rm {p}}={\frac {\omega }{k}},} where: The phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency. The angular frequency ω cannot be chosen independently from

1482-587: A ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay

1560-582: Is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . However, in many cases one can ignore one dimension, and let x {\displaystyle x} be a point of the Cartesian plane R 2 {\displaystyle \mathbb {R} ^{2}} . This

1638-473: Is a source of ubiquitous scientific patterns or patterns of observation. The sun rising and falling pattern each day results from the rotation of the earth while in orbit around the sun. Likewise, the moon's path through the sky is due to its orbit of the earth. These examples, while perhaps trivial, are examples of the "unreasonable effectiveness of mathematics" which obtain due to the differential equations whose application within physics function to describe

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1716-440: Is almost always confined to some finite region of space, called its domain . For example, the seismic waves generated by earthquakes are significant only in the interior and surface of the planet, so they can be ignored outside it. However, waves with infinite domain, that extend over the whole space, are commonly studied in mathematics, and are very valuable tools for understanding physical waves in finite domains. A plane wave

1794-494: Is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific direction of travel. Mathematically, the simplest wave is a sinusoidal plane wave in which at any point the field experiences simple harmonic motion at one frequency. In linear media, complicated waves can generally be decomposed as the sum of many sinusoidal plane waves having different directions of propagation and/or different frequencies . A plane wave

1872-538: Is classified as a transverse wave if the field disturbance at each point is described by a vector perpendicular to the direction of propagation (also the direction of energy transfer); or longitudinal wave if those vectors are aligned with the propagation direction. Mechanical waves include both transverse and longitudinal waves; on the other hand electromagnetic plane waves are strictly transverse while sound waves in fluids (such as air) can only be longitudinal. That physical direction of an oscillating field relative to

1950-501: Is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant wellbeing. Wave In physics , mathematics , engineering , and related fields,

2028-494: Is found in the echinoderms , including starfish , sea urchins , and sea lilies . Among non-living things, snowflakes have striking sixfold symmetry : each flake is unique, its structure recording the varying conditions during its crystallisation similarly on each of its six arms. Crystals have a highly specific set of possible crystal symmetries ; they can be cubic or octahedral , but cannot have fivefold symmetry (unlike quasicrystals ). Spiral patterns are found in

2106-448: Is meant to signify that, in the derivative with respect to some variable, all other variables must be considered fixed.) This equation can be derived from the laws of physics that govern the diffusion of heat in solid media. For that reason, it is called the heat equation in mathematics, even though it applies to many other physical quantities besides temperatures. For another example, we can describe all possible sounds echoing within

2184-458: Is the wavelength of the emitted note, and f = c / λ {\displaystyle f=c/\lambda } is its frequency .) Many general properties of these waves can be inferred from this general equation, without choosing specific values for the parameters. As another example, it may be that the vibrations of a drum skin after a single strike depend only on the distance r {\displaystyle r} from

2262-520: Is the (first) derivative of F {\displaystyle F} with respect to t {\displaystyle t} ; and ∂ 2 F / ∂ x i 2 {\displaystyle \partial ^{2}F/\partial x_{i}^{2}} is the second derivative of F {\displaystyle F} relative to x i {\displaystyle x_{i}} . (The symbol " ∂ {\displaystyle \partial } "

2340-648: Is the amplitude envelope of the wave, k {\displaystyle k} is the wavenumber and ϕ {\displaystyle \phi } is the phase . If the group velocity v g {\displaystyle v_{g}} (see below) is wavelength-independent, this equation can be simplified as: u ( x , t ) = A ( x − v g t ) sin ⁡ ( k x − ω t + ϕ ) , {\displaystyle u(x,t)=A(x-v_{g}t)\sin \left(kx-\omega t+\phi \right),} showing that

2418-417: Is the case, for example, when studying vibrations of a drum skin. One may even restrict x {\displaystyle x} to a point of the Cartesian line R {\displaystyle \mathbb {R} } – that is, the set of real numbers . This is the case, for example, when studying vibrations in a violin string or recorder . The time t {\displaystyle t} , on

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2496-549: Is the heat that is being generated per unit of volume and time in the neighborhood of x {\displaystyle x} at time t {\displaystyle t} (for example, by chemical reactions happening there); x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} are the Cartesian coordinates of the point x {\displaystyle x} ; ∂ F / ∂ t {\displaystyle \partial F/\partial t}

