A report on Wave equation

French scientist Jean-Baptiste le Rond d'Alembert discovered the wave equation in one space dimension.

1-d standing wave as a superposition of two waves traveling in opposite directions

Swiss mathematician and physicist Leonhard Euler (b. 1707) discovered the wave equation in three space dimensions.

Cut-away of spherical wavefronts, with a wavelength of 10 units, propagating from a point source.

Figure 1: Three consecutive mass points of the discrete model for a string

Second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves).

- Wave equation

14 related topics with Alpha

Wave

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Propagating dynamic disturbance of one or more quantities.

Example of biological waves expanding over the brain cortex, an example of spreading depolarizations.

Wavelength λ, can be measured between any two corresponding points on a waveform

Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t)

Sine, square, triangle and sawtooth waveforms.

Amplitude modulation can be achieved through f(x,t) = 1.00×sin(2π/0.10×(x−1.00×t)) and g(x,t) = 1.00×sin(2π/0.11×(x−1.00×t))only the resultant is visible to improve clarity of waveform.

Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.

The red square moves with the phase velocity, while the green circles propagate with the group velocity

A wave with the group and phase velocities going in different directions

Standing wave. The red dots represent the wave nodes

$Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism$

$Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.$

Identical waves from two sources undergoing interference. Observed at the bottom one sees 5 positions where the waves add in phase, but in between which they are out of phase and cancel.

Schematic of light being dispersed by a prism. Click to see animation.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.

Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

One-dimensional standing waves; the fundamental mode and the first 5 overtones.

A two-dimensional standing wave on a disk; this is the fundamental mode.

A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.

Waves are often described by a wave equation (standing wave field of two opposite waves) or a one-way wave equation for single wave propagation in a defined direction.

A visualisation of a solution to the two-dimensional heat equation with temperature represented by the vertical direction and color.

Partial differential equation

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Equation which imposes relations between the various partial derivatives of a multivariable function.

The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation, the heat equation, and the wave equation.

Shows a region where a differential equation is valid and the associated boundary values

Boundary value problem

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Differential equation together with a set of additional constraints, called the boundary conditions.

Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem with Dirichlet boundary conditions. Any solution function will both solve the heat equation, and fulfill the boundary conditions of a temperature of 0 K on the left boundary and a temperature of 273.15 K on the right boundary.

Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.

Standing wave

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Wave that oscillates in time but whose peak amplitude profile does not move in space.

Transient analysis of a damped traveling wave reflecting at a boundary

Standing wave in stationary medium. The red dots represent the wave nodes.

A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).

Electric force vector (E) and magnetic force vector (H) of a standing wave.

Standing waves in a string – the fundamental mode and the first 5 harmonics.

A standing wave on a circular membrane, an example of standing waves in two dimensions. This is the fundamental mode.

A higher harmonic standing wave on a disk with two nodal lines crossing at the center.

Equivalently, this boundary condition of the "free end" can be stated as ∂y/∂x = 0 at, which is in the form of the Sturm–Liouville formulation.

Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.

Superposition principle

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Sum of the responses that would have been caused by each stimulus individually.

Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: translation without rotation, and rotation without translation.

Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave.

two waves permeate without influencing each other

green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.

In any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system.

Separation of variables

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Any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

Solve linear first order differential equation by separation of variables.

The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.

Helmholtz equation

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[[Image:Helmholtz source.png|right|thumb|Two sources of radiation in the plane, given mathematically by a function

The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences.

Pastel portrait of d'Alembert by Maurice Quentin de La Tour, 1753

Jean le Rond d'Alembert

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French mathematician, mechanician, physicist, philosopher, and music theorist.

Nouvelles expériences sur la résistance des fluides

Portrait of Jean Le Rond d'Alembert, 1777, by Catherine Lusurier.

Front page of a 1758 copy of Traité de dynamique

D'Alembert's formula for obtaining solutions to the wave equation is named after him.

Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of relativity

Inhomogeneous electromagnetic wave equation

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Retarded spherical wave. The source of the wave occurs at time t. The wavefront moves away from the source as time increases for t > t. For advanced solutions, the wavefront moves backwards in time from the source t < t.

In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents.

One-way wave equation

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First-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity.

It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions.