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Topic: Wave equation

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Differential equations of mathematical physics

Partial differential equation

Wave

Transverse wave

Ordinary differential equation

Differential equations from outside physics

Examples of differential equations

Heat equation

Boundary value problem

Operator

Dirac equation

Finite difference

Finite element method

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	Wave equation - Wikipedia, the free encyclopedia
		The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves.
		The wave equation is the prototypical example of a hyperbolic partial differential equation.
		The basic wave equation is a linear differential equation which means that the amplitude of two waves interacting is simply the sum of the waves.
	en.wikipedia.org /wiki/Wave_equation (1765 words)

	What is a Wave?
		In the case of a sound wave moving from the church choir to the pews, the medium through which the sound wave travels is the air in the room.
		And in the case of the stadium wave, the medium through which the stadium wave travels is the fans who are in the stadium.
		When a wave is present in a medium (that is, when there is a disturbance moving through a medium), the individual particles of the medium are only temporarily displaced from their rest position.
	www.glenbrook.k12.il.us /gbssci/phys/Class/waves/u10l1b.html (1648 words)

	The Physics Classroom
		As was discussed in Lesson 1, a wave is produced when a vibrating source displaces the first particle of a medium.
		It also illustrates the principle that wave speed is dependent upon medium properties and independent of wave properties.
		A transverse wave is found to have a distance of 4 cm from a trough to a crest, a frequency of 12 Hz, and a distance of 5 cm from a crest to the nearest trough.
	www.physicsclassroom.com /Class/waves/U10L2e.html (804 words)

	Quantum Mechanics 1
		When Born showed that the solution of Schrodinger's wave equation could be interpreted as a non-observable object used only as an intermediary to calculate relative probabilities of the various allowed outcomes of an experiment, and when Bohr and Heisenberg solidified this interpretation (the so-called "Copenhagen interpretation") Schrödinger objected strongly.
		Whatever the correct wave equation might be, it should contain the relations between wave and particle properties that seem to apply for particles (or waves) moving freely through space.
		Solutions of the wave equation in the e-m case are the E and B fields of the wave.
	www.phy.duke.edu /courses/100/lectures/QM1/QM1.html (1249 words)

	The Wave Equation and Permutation of Rays
		Thus the wave equation simply expresses the requirement that the sum of the values of the wave function just before and just after a given event equals the sum of the values on either side of the event.
		It's interesting that partial differential equations are often regarded as characteristic of local processes, and the wave equation is the archetype partial differential equation, but we find that the basic relationships can just as well be expressed in non-local form.
		In full 3+1 dimensional spacetime the spherically symmetrical wave equation can be reduced to the same form as the 1+1 dimensional equation, except that the space wave function is normalized by r, and the parameter x is replaced with the radius r, so each "light ray" actually represents a sequence of expanding and converging shells.
	www.mathpages.com /home/kmath251/kmath251.htm (1305 words)

	PlanetMath: wave equation
		The wave equation is a partial differential equation which describes all kinds of waves.
		The general solution of the one-dimensional wave equation can be obtained by a change of variables
		This is version 4 of wave equation, born on 2002-11-21, modified 2003-05-21.
	planetmath.org /encyclopedia/WaveEquation.html (141 words)

	Wave Functions
		In fact, most physical waves, such as those of sound in air or waves on the surface of water, or an electromagnetic wave moving along a transmission line, or light propagating down a fibre, are not precisely described by an equation of this type.
		This is quite accurately true for electromagnetic waves, including light, propagating in vacuum, and very close to true for sound waves of low intensity propagating in a homogeneous isotropic medium, for example, which explains the great utility of the simple wave equation.
		Waves on the surface of deep water are also strongly dispersive, the phase velocity being proportional to the square root of the wavelength.
	www.du.edu /~jcalvert/waves/wavefun.htm (5715 words)

	Plane and Spherical Wave Spectra
		These waves are solutions to the wave equation in cylindrical coordinates, and could be obtained directly in that manner.
		The reason is that k appears in the solution of the wave equation in spherical coordinates as a single value, so integration similar to the plane wave spectrum cannot be used.
		Equation (W12) is a rigorous solution of the wave equation for any value of N. The interesting feature of equation (W12) that we need is: within a sphere of radius N/k it produces a virtually perfect plane wave.
	www.silcom.com /~aludwig/Physics/Exact_piston/Wave_spectra.htm (1316 words)

	Wave equation Info - Encyclopedia WikiWhat.com (Site not responding. Last check: 2007-08-19)
		The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves.
		For a sound wave in air this is about 300 m/s, and we refer to the speed of sound.
		The Fourier transform breaks up a wave into sinusodal components and is useful for analyzing the wave equation.
	www.wikiwhat.com /encyclopedia/w/wa/wave_equation.html (347 words)