2574-536: Is the length of the bore; and n {\displaystyle n} is a positive integer (1,2,3,...) that specifies the number of nodes in the standing wave. (The position x {\displaystyle x} should be measured from the mouthpiece , and the time t {\displaystyle t} from any moment at which the pressure at the mouthpiece is maximum. The quantity λ = 4 L / ( 2 n − 1 ) {\displaystyle \lambda =4L/(2n-1)}

2652-495: The Belousov–Zhabotinsky reaction ; and many more. Mechanical and electromagnetic waves transfer energy , momentum , and information , but they do not transfer particles in the medium. In mathematics and electronics waves are studied as signals . On the other hand, some waves have envelopes which do not move at all such as standing waves (which are fundamental to music) and hydraulic jumps . A physical wave field

2730-463: The Helmholtz decomposition of the displacement field, which is then substituted into the wave equation . From here, the plane wave eigenmodes can be calculated. The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of the reflected SV wave is identical to the incidence wave, while

2808-454: The electric field vector E {\displaystyle E} , or the magnetic field vector H {\displaystyle H} , or any related quantity, such as the Poynting vector E × H {\displaystyle E\times H} . In fluid dynamics , the value of F ( x , t ) {\displaystyle F(x,t)} could be

2886-472: The angle of the reflected P wave is greater than the SV wave. For the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture. Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different. Wave velocity is a general concept, of various kinds of wave velocities, for

2964-460: The animals' appearance changing imperceptibly as Turing predicted. In visual art, pattern consists in regularity which in some way "organizes surfaces or structures in a consistent, regular manner." At its simplest, a pattern in art may be a geometric or other repeating shape in a painting , drawing , tapestry , ceramic tiling or carpet , but a pattern need not necessarily repeat exactly as long as it provides some form or organizing "skeleton" in

3042-468: The argument x − vt . Constant values of this argument correspond to constant values of F , and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x -direction at velocity v (and G will propagate at the same speed in the negative x -direction). In the case of a periodic function F with period λ , that is, F ( x + λ − vt ) = F ( x − vt ),

3120-603: The artwork. In mathematics, a tessellation is the tiling of a plane using one or more geometric shapes (which mathematicians call tiles), with no overlaps and no gaps. In architecture, motifs are repeated in various ways to form patterns. Most simply, structures such as windows can be repeated horizontally and vertically (see leading picture). Architects can use and repeat decorative and structural elements such as columns , pediments , and lintels . Repetitions need not be identical; for example, temples in South India have

3198-400: The bar. Then the temperatures at later times can be expressed by a function F {\displaystyle F} that depends on the function h {\displaystyle h} (that is, a functional operator ), so that the temperature at a later time is F ( h ; x , t ) {\displaystyle F(h;x,t)} Another way to describe and study

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3276-576: The body plans of animals including molluscs such as the nautilus , and in the phyllotaxis of many plants, both of leaves spiralling around stems, and in the multiple spirals found in flowerheads such as the sunflower and fruit structures like the pineapple . Chaos theory predicts that while the laws of physics are deterministic , there are events and patterns in nature that never exactly repeat because extremely small differences in starting conditions can lead to widely differing outcomes. The patterns in nature tend to be static due to dissipation on

3354-430: The center of the skin to the strike point, and on the strength s {\displaystyle s} of the strike. Then the vibration for all possible strikes can be described by a function F ( r , s ; x , t ) {\displaystyle F(r,s;x,t)} . Sometimes the family of waves of interest has infinitely many parameters. For example, one may want to describe what happens to

3432-435: The combination n ^ ⋅ x → {\displaystyle {\hat {n}}\cdot {\vec {x}}} , any displacement in directions perpendicular to n ^ {\displaystyle {\hat {n}}} cannot affect the value of the field. Plane waves are often used to model electromagnetic waves far from a source. For electromagnetic plane waves,

3510-611: The dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instance electromagnetic , sound or water waves). The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation : v g = ∂ ω ∂ k {\displaystyle v_{\rm {g}}={\frac {\partial \omega }{\partial k}}} In almost all cases,

3588-435: The electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other. A standing wave, also known as a stationary wave , is a wave whose envelope remains in a constant position. This phenomenon arises as a result of interference between two waves traveling in opposite directions. The sum of two counter-propagating waves (of equal amplitude and frequency) creates

