|
|
| Line 1: |
Line 1: |
| {{about|waves in the scientific sense|waves on the surface of the ocean or lakes|Wind wave|other uses of wave or waves|Wave (disambiguation)}}
| | Im Randall and was born on 16 November 1978. My hobbies are Petal collecting and pressing and Aircraft spotting.<br><br>my site; [http://www.200machinery.com/%E0%B8%A3%E0%B8%96%E0%B9%81%E0%B8%9A%E0%B8%84%E0%B9%82%E0%B8%AE/ รถแบคโฮ] |
| | |
| [[File:2006-01-14 Surface waves.jpg|thumb|right|300px|[[Surface wave]]s in [[water]]]]
| |
| | |
| In [[physics]], a '''wave''' is a disturbance or oscillation that travels through space and matter, accompanied by a transfer of [[energy]]. '''Wave motion''' transfers [[energy]] from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist, instead, of [[oscillation]]s or vibrations around almost fixed locations. Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave.
| |
| | |
| There are two main types of waves. [[Mechanical wave]]s propagate through a medium, and the substance of this medium is deformed. The deformation reverses itself owing to [[restoring force]]s resulting from its deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When air molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave.
| |
| | |
| The second main type of wave, [[electromagnetic wave]]s, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields generated by charged particles, and can therefore travel through a [[vacuum]]. These types of waves vary in wavelength, and include [[radio wave]]s, [[microwave]]s, [[infrared radiation]], [[visible light]], [[ultraviolet radiation]], [[X-ray]]s, and [[gamma ray]]s.
| |
| | |
| Further, the behavior of particles in [[quantum mechanics]] are described by waves, and researchers believe that [[gravitational radiation|gravitational waves]] also travel through space, although gravitational waves have never been directly detected.
| |
| | |
| A wave can be [[transverse wave|transverse]] or [[longitudinal wave|longitudinal]] depending on the direction of its oscillation. Transverse waves occur when a disturbance creates oscillations perpendicular (at right angles) to the propagation (the direction of energy transfer). Longitudinal waves occur when the oscillations are [[parallel (geometry)|parallel]] to the direction of propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse.
| |
| | |
| == General features ==
| |
| A single, all-encompassing definition for the term ''wave'' is not straightforward. A [[vibration]] can be defined as a ''back-and-forth'' motion around a reference value. However, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a [[phenomenon]] to be called a ''wave'' results in a fuzzy border line.
| |
| | |
| The term ''wave'' is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the [[energy]] of a [[vibration]] is moving away from the source in the form of a disturbance within the surrounding medium {{Harv|Hall|1980| p=8}}. However, this notion is problematic for a [[standing wave]] (for example, a wave on a string), where [[energy]] is moving in both directions equally, or for electromagnetic (e.g., light) waves in a [[vacuum]], where the concept of medium does not apply and interaction with a target is the key to wave detection and practical applications. There are [[water waves]] on the ocean surface; [[gamma waves]] and [[light waves]] emitted by the Sun; [[microwaves]] used in microwave ovens and in [[radar]] equipment; [[radio waves]] broadcast by radio stations; and [[sound waves]] generated by radio receivers, telephone handsets and living creatures (as voices), to mention only a few wave phenomena.
| |
| | |
| It may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, [[acoustics]] is distinguished from [[optics]] in that sound waves are related to a mechanical rather than an electromagnetic wave transfer caused by [[vibration]]. Concepts such as [[mass]], [[momentum]], [[inertia]], or [[Elasticity (physics)|elasticity]], become therefore crucial in describing acoustic (as distinct from 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: [[vortex|vortices]], [[radiation pressure]], [[shock waves]] etc.; in the case of solids: [[Rayleigh waves]], [[Dispersion (chemistry)|dispersion]]; and so on.
| |
| | |
| Other properties, however, although usually described in terms of origin, may be generalized to all waves. For such reasons, wave theory represents a particular branch of [[physics]] that is concerned with the properties of wave processes independently of their physical origin.<ref name=Ostrovsky>
| |
| | |
| {{cite book |title = Modulated waves: theory and application |url = http://www.amazon.com/gp/product/0801873258 |author = Lev A. Ostrovsky & Alexander I. Potapov |publisher = Johns Hopkins University Press |isbn = 0-8018-7325-8 |year = 2002 }}
| |
| | |
| </ref> For example, based on the mechanical origin of acoustic waves, a moving disturbance in space–time can exist 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. 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. 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 (waves)|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 waves other than sound waves can be significant to the understanding of sound phenomena. A relevant example is [[Thomas Young (scientist)|Thomas Young]]'s principle of interference (Young, 1802, in {{Harvnb|Hunt|1992| p=132}}). This principle was first introduced in Young's study of [[light]] and, within some specific contexts (for example, [[scattering]] of sound by sound), is still a researched area in the study of sound.
| |
| | |
| == Mathematical description of one-dimensional waves ==
| |
| | |
| === Wave equation ===
| |
| {{Main|Wave equation|D'Alembert's formula}}
| |
| | |
| Consider a traveling [[transverse wave]] (which may be a [[pulse (physics)|pulse]]) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling
| |
| | |
| [[File:Nonsinusoidal wavelength.JPG|thumb|right|200 px|Wavelength ''λ'', can be measured between any two corresponding points on a waveform]]
| |
| | |
| *in the <math>x</math> direction in space. E.g., let the positive <math>x</math> direction be to the right, and the negative <math>x</math> direction be to the left.
