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Wave

2007 Schools Wikipedia Selection. Related subjects: General Physics

   A wave is a disturbance that propagates through space or spacetime,
   often transferring energy. While a mechanical wave exists in a medium
   (which on deformation is capable of producing elastic restoring
   forces), waves of electromagnetic radiation (and probably gravitational
   radiation) can travel through vacuum, that is, without a medium. Waves
   travel and transfer energy from one point to another, with little or no
   permanent displacement of the particles of the medium (there is little
   or no associated mass transport); instead there are oscillations around
   fixed positions.

   For many years, scientists have been trying to work out the problem of
   energy transfer from one place to another - especially sound and light
   energy.

Characteristics

   Surface waves in water
   Enlarge
   Surface waves in water

   Periodic waves are characterized by crests (highs) and troughs (lows),
   and may usually be categorized as either longitudinal or transverse.
   Transverse waves are those with vibrations perpendicular to the
   direction of the propagation of the wave; examples include waves on a
   string and electromagnetic waves. Longitudinal waves are those with
   vibrations parallel to the direction of the propagation of the wave;
   examples include most sound waves.

   When an object bobs up and down on a ripple in a pond, it experiences
   an orbital trajectory because ripples are not simple transverse
   sinusoidal waves.
   A = At deep water. B = At shallow water. The circular movement of a
   surface particle becomes elliptical with decreasing depth. 1 =
   Progression of wave 2 = Crest 3 = Trough
   Enlarge
   A = At deep water.
   B = At shallow water. The circular movement of a surface particle
   becomes elliptical with decreasing depth.
   1 = Progression of wave
   2 = Crest
   3 = Trough

   Ripples on the surface of a pond are actually a combination of
   transverse and longitudinal waves; therefore, the points on the surface
   follow orbital paths.

   All waves have common behaviour under a number of standard situations.
   All waves can experience the following:
     * Reflection – the change of direction of waves, due to hitting a
       reflective surface.
     * Refraction – the change of direction of waves due to them entering
       a new medium.
     * Diffraction – the circular spreading of waves that happens when the
       distance between waves move through an opening of equal distance.
     * Interference – the superposition of two waves that come into
       contact with each other.
     * Dispersion – the splitting up of waves by frequency.
     * Rectilinear propagation – the movement of waves in straight lines.

Polarization

   A wave is polarized if it can only oscillate in one direction. The
   polarization of a transverse wave describes the direction of
   oscillation, in the plane perpendicular to the direction of travel.
   Longitudinal waves such as sound waves do not exhibit polarization,
   because for these waves the direction of oscillation is along the
   direction of travel. A wave can be polarized by using a polarizing
   filter.

Examples

   Examples of waves include:
     * Ocean surface waves, which are perturbations that propagate through
       water.
     * Radio waves, microwaves, infrared rays, visible light, ultraviolet
       rays, x-rays, and gamma rays make up electromagnetic radiation. In
       this case, propagation is possible without a medium, through
       vacuum. These electromagnetic waves travel at 299,792,458 m/s in a
       vacuum.
     * Sound - a mechanical wave that propagates through air, liquid or
       solids.
     * Seismic waves in earthquakes, of which there are three types,
       called S, P, and L.
     * Gravitational waves, which are fluctuations in the gravitational
       field predicted by general Relativity. These waves are nonlinear,
       and have yet to be observed empirically.
     * Inertial waves, which occur in rotating fluids and are restored by
       the Coriolis force.

Mathematical description

   Waves can be described mathematically using a series of parameters.

   The amplitude of a wave (commonly notated as A, or another letter) is a
   measure of the maximum disturbance in the medium during one wave cycle.
   In the illustration to the right, this is the maximum vertical distance
   between the baseline and the wave. The units of the amplitude depend on
   the type of wave — waves on a string have an amplitude expressed as a
   distance (meters), sound waves as pressure (pascals) and
   electromagnetic waves as the amplitude of the electric field
   (volts/meter). The amplitude may be constant (in which case the wave is
   a c.w. or continuous wave), or may vary with time and/or position. The
   form of the variation of amplitude is called the envelope of the wave.