3666-412: The emergence process, but when there is interplay between injection of energy and dissipation there can arise a complex dynamic. Many natural patterns are shaped by this complexity, including vortex streets , other effects of turbulent flow such as meanders in rivers. or nonlinear interaction of the system Waves are disturbances that carry energy as they move. Mechanical waves propagate through

3744-428: The envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation . There are two velocities that are associated with waves, the phase velocity and the group velocity . Phase velocity is the rate at which the phase of the wave propagates in space : any given phase of

3822-495: The equation. This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve. For example, if F ( x , t ) {\displaystyle F(x,t)} is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation where Q ( p , f ) {\displaystyle Q(p,f)}

3900-445: The formula Here P ( x , t ) {\displaystyle P(x,t)} is some extra compression force that is being applied to the gas near x {\displaystyle x} by some external process, such as a loudspeaker or piston right next to p {\displaystyle p} . This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in

3978-435: The impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant wellbeing. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create

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4056-415: The local pressure and particle motion that propagate through the medium. Other examples of mechanical waves are seismic waves , gravity waves , surface waves and string vibrations . In an electromagnetic wave (such as light), coupling between the electric and magnetic fields sustains propagation of waves involving these fields according to Maxwell's equations . Electromagnetic waves can travel through

4134-426: The mathematical biologist James D. Murray and other scientists, described a mechanism that spontaneously creates spotted or striped patterns, for example in the skin of mammals or the plumage of birds: a reaction–diffusion system involving two counter-acting chemical mechanisms, one that activates and one that inhibits a development, such as of dark pigment in the skin. These spatiotemporal patterns slowly drift,

4212-589: The medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation 1 v 2 ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 . {\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}.} General solutions are based upon Duhamel's principle . The form or shape of F in d'Alembert's formula involves

4290-624: The medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. Wave propagation is any of the ways in which waves travel. With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves . Electromagnetic waves propagate in vacuum as well as in material media. Propagation of other wave types such as sound may occur only in

4368-443: The most general empirical patterns of the universe . Daniel Dennett 's notion of real patterns , discussed in his 1991 paper of the same name, provides an ontological framework aiming to discern the reality of patterns beyond mere human interpretation, by examining their predictive utility and the efficiency they provide in compressing information. For example, centre of gravity is a real pattern because it allows us to predict

4446-419: The motion of a drum skin , one can consider D {\displaystyle D} to be a disk (circle) on the plane R 2 {\displaystyle \mathbb {R} ^{2}} with center at the origin ( 0 , 0 ) {\displaystyle (0,0)} , and let F ( x , t ) {\displaystyle F(x,t)} be the vertical displacement of

4524-450: The movements of a bodies such as the earth around the sun, and it compresses all the information about all the particles in the sun and the earth that allows us to make those predictions. Some mathematical rule-patterns can be visualised, and among these are those that explain patterns in nature including the mathematics of symmetry, waves, meanders, and fractals. Fractals are mathematical patterns that are scale invariant. This means that

4602-413: The other hand, is always assumed to be a scalar ; that is, a real number. The value of F ( x , t ) {\displaystyle F(x,t)} can be any physical quantity of interest assigned to the point x {\displaystyle x} that may vary with time. For example, if F {\displaystyle F} represents the vibrations inside an elastic solid,

4680-609: The overall shape of the waves' amplitudes—modulation or envelope of the wave. A sine wave , sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function . In mechanics , as a linear motion over time, this is simple harmonic motion ; as rotation , it corresponds to uniform circular motion . Sine waves occur often in physics , including wind waves , sound waves, and light waves, such as monochromatic radiation . In engineering , signal processing , and mathematics , Fourier analysis decomposes general functions into

4758-476: The periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F ( x − v ( t + T )) = F ( x − vt ) provided vT = λ , so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ / v . The amplitude of

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4836-438: The propagation direction is also referred to as the wave's polarization , which can be an important attribute. A wave can be described just like a field, namely as a function F ( x , t ) {\displaystyle F(x,t)} where x {\displaystyle x} is a position and t {\displaystyle t} is a time. The value of x {\displaystyle x}

4914-407: The same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference

4992-441: The same, so the wave form will change over time and space. Sometimes one is interested in a single specific wave. More often, however, one needs to understand large set of possible waves; like all the ways that a drum skin can vibrate after being struck once with a drum stick , or all the possible radar echoes one could get from an airplane that may be approaching an airport . In some of those situations, one may describe such

5070-483: The search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict regularities in the world. In many areas of the decorative arts , from ceramics and textiles to wallpaper , "pattern" is used for an ornamental design that is manufactured, perhaps for many different shapes of object. In art and architecture, decorations or visual motifs may be combined and repeated to form patterns designed to have

5148-519: The shape of the pattern does not depend on how closely you look at it. Self-similarity is found in fractals. Examples of natural fractals are coast lines and tree shapes, which repeat their shape regardless of what magnification you view at. While self-similar patterns can appear indefinitely complex, the rules needed to describe or produce their formation can be simple (e.g. Lindenmayer systems describing tree shapes). In pattern theory , devised by Ulf Grenander , mathematicians attempt to describe

5226-425: The skeletons of silicoflagellates and sea urchins . Cracks form in materials to relieve stress: with 120 degree joints in elastic materials, but at 90 degrees in inelastic materials. Thus the pattern of cracks indicates whether the material is elastic or not. Cracking patterns are widespread in nature, for example in rocks, mud, tree bark and the glazes of old paintings and ceramics. Alan Turing , and later

5304-420: The skin at the point x {\displaystyle x} of D {\displaystyle D} and at time t {\displaystyle t} . Waves of the same type are often superposed and encountered simultaneously at a given point in space and time. The properties at that point are the sum of the properties of each component wave at that point. In general, the velocities are not

5382-423: The sound pressure inside a recorder that is playing a "pure" note is typically a standing wave , that can be written as The parameter A {\displaystyle A} defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); c {\displaystyle c} is the speed of sound; L {\displaystyle L}

5460-463: The temperature in a metal bar when it is initially heated at various temperatures at different points along its length, and then allowed to cool by itself in vacuum. In that case, instead of a scalar or vector, the parameter would have to be a function h {\displaystyle h} such that h ( x ) {\displaystyle h(x)} is the initial temperature at each point x {\displaystyle x} of

5538-415: The two-dimensional functions or, more generally, by d'Alembert's formula : u ( x , t ) = F ( x − v t ) + G ( x + v t ) . {\displaystyle u(x,t)=F(x-vt)+G(x+vt).} representing two component waveforms F {\displaystyle F} and G {\displaystyle G} traveling through

5616-418: The value of F ( x , t ) {\displaystyle F(x,t)} is usually a vector that gives the current displacement from x {\displaystyle x} of the material particles that would be at the point x {\displaystyle x} in the absence of vibration. For an electromagnetic wave, the value of F {\displaystyle F} can be

5694-485: The velocity vector of the fluid at the point x {\displaystyle x} , or any scalar property like pressure , temperature , or density . In a chemical reaction, F ( x , t ) {\displaystyle F(x,t)} could be the concentration of some substance in the neighborhood of point x {\displaystyle x} of the reaction medium. For any dimension d {\displaystyle d} (1, 2, or 3),

5772-441: The wave (for example, the crest ) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as v p = λ T . {\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.} Group velocity is a property of waves that have a defined envelope, measuring propagation through space (that is, phase velocity) of

5850-458: The wave varies in, and a wave profile describing how the wave varies as a function of the displacement along that direction ( n ^ ⋅ x → {\displaystyle {\hat {n}}\cdot {\vec {x}}} ) and time ( t {\displaystyle t} ). Since the wave profile only depends on the position x → {\displaystyle {\vec {x}}} in

5928-419: The wave's domain is then a subset D {\displaystyle D} of R d {\displaystyle \mathbb {R} ^{d}} , such that the function value F ( x , t ) {\displaystyle F(x,t)} is defined for any point x {\displaystyle x} in D {\displaystyle D} . For example, when describing

6006-438: The wavenumber k , but both are related through the dispersion relationship : ω = Ω ( k ) . {\displaystyle \omega =\Omega (k).} In the special case Ω( k ) = ck , with c a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed c . For instance electromagnetic waves in vacuum are non-dispersive. In case of other forms of

6084-729: The world in terms of patterns. The goal is to lay out the world in a more computationally friendly manner. In the broadest sense, any regularity that can be explained by a scientific theory is a pattern. As in mathematics, science can be taught as a set of patterns. A recent study from Aesthetics and Psychological Effects of Fractal Based Design suggested that fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on

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