| |
| *with constant [[amplitude]] <math>u</math>
| |
| *with constant velocity <math>v</math>, where <math>v</math> is
| |
| **independent of [[wavelength]] (no [[dispersion relation|dispersion]])
| |
| **independent of amplitude ([[linear]] media, not [[nonlinear]]).<ref name=Helbig>
| |
| | |
| {{cite book |title = Seismic waves and rays in elastic media |url = http://books.google.com/?id=s7bp6ezoRhcC&pg=PA134 |pages = 131 ''ff'' |author = Michael A. Slawinski |chapter = Wave equations |isbn = 0-08-043930-6 |year = 2003 |publisher = Elsevier }}
| |
| | |
| </ref>
| |
| *with constant [[waveform]], or shape
| |
| | |
| This wave can then be described by the two-dimensional functions
| |
| | |
| : <math>u(x,t) = F(x - v \ t)</math> (waveform <math>F</math> traveling to the right)
| |
| : <math>u(x,t) = G(x + v \ t)</math> (waveform <math>G</math> traveling to the left)
| |
| | |
| or, more generally, by [[d'Alembert's formula]]:<ref name=Graaf>
| |
| | |
| {{cite book |title = Wave motion in elastic solids |author = Karl F Graaf |edition = Reprint of Oxford 1975 |publisher = Dover |year = 1991 |url = http://books.google.com/?id=5cZFRwLuhdQC&printsec=frontcover |pages = 13–14 |isbn = 978-0-486-66745-4 }}
| |
| | |
| </ref>
| |
| | |
| :<math>
| |
| u(x,t) = F(x-vt) + G(x+vt). \,
| |
| </math>
| |
| | |
| representing two component waveforms <math>F</math> and <math>G</math> traveling through the medium in opposite directions. A generalized representation of this wave can be obtained<ref>
| |
| | |
| For an example derivation, see the steps leading up to eq. (17) in {{cite web |url = http://prism.texarkanacollege.edu/physicsjournal/wave-eq.html |title = Kinematic Derivation of the Wave Equation |author = Francis Redfern |work = Physics Journal }}
| |
| | |
| </ref> as the [[partial differential equation]]
| |
| | |
| :<math>
| |
| \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}. \,
| |
| </math>
| |
| | |
| General solutions are based upon [[Duhamel's principle]].<ref name=Struwe>
| |
| | |
| {{cite book |title = Geometric wave equations |author = Jalal M. Ihsan Shatah, Michael Struwe |url = http://books.google.com/?id=zsasG2axbSoC&pg=PA37 |chapter = The linear wave equation |pages = 37 ''ff'' |isbn = 0-8218-2749-9 |year = 2000 |publisher = American Mathematical Society Bookstore }}
| |
| | |
| </ref>
| |
| | |
| === Wave forms ===
| |
| [[File:Waveforms.svg|thumb|right|280 px|[[Sine wave|Sine]], [[Square wave|square]], [[Triangle wave|triangle]] and [[Sawtooth wave|sawtooth]] waveforms.]]
| |
| | |
| The form or shape of ''F'' in [[d'Alembert's formula]] involves 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).<ref name=Lyons>
| |
| | |
| {{cite book |url = http://books.google.com/?id=WdPGzHG3DN0C&pg=PA128 |pages = 128 ''ff'' |title = All you wanted to know about mathematics but were afraid to ask |author = Louis Lyons |isbn = 0-521-43601-X |publisher = Cambridge University Press |year = 1998 }}</ref>
| |
| | |
| In the case of a periodic function ''F'' with period ''λ'', that is, ''F''(''x + λ'' − ''vt'') = ''F''(''x '' − ''vt''), 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''.<ref name="McPherson0">
| |
| | |
| {{cite book |title = Introduction to Macromolecular Crystallography |author = Alexander McPherson |url = http://books.google.com/?id=o7sXm2GSr9IC&pg=PA77 |page = 77 |chapter = Waves and their properties |isbn = 0-470-18590-2 |year = 2009 |edition = 2 |publisher = Wiley }}
| |
| | |
| </ref>
| |
| | |
| === Amplitude and modulation ===
| |
| | |
| [[File:Wave packet.svg|right|thumb|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.]]
| |
| | |
| {{Main|Amplitude modulation}}
| |
| | |
| {{See also|Frequency modulation|Phase modulation}}
| |
| | |
| The amplitude of 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 [[Amplitude modulation|modulated wave]] can be written in the form:<ref name=Jirauschek>
| |
| | |
| {{cite book |url = http://books.google.com/?id=6kOoT_AX2CwC&pg=PA9 |page = 9 |title = FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection |author = Christian Jirauschek |isbn = 3-86537-419-0 |year = 2005 |publisher = Cuvillier Verlag }}
| |
| | |
| </ref><ref name=Kneubühl>
| |
| | |
| {{cite book |title = Oscillations and waves |author = Fritz Kurt Kneubühl |url = http://books.google.com/?id=geYKPFoLgoMC&pg=PA365 |page = 365 |year = 1997 |isbn = 3-540-62001-X |publisher = Springer }}
| |
| | |
| </ref><ref name=Lundstrom>
| |
| | |
| {{cite book |url = http://books.google.com/?id=FTdDMtpkSkIC&pg=PA33 |page = 33 |author = Mark Lundstrom |isbn = 0-521-63134-3 |year = 2000 |title = Fundamentals of carrier transport |publisher = Cambridge University Press }}
| |
| | |
| </ref>
| |
| | |
| :<math>u(x,t) = A(x,t)\sin (kx - \omega t + \phi) \ , </math>
| |
| | |
| where <math>A(x,\ t)</math> is the amplitude envelope of the wave, <math>k</math> is the ''[[wavenumber]]'' and <math>\phi</math> is the ''[[phase (waves)|phase]]''. If the [[group velocity]] <math>v_g</math> (see below) is wavelength-independent, this equation can be simplified as:<ref name=Chen>
| |
| | |
| {{cite book |url = http://books.google.com/?id=LxzWPskhns0C&pg=PA363 |author = Chin-Lin Chen |title = Foundations for guided-wave optics |page = 363 |chapter = §13.7.3 Pulse envelope in nondispersive media |isbn = 0-471-75687-3 |year = 2006 |publisher = Wiley }}
| |
| | |
| </ref>
| |
| | |
| :<math>u(x,t) = A(x - v_g \ t)\sin (kx - \omega t + \phi) \ , </math>
| |
| | |
| showing that 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''.<ref name=Chen/><ref name=Recami>
| |
| | |
| {{cite book |title = Localized Waves |chapter = Localization and Wannier wave packets in photonic crystals |author = Stefano Longhi, Davide Janner |editor = Hugo E. Hernández-Figueroa, Michel Zamboni-Rached, Erasmo Recami |url = http://books.google.com/?id=xxbXgL967PwC&pg=PA329 |page = 329 |isbn = 0-470-10885-1 |year = 2008 |publisher = Wiley-Interscience }}
| |
| | |
| </ref>
| |
| | |
| === Phase velocity and group velocity ===
| |
| [[Image:Wave group.gif|thumb|frame|right|[[Dispersion (water waves)|Frequency dispersion]] in groups of [[gravity waves]] on the surface of deep water. The red dot moves with the [[phase velocity]], and the green dots propagate with the [[group velocity]].]]