	Wave Equation
		where v is the phase velocity of the wave and y represents the variable which is changing as the wave passes.
		This is the form of the wave equation which applies to a stretched string or a plane electromagnetic wave.
		The wave equation for a wave in an ideal string can be obtained by applying Newton's 2nd Law to an infinitesmal segment of a string.
	hyperphysics.phy-astr.gsu.edu /hbase/waves/waveq.html (265 words)

	Wave Equations
		The purpose of this section is to give a plausible derivation of such an equation by examining how the Maxwell wave equation works for a single-particle (photon) wave, and constructing parallel equations for particles which, unlike photons, have nonzero rest mass.
		The wave equation operator applied to the plane wave describing the particle propagation yields the energy-momentum relationship for the particle.
		The effect of a potential on a de Broglie wave was considered by Sommerfeld in an attempt to generalize the rather restrictive conditions in Bohr's model of the atom.
	galileo.phys.virginia.edu /classes/252/wave_equations.html (1267 words)

	Discretizing the wave equation
		The equation says the rate at which the level is accelerating in time at a give point is proportional to the sum of the rates at which the level is accelerating across each dimension in space at that point.
		The wave equation is the universal equation of physics.
		It is the relativistic Schrödinger equation that describes the quantum mechanical evolution of the wave function of a single particle with zero rest mass
	www.mtnmath.com /whatrh/node66.html (708 words)

	Chapter 7: Wave Mechanics and Wave Particle Duality (Site not responding. Last check: 2007-08-19)
		This wave equation describes how a wave associated with an electron or other subatomic particle varies in space and time as the particle moves under various forces.
		He worked out the exact solutions of the wave equation for the hydrogen atom, and the results perfectly agreed with the known energy levels of these atoms, seemingly without any of the complications and metaphysical speculations associated with the uncertainty principle.
		This indicated that the simple interpretation of the wave equation as a description of the physical matter waves in ordinary space, as was originally assumed by Schrödinger himself, is incorrect.
	www.faqs.org /docs/qp/chap07.html (827 words)

	What is the Schrodinger equation, and how is it used?
		The eigenvalues of the wave equation were shown to be equal to the energy levels of the quantum mechanical system, and the best test of the equation was when it was used to solve for the energy levels of the Hydrogen atom, and the energy levels were found to be in accord with Rydberg's Law.
		Schrodinger's equation shows all of the wave like properties of matter and was one of greatest achievements of 20th century science.
		Answered by: Simon Hooks, Physics A-Level Student, Gosport, UK The Schrodinger equation is the name of the basic non-relativistic wave equation used in one version of quantum mechanics to describe the behaviour of a particle in a field of force.
	www.physlink.com /Education/AskExperts/ae329.cfm (578 words)

	The Schrodinger Wave Equation (Site not responding. Last check: 2007-08-19)
		Further, to illustrate the equation, the particle in a rigid box model is examined.
		The troubling dichotomy of waves and particle of light quanta cannot be escaped.
		It is replete throughout nature: in packets of light quanta, photons; in subatomic particles such as electrons and protons; in the flight of a baseball; in the orbit of a planet.
	online.redwoods.cc.ca.us /DEPTS/science/chem/storage/Schrod (1026 words)

	ECE 371 Lecture 2: The Wave Equation and the Optical Field
		The second equation is Ampere's Circuital Law and expresses the experimental fact that a current or time varying electric field induces a magnetic field.
		Since a wave disturbance propagates without dissipation, there are no spatially bounded solutions to the wave equation in source and absorber free space.
		Plane waves are unbounded and constant in two dimensions and propagate along a third dimension.
	www.disp.duke.edu /~dbrady/ece371/notes/l2 (1365 words)

	Wave-equation for sound propagation in a fluid
		Sound waves are traveling pressure waves in a fluid (gas or a liquid).
		Corresponding to the pressure wave is a displacement wave s(x-vt) of atoms from where they would be if there had been no sound propagating.
		The relationship between the displacement wave and the deviation from average pressure is a characteristic property of the fluid:
	www.pha.jhu.edu /~broholm/l28/node4.html (312 words)

	Wave Equation Studies
		The primary objective of dynamic monitoring with the PDA is to collect supporting data for the development of a wave equation based field acceptance criteria for use by Office of Structure Construction field personnel during the installation of production piles.
		The primary advantage of the wave equation based field acceptance criteria is its site-specific and hammer-specific applicability and treatment of time dependent capacity relationships.
		In the wave equation based field acceptance criteria, pile capacity is determined as a function of blow count, hammer stroke, and time.
	www.dot.ca.gov /hq/esc/geotech/ft/wave.htm (317 words)