   The wavelength (denoted as λ) is the distance between two sequential
   crests (or troughs). This generally has the unit of metres; it is also
   commonly measured in nanometres for the optical part of the
   electromagnetic spectrum.

   A wavenumber k can be associated with the wavelength by the relation

          k = \frac{2 \pi}{\lambda} .

   Waves can be represented by simple harmonic motion.
   Enlarge
   Waves can be represented by simple harmonic motion.

   The period T is the time for one complete cycle for an oscillation of a
   wave. The frequency f (also frequently denoted as ν) is how many
   periods per unit time (for example one second) and is measured in
   hertz. These are related by:

          f=\frac{1}{T} .

   In other words, the frequency and period of a wave are reciprocals of
   each other.

   The angular frequency ω represents the frequency in terms of radians
   per second. It is related to the frequency by:

          \omega = 2 \pi f = \frac{2 \pi}{T} .

   There are two velocities that are associated with waves. The first is
   the phase velocity, which gives the rate at which the wave propagates,
   is given by

          v_p = \frac{\omega}{k} .

   The second is the group velocity, which gives the velocity at which
   variations in the shape of the wave's amplitude propagate through
   space. This is the rate at which information can be transmitted by the
   wave. It is given by

          v_g = \frac{\partial \omega}{\partial k}

The wave equation

   The wave equation is a differential equation that describes the
   evolution of a harmonic wave over time. The equation has slightly
   different forms depending on how the wave is transmitted, and the
   medium it is traveling through. Considering a one-dimensional wave that
   is travelling down a rope along the x-axis with velocity v and
   amplitude u (which generally depends on both x and t), the wave
   equation is

          \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2
          u}{\partial x^2}. \

   In three dimensions, this becomes

          \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2} = \nabla^2 u ,

   where \nabla^2 is the Laplacian.

   The velocity v will depend on both the type of wave and the medium
   through which it is being transmitted.

   A general solution for the wave equation in one dimension was given by
   d'Alembert. It is

          u(x,t)=F(x-vt)+G(x+vt). \

   This can be viewed as two pulses travelling down the rope in opposite
   directions; F in the +x direction, and G in the -x direction. If we
   substitute for x above, replacing it with directions x, y, z, we then
   can describe a wave propagating in three dimensions.

   The Schrödinger equation describes the wave-like behaviour of particles
   in quantum mechanics. Solutions of this equation are wave functions
   which can be used to describe the probability density of a particle.
   Quantum mechanics also describes particle properties that other waves,
   such as light and sound, have on the atomic scale and below.

Traveling waves

   Waves that remain in one place are called standing waves - e.g.
   vibrations on a violin string. Waves that are moving are called
   traveling waves, and have a disturbance that varies both with time t
   and distance z. This can be expressed mathematically as:

          u = A(z,t) \cos (\omega t - kz + \phi)\,

   where A(z,t) is the amplitude envelope of the wave, k is the wave
   number and φ is the phase. The phase velocity v[p] of this wave is
   given by:

          v_p = \frac{\omega}{k}= \lambda f,

   where λ is the wavelength of the wave.

Propagation through strings

   The speed of a wave traveling along a string (v) is directly
   proportional to the square root of the tension (T) over the linear
   density (μ):

          v=\sqrt{\frac{T}{\mu}}.

Transmission medium

   The medium that carries a wave is called a transmission medium. It can
   be classified into one or more of the following categories:
     * A linear medium if the amplitudes of different waves at any
       particular point in the medium can be added.
     * A bounded medium if it is finite in extent, otherwise an unbounded
       medium.
     * A uniform medium if its physical properties are unchanged at
       different locations in space.
     * An isotropic medium if its physical properties are the same in
       different directions.

   Retrieved from " http://en.wikipedia.org/wiki/Wave"
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   with only minor checks and changes (see www.wikipedia.org for details
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