| |
| | |
| {{Main|Phase velocity|Group velocity}}
| |
| | |
| There are two velocities that are associated with waves, the [[phase velocity]] and the [[group velocity]]. To understand them, one must consider several types of waveform. For simplification, examination is restricted to one dimension.
| |
| | |
| [[Image:Wave opposite-group-phase-velocity.gif|thumb|frame|right|This shows a wave with the Group velocity and Phase velocity going in different directions.]]
| |
| | |
| The most basic wave (a form of [[plane wave]]) may be expressed in the form:
| |
| | |
| :<math> \psi (x,t) = A e^{i \left( kx - \omega t \right)} \ , </math>
| |
| | |
| which can be related to the usual sine and cosine forms using [[Euler's formula]]. Rewriting the argument, <math>kx-\omega t = \left(\frac{2\pi}{\lambda}\right)(x - vt)</math>, makes clear that this expression describes a vibration of wavelength <math>\lambda = \frac{2\pi}{k}</math> traveling in the ''x''-direction with a constant ''phase velocity'' <math>v_p = \frac{\omega}{k}\,</math>.<ref name=Messiah>
| |
| | |
| {{cite book |author = Albert Messiah |title = Quantum Mechanics |pages = 50–52 |isbn = 978-0-486-40924-5 |year = 1999 |publisher = Courier Dover |edition = Reprint of two-volume Wiley 1958 |url = http://books.google.com/?id=mwssSDXzkNcC&pg=PA52&dq=intitle:quantum+inauthor:messiah+%22group+velocity%22+%22center+of+the+wave+packet%22 }}
| |
| | |
| </ref>
| |
| | |
| The other type of wave to be considered is one with localized structure described by an [[envelope detector|envelope]], which may be expressed mathematically as, for example:
| |
| | |
| :<math> \psi (x,t) = \int_{-\infty} ^{\infty}\ dk_1 \ A(k_1)\ e^{i\left(k_1x - \omega t \right)} \ , </math>
| |
| | |
| where now ''A(k''<sub>1</sub>'')'' (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δ''k'' surrounding the point ''k''<sub>1</sub> = ''k''. In exponential form:
| |
| | |
| :<math> A = A_o (k_1) e^ {i \alpha (k_1)} \ , </math>
| |
| | |
| with ''A''<sub>o</sub> the magnitude of ''A''. For example, a common choice for ''A''<sub>o</sub> is a [[Wave packet|Gaussian wave packet]]:<ref name=Bromley0>
| |
| | |
| See, for example, Eq. 2(a) in
| |
| | |
| {{cite book |title = Quantum Mechanics: An introduction |author = Walter Greiner, D. Allan Bromley |url = http://books.google.com/?id=7qCMUfwoQcAC&pg=PA61 |pages = 60–61 |isbn = 3-540-67458-6 |year = 2007 |edition = 2nd |publisher = Springer }}
| |
| | |
| </ref>
| |
| | |
| :<math>A_o (k_1) = N\ e^{-\sigma^2 (k_1-k)^2 / 2} \ , </math>
| |
| | |
| where σ determines the spread of ''k''<sub>1</sub>-values about ''k'', and ''N'' is the amplitude of the wave.
| |
| | |
| The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(''k''<sub>1</sub>), and where it varies rapidly, the exponentials cancel each other out, [[Interference (wave propagation)|interfere]] destructively, contributing little to ψ.<ref name=Messiah/> However, an exception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basis for the method of [[Stationary phase approximation|stationary phase]] for evaluation of such integrals.<ref name=Orland>
| |
| | |
| {{cite book |title = Quantum many-particle systems |author = John W. Negele, Henri Orland |url = http://books.google.com/?id=mx5CfeeEkm0C&pg=PA121 |page = 121 |isbn = 0-7382-0052-2 |year = 1998 |publisher = Westview Press |edition = Reprint in Advanced Book Classics }}
| |
| | |
| </ref>) The condition for φ to vary slowly is that its rate of change with ''k''<sub>1</sub> be small; this rate of variation is:<ref name=Messiah/>
| |
| | |
| :<math>\left . \frac{d \varphi }{d k_1} \right | _{k_1 = k } = x - t \left . \frac{d \omega}{dk_1}\right | _{k_1 = k } +\left . \frac{d \alpha}{d k_1}\right | _{k_1 = k } \ ,</math>
| |
| | |
| where the evaluation is made at ''k''<sub>1</sub> = ''k'' because ''A(k''<sub>1</sub>'')'' is centered there. This result shows that the position ''x'' where the phase changes slowly, the position where ψ is appreciable, moves with time at a speed called the ''group velocity'':
| |
| | |
| :<math>v_g = \frac{d \omega}{dk} \ . </math>
| |
| | |
| The group velocity therefore depends upon the [[dispersion relation]] connecting ω and ''k''. For example, in quantum mechanics the energy of a particle represented as a wave packet is ''E'' = ħω = (ħ''k'')<sup>2</sup>/(2''m''). Consequently, for that wave situation, the group velocity is
| |
| | |
| :<math> v_g = \frac {\hbar k}{m} \ , </math>
| |
| | |
| showing that the velocity of a localized particle in quantum mechanics is its group velocity.<ref name=Messiah/> Because the group velocity varies with ''k'', the shape of the wave packet broadens with time, and the particle becomes less localized.<ref name=Fitt>
| |
| | |
| {{cite book |title = Principles of quantum mechanics: as applied to chemistry and chemical physics |author = Donald D. Fitts |url = http://books.google.com/?id=8t4DiXKIvRgC&pg=PA15 |pages = 15 ''ff'' |isbn = 0-521-65841-1 |year = 1999 |publisher = Cambridge University Press }}
| |
| | |
| </ref> In other words, the velocity of the constituent waves of the wave packet travel at a rate that varies with their wavelength, so some move faster than others, and they cannot maintain the same [[interference (wave propagation)|interference pattern]] as the wave propagates.
| |
| | |
| == Sinusoidal waves ==
| |
| [[File:Simple harmonic motion animation.gif|thumb|right|Sinusoidal waves correspond to [[simple harmonic motion]].]]
| |
| | |
| Mathematically, the most basic wave is the (spatially) one-dimensional [[sine wave]] (or ''harmonic wave'' or ''sinusoid'') with an amplitude <math>u</math> described by the equation:
| |
| | |
| :<math>u(x,t)= A \sin (kx- \omega t + \phi) \ , </math>
| |
| | |
| where
| |
| *<math>A</math> is the maximum [[amplitude]] of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.
| |
| *<math>x</math> is the space [[coordinate]]
| |
| *<math>t</math> is the time coordinate
| |
| *<math>k</math> is the [[wavenumber]]
| |
| *<math>\omega</math> is the [[angular frequency]]
| |
| *<math>\phi</math> is the [[phase (waves)|phase constant]].
| |
| | |
| The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) have an amplitude expressed as a [[distance]] (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its [[electric field]] (e.g., volts/meter).
| |
| | |
| The [[wavelength]] <math>\lambda</math> is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A [[wavenumber]] <math>k</math>, the spatial frequency of the wave in [[radian]]s per unit distance (typically per meter), can be associated with the wavelength by the relation
| |
| | |
| :<math>
| |
| k = \frac{2 \pi}{\lambda}. \,
| |
| </math>
| |
| | |
| The [[period (physics)|period]] <math>T</math> is the time for one complete cycle of an oscillation of a wave. The [[frequency]] <math>f</math> is the number of periods per unit time (per second) and is typically measured in [[hertz]]. These are related by:
| |
| | |
| :<math>
| |
| f=\frac{1}{T}. \,
| |
| </math>
| |
| | |
| In other words, the frequency and period of a wave are reciprocals.
| |
| | |
| The [[angular frequency]] <math>\omega</math> represents the frequency in radians per second. It is related to the frequency or period by
| |
| | |
| :<math>
| |
| \omega = 2 \pi f = \frac{2 \pi}{T}. \,
| |
| </math>
| |
| | |
| The wavelength <math>\lambda</math> of a sinusoidal waveform traveling at constant speed <math>v</math> is given by:<ref name=Cassidy>
| |
| | |
| {{cite book |title = Understanding physics |author = David C. Cassidy, Gerald James Holton, Floyd James Rutherford |url = http://books.google.com/?id=rpQo7f9F1xUC&pg=PA340 |pages = 339 ''ff'' |isbn = 0-387-98756-8 |year = 2002 |publisher = Birkhäuser }}
| |
| | |
| </ref>
| |
| :<math>\lambda = \frac{v}{f},</math>
| |
| | |
| where <math>v</math> is called the phase speed (magnitude of the [[phase velocity]]) of the wave and <math>f</math> is the wave's frequency.
| |
| | |
| Wavelength can be a useful concept even if the wave is not [[periodic function|periodic]] in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying ''local'' wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.<ref name=Pinet2>
| |
| | |
| {{cite book |title = op. cit. |author = Paul R Pinet |url = http://books.google.com/?id=6TCm8Xy-sLUC&pg=PA242 |page = 242 |isbn = 0-7637-5993-7 |year = 2009 }}
| |
| | |
| </ref>
| |
| | |
| Although arbitrary wave shapes will propagate unchanged in lossless [[linear time-invariant system]]s, in the presence of dispersion the [[sine wave]] is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze.<ref>
| |
| | |
| {{cite book |title = Communication Systems and Techniques |author = Mischa Schwartz, William R. Bennett, and Seymour Stein |publisher = John Wiley and Sons |year = 1995 |isbn = 978-0-7803-4715-1 |page = 208 |url = http://books.google.com/?id=oRSHWmaiZwUC&pg=PA208&dq=sine+wave+medium++linear+time-invariant }}
| |
| | |
| </ref> Due to the [[Kramers–Kronig relation]]s, a linear medium with dispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium.<ref name=Tielens>
| |
| | |
| See Eq. 5.10 and discussion in {{cite book |author = A. G. G. M. Tielens |title = The physics and chemistry of the interstellar medium |url = http://books.google.com/?id=wMnvg681JXMC&pg=PA119 |pages = 119 ''ff'' |isbn = 0-521-82634-9 |year = 2005 |publisher = Cambridge University Press }}; Eq. 6.36 and associated discussion in {{cite book |title = Introduction to solid-state theory |author = Otfried Madelung |url = http://books.google.com/?id=yK_J-3_p8_oC&pg=PA261 |pages = 261 ''ff'' |isbn = 3-540-60443-X |year = 1996 |edition = 3rd |publisher = Springer }}; and Eq. 3.5 in {{cite book |author = F Mainardi |chapter = Transient waves in linear viscoelastic media |editor = Ardéshir Guran, A. Bostrom, Herbert Überall, O. Leroy |title = Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering |url = http://books.google.com/?id=UfSk45nCVKMC&pg=PA134 |page = 134 |isbn = 981-02-4271-9 |year = 1996 |publisher = World Scientific }}
| |
| | |
| </ref>
| |
| The [[sine function]] is periodic, so the [[sine wave]] or sinusoid has a [[wavelength]] in space and a period in time.<ref name=Filippov>
| |
| {{cite book |url = http://books.google.com/?id=TC4MCYBSJJcC&pg=PA106 |page = 106 |author = Aleksandr Tikhonovich Filippov |title = The versatile soliton |year = 2000 |publisher = Springer |isbn = 0-8176-3635-8 }}
| |
| | |
| </ref><ref name=Stein1>
| |
| {{cite book |title = An introduction to seismology, earthquakes, and earth structure |author = Seth Stein, [[Michael E. Wysession]] |page = 31 |url = http://books.google.com/?id=Kf8fyvRd280C&pg=PA31 |isbn = 0-86542-078-5 |year = 2003 |publisher = Wiley-Blackwell }}
| |
| | |
| </ref>
| |
| | |
| The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of [[Fourier analysis]]. As a result, the simple case of a single sinusoidal wave can be applied to more general cases.<ref name=Stein2>
| |
| | |
| {{cite book |title = ''op. cit.'' |author = Seth Stein, [[Michael E. Wysession]] |page = 32 |url = http://books.google.com/?id=Kf8fyvRd280C&pg=PA32 |isbn = 0-86542-078-5 |year = 2003 }}
| |
| | |
| </ref><ref name=Schwinger>
| |
| | |
| {{cite book |title = Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators |author = Kimball A. Milton, Julian Seymour Schwinger |url = http://books.google.com/?id=x_h2rai2pYwC&pg=PA16 |page = 16 |isbn = 3-540-29304-3 |publisher = Springer |year = 2006 |quote = Thus, an arbitrary function ''f''('''''r''''', ''t'') can be synthesized by a proper superposition of the functions ''exp''[i ('''''k·r'''''−ω''t'')]... }}
| |
| | |
| </ref> In particular, many media are [[linear]], or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the [[superposition principle]] to find the solution for a general waveform.<ref name=Jewett>
| |
| | |
| {{cite book |url = http://books.google.com/?id=1DZz341Pp50C&pg=PA433 |page = 433 |title = Principles of physics |author = Raymond A. Serway and John W. Jewett |chapter = §14.1 The Principle of Superposition |isbn = 0-534-49143-X |year = 2005 |edition = 4th |publisher = Cengage Learning }}
| |
| | |
| </ref> When a medium is [[nonlinear]], the response to complex waves cannot be determined from a sine-wave decomposition.
| |
| | |
| == Plane waves ==
| |
| {{Main|Plane wave}}
| |
| | |
| == Standing waves ==
| |
| {{Main|Standing wave|Acoustic resonance|Helmholtz resonator|Organ pipe}}
| |
| | |
| [[File:Standing wave.gif|thumb|right|300px|Standing wave in stationary medium. The red dots represent the wave [[Node (physics)|nodes]]]]
| |
| | |
| 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 [[Interference (wave propagation)|interference]] between 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 [[violin]] string is displaced, transverse waves propagate out to where the string is held in place at the [[Bridge (instrument)|bridge]] and the [[Nut (string instrument)|nut]], where the waves are reflected back. At the bridge and nut, the two opposed waves are in [[antiphase]] and cancel each other, producing a [[node (physics)|node]]. Halfway between two nodes there is an [[antinode]], where the two counter-propagating waves ''enhance'' each other maximally. There is no net [[Energy transfer|propagation of energy]] over time.
| |
| | |
| <gallery>
| |
| Image:Standing waves on a string.gif|One-dimensional standing waves; the [[fundamental frequency|fundamental]] mode and the first 5 [[overtone]]s.
| |
| Image:Drum vibration mode01.gif|A two-dimensional [[Vibrations of a circular drum|standing wave on a disk]]; this is the fundamental mode.
| |
| Image:Drum vibration mode21.gif|A [[Vibrations of a circular drum|standing wave on a disk]] with two nodal lines crossing at the center; this is an overtone.
| |
| </gallery>
| |
| | |
| == Physical properties ==
| |
| | |
| [[File:Light dispersion of a mercury-vapor lamp with a flint glass prism IPNr°0125.jpg|thumb|right|upright|Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism]]
| |
| | |
| Waves exhibit common behaviors under a number of standard situations, e. g.
| |
| | |
| === Transmission and media ===
| |
| {{Main|Rectilinear propagation|Transmittance|Transmission medium}}
| |
| | |
| Waves normally move in a straight line (i.e. rectilinearly) through a ''[[transmission medium]]''. Such media 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'' or ''homogeneous medium'' if its physical properties are unchanged at different locations in space
| |
| * An ''anisotropic medium'' if one or more of its physical properties differ in one or more directions
| |
| * An ''isotropic medium'' if its physical properties are the ''same'' in all directions
| |
| | |
| === Absorption ===
| |
| {{Main|Absorption (acoustics)|Absorption (electromagnetic radiation)}}
| |
| Absorption of waves mean, if a kind of wave strikes a matter, it will be absorbed by the matter. When a wave with that same natural frequency impinges upon an atom, then the electrons of that atom will be set into vibrational motion. If a wave of a given frequency strikes a material with electrons having the same vibrational frequencies, then those electrons will absorb the energy of the wave and transform it into vibrational motion.
| |
| | |
| === Reflection ===
| |
| {{Main|Reflection (physics)}}
| |
| | |
| When a wave strikes a reflective surface, it changes direction, such that the angle made by the [[incident ray|incident wave]] and line [[perpendicular|normal]] to the surface equals the angle made by the reflected wave and the same normal line.