	Electromagnetic Waves (Site not responding. Last check: 2007-08-19)
		The wave equation for a plane electric wave traveling in the x direction in space is
		The wave equation for electromagnetic waves arises from Maxwell's equations.
		The rate of energy transport S is perpendicular to both E and B and in the direction of propagation of the wave.
	hyperphysics.phy-astr.gsu.edu /hbase/waves/emwv.html (256 words)

	Schrodinger's Equation
		The wave description of de Broglie is not adequate to explain the properties of the electron under different types of external potential fields.
		This very important equation is a result of conservation of energy and the relations between total energy and w and between the momentum and the wave number, k.
		The last equation reduces to two equations: one for the real part and one for the imaginary part.
	www.missioncollege.org /depts/physics/P4poe/P4D/Schrodinger.htm (1139 words)

	wave equation (Site not responding. Last check: 2007-08-19)
		E both E and B satisfy the damped wave equation.
		For waves propagating across a dielectric-dielectric boundary or from a dielectric into a conductor we use
		A linear polarized wave has p-polarization if E lies in the plane of incidence, and it has s-polarization if E is perpendicular to the plane of incidence.
	electron6.phys.utk.edu /phys594/Tools/e&m/summary/waves/waves.html (178 words)

	Derivation of the acoustic wave equation (Site not responding. Last check: 2007-08-19)
		The acoustic wave equation describes sound waves in a liquid or gas.
		Later an equation like (20) will be developed by considering only the geometrical behavior of waves.
		Try again assuming space variability, and the resulting equation is still a differential equation, not an algebraic equation like (20).
	sepwww.stanford.edu /public/docs/iei/xrf/paper_html/node13.html (442 words)

	The Quantum Wave Equation
		The initial complex phase of a quantum wave plays the same role as the initial transverse velocity for a wave on a string.
		The only parameters in the final form of the equation are the width L of the well and the ground state period.
		The direct numerical solution of the Schrodinger equation, even with 300 time steps between each of the plotted 640 frames, cannot be expected to give results as accurate as the FFSS over an extended period of time.
	www.kw.igs.net /~jackord/bp/n4.html (1510 words)

	Series Solutions of the Wave Equation
		The spherically symmetrical wave equation in N space dimensions and 1 time dimension is
		The second column is essentially the rate of change of the wave function at the initial instant as a function of radial position.
		The form of the recursive relation implies that we can fully specify the entire wave function (for the case p = q = 0) simply by specifying the value at the origin for all time.
	www.mathpages.com /home/kmath247/kmath247.htm (1160 words)

	The Wave Equation
		As the wavelength of a wave in a uniform medium increases, its speed will _____.
		As the wavelength of a wave in a uniform medium increases, its frequency will _____.
		There are never any wave crests between the boats.
	www.glenbrook.k12.il.us /gbssci/phys/Class/waves/u10l2e.html (750 words)

	The Schrodinger Wave Equation -- Page 2 (Site not responding. Last check: 2007-08-19)
		What came of that illicit vacation to the Swiss Alps was a model that was not derived from any other, a model that can be called an intuitive guess, a leap of imagination, a model that is astonishingly accurate.
		Therefore, the wave equation corresponding to the particle in the box is given by
		Gasping for breath, we at last have unearthed the solution to the wave equation for the particle of the nth quantum state in the rigid box.
	online.redwoods.cc.ca.us /depts/science/chem/storage/Schrod/page2.htm (695 words)

	The one dimensional wave equation (Site not responding. Last check: 2007-08-19)
		F(x-ct) represents a wave of shape F(x) (at time t=0) moving in the direction of increasing x at constant speed c.
		That is, F(x-ct) represents the same shape as F(x), but with the origin moved to x=ct, ie the same shape as F(x) translated by ct units in the positive x direction.
		E(x+ct) represents a wave of shape E(x) (at time t=0) moving in the direction of decreasing x at constant speed c.
	www.geocities.com /aboutwaves/node71.html (160 words)

	Wave equation:
		The wave equation expressed in polar coordinates with no damping is given by
		For simplicity, the initial conditions are assumed independent of theta, and the equation becomes
		The vibration of a guitar string is solved by the 1-dimensional wave equation.
	amath.colorado.edu /courses/4350/2002fall/projects/present12.htm (764 words)

	Solution of the inhomogeneous wave equation
		We are able to find solutions of the inhomogeneous wave equation (2.103) which propagate backward in time because this equation is time symmetric (i.e.
		In conclusion, the most general solution of the inhomogeneous wave equation (2.103) which satisfies sensible boundary conditions at infinity and is consistent with causality is
		(2.122) represents the unique solution of the wave equation (2.103) which is consistent with causality.
	farside.ph.utexas.edu /teaching/jk1/lectures/node19.html (646 words)

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