thumb in water
A wave
is a disturbance that propagates through space and time, usually with transference of energy. A mechanical wave is a wave that propagates or travels through a medium due to the restoring forces it produces upon deformation. There also exist waves capable of traveling through a vacuum, including electromagnetic radiation and probably [1] gravitational radiation. Waves travel and transfer 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 oscillations or vibrations around almost fixed locations.
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WAVES TICKETS
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Definitions
Diving grebe creates surface waves|thumb|250px
Agreeing on a single, all-encompassing definition for the term
wave
is
non-trivial. A
vibration can be defined as a
back-and-forth
motion around a reference value. However, a vibration is not necessarily a wave. Defining the necessary and sufficient characteristics that qualify a
phenomenon to 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
energy of a
vibration is moving away from the source in the form of a disturbance within the surrounding medium
(Hall et al.
,
). 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 / light waves in a
vacuum, where the concept of medium does not apply. There are
water waves in the ocean;
light waves from the sun;
microwaves inside the microwave oven;
radio waves transmitted to the radio; and
sound waves from the radio, telephone, and voices.
It may be seen that the description of waves is accompanied by a heavy reliance on physical origin when describing any 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-like transfer / transformation of vibratory
energy. Concepts such as
mass,
momentum,
inertia, or
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: 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 such reasons, wave theory represents a particular branch of
physics that is concerned with the properties of wave processes independently from their physical origin.
[2] 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 waves different from that of sound waves can be significant to the understanding of sound phenomena. A relevant example is
Thomas Young's principle of interference (Young, 1802, in Hunt et al.
,
). 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.
Characteristics
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.
Image:Wave motion-i18n-mod.svg|thumb|300px|right|
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 — change in wave direction after it strikes a reflective surface, causing the angle the wave makes with the reflective surface in relation to a normal line to the surface to equal the angle the reflected wave makes with the same normal line
- Refraction — change in wave direction because of a change in the wave's speed from entering a new medium
- Diffraction — bending of waves as they interact with obstacles in their path, which is more 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
Polarization
A wave is
polarized if it can only oscillate in one direction or plane. 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. A wave can be polarized by the use of a polarizing filter.
Examples
thumb crashing into
rocks
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, which make up electromagnetic radiation; can be propagated without a medium, through vacuum; and travel at 299,792,458 m/s in a vacuum
- Sound — a mechanical wave that propagates through air, liquid or solids
- Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves, as first presented by Sir M. J. Lighthill [3]
- Seismic waves in earthquakes, of which there are three types, called S, P, and L
- Gravitational waves, which are fluctuations in the curvature of spacetime predicted by general Relativity, and which are nonlinear, and which have yet to be observed empirically
- Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect
Mathematical description
thumb, ?, can be measured between any two points with the same
phase, such as between crests, or troughs, or corresponding
zero crossings as shown.
Sinusoidal waves
Mathematically, the most basic wave is the
sine wave (or harmonic wave or sinusoid), with an amplitude
u
described by the equation:
where
A
is the
semi-amplitude of the wave, half the peak-to-peak amplitude, often called simply the amplitude – the 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;
x
is the space coordinate,
t
is the time coordinate,
k
is the
wavenumber (spatial frequency), ? is the temporal frequency, and f is a phase offset.
The units of the semi-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
wavelength (denoted as ?) is the distance between two sequential crests (or troughs), and generally is measured in meters.
A
wavenumber k
, the spatial frequency of the wave in
radians per unit distance (typically per meter), can be associated with the wavelength by the relation
k = \frac{2 \pi}{\lambda}. \,
thumb.
The
period T
is the time for one complete cycle of an oscillation of a wave. The
frequency f
(also frequently denoted as ? ) is the number of periods per unit time (per 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.
The
angular frequency ? represents the frequency in radians per second. It is related to the frequency by
\omega = 2 \pi f = \frac{2 \pi}{T}. \,
thumb
Wavelength can be a useful concept even if the wave is not
periodic in space. For example, in an ocean wave approaching shore, shown in the figure, 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.
[4]
Although arbitrary wave shapes will propagate unchanged in lossless
linear time-invariant systems, 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.
[5] (Due to the
Kramers–Kronig relations, a 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.