| |
| | |
| === Interference ===
| |
| {{Main|Interference (wave propagation)}}
| |
| | |
| Waves that encounter each other combine through [[superposition principle|superposition]] to create a new wave called an [[Interference (wave propagation)|interference pattern]]. Important interference patterns occur for waves that are in [[phase (waves)|phase]].
| |
| | |
| === Refraction ===
| |
| {{Main|Refraction}}
| |
| | |
| [[File:Wave refraction.gif|thumb|right|200 px|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.]]
| |
| | |
| Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the [[phase velocity]] changes. Typically, refraction occurs when a wave passes from one [[Transmission medium|medium]] into another. The amount by which a wave is refracted by a material is given by the [[refractive index]] of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by [[Snell's law]].
| |
| | |
| === Diffraction ===
| |
| {{Main|Diffraction}}
| |
| | |
| A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.
| |
| | |
| === Polarization ===
| |
| {{Main|Polarization (waves)}}
| |
| | |
| [[File:Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg|thumb|left]]
| |
| | |
| A wave is polarized if it oscillates in one direction or plane. A wave can be polarized by the use of a polarizing filter. 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. For these waves the direction of oscillation is along the direction of travel.
| |
| | |
| === Dispersion ===
| |
| | |
| [[File:Light dispersion conceptual waves.gif|thumb|right|270 px|Schematic of light being dispersed by a prism. Click to see animation.]]
| |
| | |
| {{Main|Dispersion (optics)|Dispersion (water waves)}}
| |
| | |
| A wave undergoes dispersion when either the [[phase velocity]] or the [[group velocity]] depends on the wave frequency.
| |
| Dispersion is most easily seen by letting white light pass through a [[prism (optics)|prism]], the result of which is to produce the spectrum of colours of the rainbow. [[Isaac Newton]] performed experiments with light and prisms, presenting his findings in the ''[[Opticks]]'' (1704) that white light consists of several colours and that these colours cannot be decomposed any further.<ref name=Newton>
| |
| | |
| {{cite book |last = Newton |first = Isaac |year = 1704 |authorlink = Isaac Newton |title = Opticks: Or, A treatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of Curvilinear Figures |page = 118 |location = London |chapter = Prop VII Theor V |quote = All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, or compounded of these... |volume = 1 |url = http://gallica.bnf.fr/ark:/12148/bpt6k3362k.image.f128.pagination }}
| |
| | |
| </ref>
| |
| | |
| == Mechanical waves ==
| |
| {{Main|Mechanical wave}}
| |
| | |
| === Waves on strings ===
| |
| {{Main|Vibrating string}}
| |
| | |
| The speed of a transverse wave traveling along a [[vibrating string]] ('' v '') is directly proportional to the square root of the [[Tension (mechanics)|tension]] of the string ('' T '') over the [[linear mass density]] ('' μ ''):
| |
| | |
| :<math>
| |
| v=\sqrt{\frac{T}{\mu}}, \,
| |
| </math>
| |
| | |
| where the linear density ''μ'' is the mass per unit length of the string.
| |
| | |
| === Acoustic waves ===
| |
| Acoustic or [[sound]] waves travel at speed given by
| |
| | |
| :<math>
| |
| v=\sqrt{\frac{B}{\rho_0}}, \,
| |
| </math>
| |
| | |
| or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see [[speed of sound]]).
| |
| | |
| === Water waves ===
| |
| [[File:Shallow water wave.gif|thumb]]
| |
| | |
| {{Main|Water waves}}
| |
| | |
| * [[ripple tank|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.
| |
| * [[Sound]]—a mechanical wave that propagates through gases, liquids, solids and plasmas;
| |
| * [[Inertial waves]], which occur in rotating fluids and are restored by the [[Coriolis effect]];
| |
| * [[Ocean surface wave]]s, which are perturbations that propagate through water.
| |
| | |
| === Seismic waves ===
| |
| {{Main|Seismic waves}}
| |
| | |
| === Shock waves ===
| |
| [[File:Transonico-en.svg|thimb|right|300 px|Formation of a shock wave by a plane.]]
| |
| | |
| {{Main|Shock wave}}
| |
| {{See also|Sonic boom|Cherenkov radiation}}
| |
| | |
| === Other ===
| |
| * Waves of [[Traffic wave|traffic]], that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves<ref name=Lighthill>
| |
| | |
| {{cite journal |author1 = M. J. Lighthill | author1-link=James Lighthill |author2 = G. B. Whitham | author2-link=Gerald B. Whitham |year = 1955 |title = On kinematic waves. II. A theory of traffic flow on long crowded roads |journal = Proceedings of the Royal Society of London. Series A |volume = 229 |pages = 281–345 |ref = harv |postscript = . |bibcode = 1955RSPSA.229..281L |doi = 10.1098/rspa.1955.0088 }} And: {{cite journal |doi = 10.1287/opre.4.1.42 |author = P. I. Richards |year = 1956 |title = Shockwaves on the highway |journal = Operations Research |volume = 4 |issue = 1 |pages = 42–51 |ref = harv |postscript = . }}
| |
| | |
| </ref>
| |
| | |
| * [[metachronal rhythm|Metachronal wave]] refers to the appearance of a traveling wave produced by coordinated sequential actions.
| |
| | |
| * It is worth noting that the [[mass-energy equivalence]] equation can be solved for this form: <math>c=\sqrt{\frac{e}{m}}</math>.
| |
| | |
| == Electromagnetic waves ==
| |
| [[File:Onde electromagnétique.png|thumb|right|200 px]]
| |
| | |
| {{Main|Electromagnetic radiation|Electromagnetic spectrum}}
| |
| (radio, micro, infrared, visible, uv)
| |
| | |
| An electromagnetic wave consists of two waves that are oscillations of the [[electric field|electric]] and [[magnetic field|magnetic]] fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, [[James Clerk Maxwell]] showed that, in [[vacuum]], the electric and magnetic fields satisfy the [[wave equation]] both with speed equal to that of the [[speed of light]]. From this emerged the idea that [[visible light|light]] is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as [[radio waves]], [[microwaves]], [[infrared]], visible light, [[ultraviolet]] and [[X-rays]].
