A wave is a disturbance that propagates through
spaceand time, usually with transference of 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 energyfrom one point to another, often with little or no permanent displacement of the particles of the medium (that is, with little or no associated mass transport); instead there are oscillations around almost fixed locations.
Agreeing on a single, all-encompassing definition for the term "wave" is non-trivial. A
vibrationcan be defined as a "back-and-forth" motion around a point m around a reference value. However, defining the necessary and sufficient characteristics that qualify a phenomenonto be called a wave is, at least, flexible. The term is often understood intuitively as the transport of disturbances in space, not associated with motion of the medium occupying this space as a whole. In a wave, the energyof a vibrationis moving away from the source in the form of a disturbance within the surrounding medium (Hall, 1980: 8). However, this notion is problematic for a standing wave(for example, a wave on a string), where energyis moving in both directions equally, or for electromagnetic / light waves in a vacuum, where the concept of medium does not apply.
For such reasons, wave theory represents a peculiar branch of
physicsthat is concerned with the properties of wave processes independently from their physical origin (Ostrovsky and Potapov, 1999). The peculiarity lies in the fact that this independence from physical origin is accompanied by a heavy reliance on origin when describing any specific instance of a wave process. For example, acousticsis distinguished from opticsin that sound waves are related to a mechanical rather than an electromagnetic wave-like transfer / transformation of vibratory energy. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as opposed to optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved (for example, in the case of air: vortices, radiation pressure, shock waves, etc., in the case of solids: Rayleigh waves, dispersion, etc., and so on).
Other properties, however, although they are usually described in an origin-specific manner, may be generalized to all waves. For example, based on the mechanical origin of acoustic waves there can be a moving disturbance in space-time if and only if the medium involved is neither infinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly "bound", then they would all vibrate as one, with no delay in the transmission of the vibration and therefore no wave motion (or rather infinitely fast wave motion). On the other hand, if all the parts were independent, then there would not be any transmission of the vibration and again, no wave motion (or rather infinitely slow wave motion). Although the above statements are meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times.
Similarly, wave processes revealed from the study of wave phenomena with origins different from that of sound waves can be equally significant to the understanding of sound phenomena. A relevant example is Young's principle of interference (Young, 1802, in Hunt, 1978: 132). This principle was first introduced in Young's study of
lightand, within some specific contexts (for example, scatteringof sound by sound), is still a researched area in the study of sound.
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 = In deep water.
B = In shallow water. The elliptical movement of a surface particle becomes flatter 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 behavior under a number of standard situations. All waves can experience the following:
*Reflection - wave direction change from hitting a reflective surface
Refraction- wave direction change from entering a new medium
Diffraction- bending of waves as they interact with obstacles in their path, most pronounced for wavelengths on the order of the diffracting object size
Interference- superposition of two waves that come into contact with each other (collide)
*Dispersion - wave splitting up by frequency
Rectilinear propagation- the movement of light waves in a straight line
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 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.
traffic(that is, propagation of different densities of motor vehicles, "etc".) — these can be modelled as kinematic waves, as first presented by Sir M. J. Lighthill
Seismic waves in earthquakes, of which there are three types, called S, P, and L.
Gravitational waves, which are fluctuationsin the curvature of spacetime 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 effect.
From a mathematical point of view, the most primitive or fundamental wave is
harmonic(sinusoidal) wave which is described by the equation where is the amplitudeof a wave - a measure of the maximum disturbance in the medium during one wave cycle (the maximum distance from the highest point of the crest to the equilibrium). 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.
wavelength(denoted as ) is the distance between two sequential crests (or troughs). This generally is measured in meters; it is also commonly measured in nanometers for the optical part of the electromagnetic spectrum.
wavenumbercan be associated with the wavelength by the relation
simple harmonic motion.] The period is the time for one complete cycle for an oscillation of a wave. The frequency(also frequently denoted as ) is how many periods per unit time (for example one second) and is measured in hertz. These are related by:
In other words, the frequency and period of a wave are reciprocals of each other.
angular frequency" represents the frequency in terms of radians per second. It is related to the frequency by
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
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
The wave equation
The wave equation is a
differential equationthat 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 traveling down a rope along the "x"-axis with velocity and amplitude (which generally depends on both "x" and "t"), the wave equation is
In three dimensions, this becomes
where is the
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
This can be viewed as two pulses traveling 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.
Schrödinger equationdescribes the wave-like behavior 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.
Simple wave or a "traveling wave", also sometimes called a "progressive wave" is a disturbance that varies both with time and distance in the following way:
where is the amplitude envelope of the wave, is the "wave number" and is the "phase". The
phase velocity"v""p" of this wave is given by
where is the "
wavelength" of the wave.
A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of
interferencebetween two waves traveling in opposite directions.
The "sum" of two counter-propagating waves (of equal amplitude and frequency) creates 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
violinstring is displaced, longitudinal waves propagate out to where the string is held in place at the bridge and the "nut", where upon the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphaseand cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves "enhance" each other maximally. There is on averageno net propagation of energy.
Acoustic resonance, Helmholtz resonator, and organ pipe
Propagation through strings
The speed of a wave traveling along a
vibrating string(v) is directly proportional to the square root of the tension ("T") over the linear density(μ):
The medium that carries a wave is called a "transmission medium". It can be classified into one or more of the following categories:
*A "bounded medium" if it is finite in extent, otherwise an "unbounded medium".
*A "linear medium" if the amplitudes of different waves at any particular point in the medium can be added.
*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.
List of wave topics
Ocean surface wave
Rogue wave (oceanography)
Shallow water equations
*Campbell, M. and Greated, C. (1987). "The Musician’s Guide to Acoustics". New York: Schirmer Books.
*cite book | first = A.P. | last = French | title = Vibrations and Waves (M.I.T. Introductory physics series) | publisher = Nelson Thornes | year = 1971 | isbn = 0-393-09936-9 | oclc = 163810889
*Hall, D. E. (1980). "Musical Acoustics: An Introduction". Belmont, California: Wadsworth Publishing Company.
*Hunt, F. V. (1978). "Origins in Acoustics". New York: Acoustical Society of America Press, (1992).
*Ostrovsky, L. A. and Potapov, A. S. (1999). "Modulated Waves, Theory and Applications". Baltimore: The Johns Hopkins University Press.
* [http://www.acousticslab.org/papers/diss.htm Vassilakis, P.N. (2001)] . "Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance". Doctoral Dissertation. University of California, Los Angeles.
* [http://kestrel.nmt.edu/~raymond/classes/ph13xbook/node1.html A Radically Modern Approach to Introductory Physics - an online physics textbook that starts with waves rather than mechanics]
* [http://resonanceswavesandfields.blogspot.com/2007/08/true-waves.html Interactive Visual Representation of Waves]
* [http://www.scienceaid.co.uk/physics/waves/properties.html Science Aid: Wave properties - Concise guide aimed at teens]
* [http://www.phy.hk/wiki/englishhtm/Diffraction.htm Simulation of diffraction of water wave passing through a gap]
* [http://www.phy.hk/wiki/englishhtm/Interference.htm Simulation of interference of water waves]
* [http://www.phy.hk/wiki/englishhtm/Lwave.htm Simulation of longitudinal traveling wave]
* [http://www.phy.hk/wiki/englishhtm/StatWave.htm Simulation of stationary wave on a string]
* [http://www.phy.hk/wiki/englishhtm/TwaveA.htm Simulation of transverse traveling wave]
* [http://www.acoustics.salford.ac.uk/feschools/ Sounds Amazing - AS and A-Level learning resource for sound and waves]
* [http://www.lightandmatter.com/html_books/3vw/ch03/ch03.html Vibrations and Waves - an online textbook]
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