| |
| | |
| == Quantum mechanical waves ==
| |
| {{Main|Schrödinger equation}}
| |
| {{See also|Wave function}}
| |
| | |
| The [[Schrödinger equation]] describes the wave-like behavior of particles in [[quantum mechanics]]. Solutions of this equation are [[wave function]]s which can be used to describe the probability density of a particle.
| |
| | |
| [[File:Wave packet (dispersion).gif|thumb|A propagating wave packet; in general, the ''envelope'' of the wave packet moves at a different speed than the constituent waves.<ref name=Fromhold>{{cite book |title = Quantum Mechanics for Applied Physics and Engineering |author = A. T. Fromhold |chapter = Wave packet solutions |pages = 59 ''ff'' |quote = (p. 61) ...the individual waves move more slowly than the packet and therefore pass back through the packet as it advances |url = http://books.google.com/?id=3SOwc6npkIwC&pg=PA59 |isbn = 0-486-66741-3 |publisher = Courier Dover Publications |year = 1991 |edition = Reprint of Academic Press 1981 }}</ref>]]
| |
| | |
| === de Broglie waves ===
| |
| {{Main|Wave packet|Matter wave}}
| |
| | |
| [[Louis de Broglie]] postulated that all particles with [[momentum]] have a wavelength
| |
| | |
| :<math>\lambda = \frac{h}{p},</math>
| |
| | |
| where ''h'' is [[Planck's constant]], and ''p'' is the magnitude of the [[momentum]] of the particle. This hypothesis was at the basis of [[quantum mechanics]]. Nowadays, this wavelength is called the [[de Broglie wavelength]]. For example, the [[electron]]s in a [[cathode ray tube|CRT]] display have a de Broglie wavelength of about 10<sup>−13</sup> m.'''
| |
| | |
| A wave representing such a particle traveling in the ''k''-direction is expressed by the wave function as follows:
| |
| | |
| :<math>\psi (\mathbf{r}, \ t=0) =A\ e^{i\mathbf{k \cdot r}} \ , </math>
| |
| | |
| where the wavelength is determined by the [[wave vector]] '''k''' as:
| |
| | |
| :<math> \lambda = \frac {2 \pi}{k} \ , </math>
| |
| | |
| and the momentum by:
| |
| | |
| :<math> \mathbf{p} = \hbar \mathbf{k} \ . </math>
| |
| | |
| However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a [[wave packet]],<ref name=Marton>
| |
| | |
| {{cite book |title = Advances in Electronics and Electron Physics |page = 271 |url = http://books.google.com/?id=g5q6tZRwUu4C&pg=PA271 |isbn = 0-12-014653-3 |year = 1980 |publisher = Academic Press |volume = 53 |editor = L. Marton & Claire Marton |author = Ming Chiang Li |chapter = Electron Interference }}
| |
| | |
| </ref> a waveform often used in [[quantum mechanics]] to describe the [[wave function]] of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.
| |
| | |
| In representing the wave function of a localized particle, the [[wave packet]] is often taken to have a [[Gaussian function|Gaussian shape]] and is called a ''Gaussian wave packet''.<ref name=wavepacket>
| |
| | |
| See for example {{cite book |url = http://books.google.com/?id=7qCMUfwoQcAC&pg=PA60 |title = Quantum Mechanics |author = Walter Greiner, D. Allan Bromley |page = 60 |isbn = 3-540-67458-6 |edition = 2 |year = 2007 |publisher = Springer }} and {{cite book |title = Electronic basis of the strength of materials |author = John Joseph Gilman |url = http://books.google.com/?id=YWd7zHU0U7UC&pg=PA57 |page = 57 |year = 2003 |isbn = 0-521-62005-8 |publisher = Cambridge University Press }},{{cite book |title = Principles of quantum mechanics |author = Donald D. Fitts |url = http://books.google.com/?id=8t4DiXKIvRgC&pg=PA17 |page = 17 |isbn = 0-521-65841-1 |publisher = Cambridge University Press |year = 1999 }}.
| |
| | |
| </ref> Gaussian wave packets also are used to analyze water waves.<ref name=Mei>
| |
| | |
| {{cite book |url = http://books.google.com/?id=WHMNEL-9lqkC&pg=PA47 |page = 47 |author = Chiang C. Mei |author-link=Chiang C. Mei |title = The applied dynamics of ocean surface waves |isbn = 9971-5-0789-7 |year = 1989 |edition = 2nd |publisher = World Scientific }}
| |
| | |
| </ref>
| |
| | |
| For example, a Gaussian wavefunction ψ might take the form:<ref name=Bromley>
| |
| | |
| {{cite book |title = Quantum Mechanics |author = Walter Greiner, D. Allan Bromley |page = 60 |url = http://books.google.com/?id=7qCMUfwoQcAC&pg=PA60 |edition = 2nd |year = 2007 |publisher = Springer |isbn = 3-540-67458-6 }}
| |
| | |
| </ref>
| |
| | |
| :<math> \psi(x,\ t=0) = A\ \exp \left( -\frac{x^2}{2\sigma^2} + i k_0 x \right) \ , </math>
| |
| | |
| at some initial time ''t'' = 0, where the central wavelength is related to the central wave vector ''k''<sub>0</sub> as λ<sub>0</sub> = 2π / ''k''<sub>0</sub>. It is well known from the theory of [[Fourier analysis]],<ref name=Brandt>
| |
| | |
| {{cite book |page = 23 |url = http://books.google.com/?id=VM4GFlzHg34C&pg=PA23 |title = The picture book of quantum mechanics |author = Siegmund Brandt, Hans Dieter Dahmen |isbn = 0-387-95141-5 |year = 2001 |edition = 3rd |publisher = Springer }}
| |
| | |
| </ref> or from the [[Heisenberg uncertainty principle]] (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The [[Fourier transform]] of a Gaussian is itself a Gaussian.<ref name=Gaussian>
| |
| | |
| {{cite book |title = Modern mathematical methods for physicists and engineers |author = Cyrus D. Cantrell |page = 677 |url = http://books.google.com/?id=QKsiFdOvcwsC&pg=PA677 |isbn = 0-521-59827-3 |publisher = Cambridge University Press |year = 2000 }}
| |
| | |
| </ref> Given the Gaussian:
| |
| | |
| :<math>f(x) = e^{-x^2 / (2\sigma^2)} \ , </math>
| |
| | |
| the Fourier transform is:
| |
| | |
| :<math>\tilde{ f} (k) = \sigma e^{-\sigma^2 k^2 / 2} \ . </math>
| |
| | |
| The Gaussian in space therefore is made up of waves:
| |
| | |
| :<math>f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \ \tilde{f} (k) e^{ikx} \ dk \ ; </math>
| |
| | |
| that is, a number of waves of wavelengths λ such that ''k''λ = 2 π.
| |
| | |
| The parameter σ decides the spatial spread of the Gaussian along the ''x''-axis, while the Fourier transform shows a spread in [[wave vector]] ''k'' determined by 1/σ. That is, the smaller the extent in space, the larger the extent in ''k'', and hence in λ = 2π/''k''.
| |
| | |
| [[File:GravitationalWave CrossPolarization.gif|thumb|right|Animation showing the effect of a cross-polarized gravitational wave on a ring of [[test particles]]]]
| |
| | |
| == Gravitational waves ==
| |
| | |
| {{Main|Gravitational wave}}
| |
| | |
| Researchers believe that [[gravitational radiation|gravitational waves]] also travel through space, although gravitational waves have never been directly detected.
| |
| Not to be confused with [[gravity waves]], gravitational waves are disturbances in the curvature of [[spacetime]], predicted by Einstein's theory of [[general relativity]].
| |
| | |
| == WKB method ==
| |
| | |
| {{Main|WKB method}}
| |
| {{See also|Slowly varying envelope approximation}}
| |
| | |
| In a nonuniform medium, in which the wavenumber ''k'' can depend on the location as well as the frequency, the phase term ''kx'' is typically replaced by the integral of ''k''(''x'')''dx'', according to the [[WKB method]]. Such nonuniform traveling waves are common in many physical problems, including the mechanics of the [[cochlea]] and waves on hanging ropes.
| |
| | |
| == See also ==
| |
| {{Div col|3}}
| |
| * [[Audience wave]]
| |
| * [[Beat wave]]s
| |
| * [[Capillary waves]]
| |
| * [[Cymatics]]
| |
| * [[Doppler effect]]
| |
| * [[Envelope detector]]
| |
| * [[Group velocity]]
| |
| * [[Harmonic]]
| |
| * [[Inertial wave]]
| |
| * [[Index of wave articles]]
| |
| * [[List of waves named after people]]
| |
| * [[Ocean surface wave]]
| |
| * [[Phase velocity]]
| |
| * [[Reaction-diffusion equation]]
| |
| * [[Resonance]]
| |
| * [[Ripple tank]]
| |
| * [[Rogue wave]]
| |
| * [[Shallow water equations]]
| |
| * [[Shive wave machine]]
| |
| * [[Standing wave]]
| |
| * [[Transmission medium]]
| |
| * [[Wave turbulence]]
| |
| * [[Waves in plasmas]]
| |
| {{Div col end}}
| |
| | |
| == References ==
| |
| {{Reflist|35em}}
| |
| | |
| == Sources ==
| |
| {{Refbegin}}
| |
| *{{cite book|last=Campbell|first=Murray|title=The musician's guide to acoustics|year=2001|publisher=Oxford University Press|location=Oxford|isbn=978-0198165057|edition=Repr.|last2=Greated |first2=Clive}}
| |
| * {{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 }}
| |
| * {{cite book |last = Hall |first = D. E. |year = 1980 |title = Musical Acoustics: An Introduction |location = Belmont, California |publisher = Wadsworth Publishing Company |isbn = 0-534-00758-9 |ref = harv |postscript = . }}.
| |
| *{{cite book|last=Hunt|first=Frederick Vinton |title=Origins in acoustics.|year=1978|publisher=Published for the Acoustical Society of America through the American Institute of Physics|location=Woodbury, NY|isbn=978-0300022209}}
| |
| * {{cite book |last1 = Ostrovsky |first1 = L. A. |last2 = Potapov |first2 = A. S. |year = 1999 |title = Modulated Waves, Theory and Applications |location = Baltimore |publisher = The Johns Hopkins University Press |isbn = 0-8018-5870-4 |ref = harv |postscript = . }}.
| |
| * [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.
| |
| {{Refend}}
| |
| | |
| == External links ==
| |
| {{commons|Wave|Wave}}
| |
| {{Wiktionary}}
| |
| * [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/lm/ch19/ch19.html chapter from an online textbook]
| |
| * [http://www.physics-lab.net/applets/mechanical-waves Simulation of waves on a string]
| |
| * [http://www.cbu.edu/~jvarrian/applets/waves1/lontra_g.htm-simulation of longitudinal and transverse mechanical wave]
| |
| * [http://ocw.mit.edu/courses/physics/8-03-physics-iii-vibrations-and-waves-fall-2004/ MIT OpenCourseWare 8.03: Vibrations and Waves] Free, independent study course with video lectures, assignments, lecture notes and exams.
| |
| | |
| {{Velocities of Waves}}
| |
| | |
| [[Category:Concepts in physics]]
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Waves| ]]
| |