[6])
The
sine function is periodic, so the
sine wave or sinusoid has a
wavelength in space and a period in time.
[7] [8]
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.
[9] [10] 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.
[11]. When a medium is
nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition.
The wave equation
The
wave equation is a
partial differential equation that describes the evolution of a wave over time in a medium where the wave propagates at the same speed independent of wavelength (no
dispersion), and independent of amplitude (
linear media, not
nonlinear).
[12] General solutions are based upon
Duhamel's principle.
[13]
In particular, consider the wave equation in one dimension, for example, as applied to a string. Suppose a one-dimensional wave is traveling along the
x
axis with velocity
and amplitude
(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}. \,
The velocity
v
will depend on the medium through which the wave is moving.
The general solution for the wave equation in one dimension was given by
d'Alembert; it is known as
d'Alembert's formula:
[14]
u(x,t)=F(x-vt)+G(x+vt). \,
This formula represents two shapes traveling through the medium in opposite directions;
F
in the positive
x
direction, and
G
in the negative
x
direction, of arbitrary functional shapes
F
and
G
.
Spatial and temporal relationships
thumb
The form of the forward propagating wave
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).
[16]
In the case of a periodic function
F
with period
?
, that is,
F
(
x + ?
-
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
?
(sometimes called 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
=
?
, so an observation of the wave at a fixed location
x
finds the wave undulating periodically in time with period
T = ?
/
v
.
[17]
To summarize:
"A function F (x)
is periodic if F(x+?) = F(x)
, for all x
. The constant ?
is called a period
of the function. The smallest such period is called the fundamental period
or simply the period
of F
. If x
represents a space coordinate, then the period may instead be called the wavelength
and is often written ?
; if it represents the time coordinate, the period might instead be denoted by T
." Flowers, p. 473 [18]
The Schrödinger equation
The
Schrödinger equation describes 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.
thumb
Wave packets and the de Broglie wavelength
Louis de Broglie postulated that all particles with
momentum have a wavelength
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
electrons in a
CRT display have a de Broglie wavelength of about 10
–13 m.'''
A wave representing such a particle traveling in the
k
-direction is expressed by the wave function:
where the wavelength is determined by the
wave vector k
as:
and the momentum by:
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,
[20] 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 shape and is called a
Gaussian wave packet
.
[21]Gaussian wave packets also are used to analyze water waves.
[22]
For example, a Gaussian wavefunction ? might take the form:
[23]
at some initial time
t
= 0, where the central wavelength is related to the central wave vector
k
0 as ?
0 = 2p /
k
0. It is well known from the theory of
Fourier analysis,
[24] 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.
[25] Given the Gaussian:
the Fourier transform is:
The Gaussian in space therefore is made up of waves:
that is, a number of waves of wavelengths ? such that
k
? = 2 p.
The parameter s 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/s. That is, the smaller the extent in space, the larger the extent in
k
, and hence in ? = 2p/
k
.
right
Modulated waves
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
modulated wave can be written in the form:
[26] [27] [28]
where
is the amplitude envelope of the wave,
is the
wave number
and
is the
phase
. If the
group velocity (see below) is wavelength independent, this equation can be simplified as:
[29]
where
v
g is the group velocity, showing that the envelope moves with velocity
v
g 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
.
[ [30]
]
Phase velocity and 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.
The most basic wave (a form of plane wave) may be expressed in the form:
which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument, (kx -?t
) = (2p/?)(x - vt
), makes clear that this expression describes a vibration of wavelength ? = 2p/k traveling in the x
-direction with a constant phase velocity
v
p: [31]
The other type of wave to be considered is one with localized structure described by an envelope, which may be expressed mathematically as, for example:
where now A(k
1)
is a function exhibiting a sharp peak in a region of wave vectors ?k
surrounding the point k
1 = k
. In exponential form:
with A
o the magnitude of A
. For example, a common choice for A
o is a Gaussian wave packet: [32]
where s determines the spread of k
1-values about k
, and N
is the amplitude of the wave.
The exponential function inside the integral for ? oscillates rapidly with its argument, say f(k
1), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ?.[ However, an exception occurs at the location where the argument f of the exponential varies slowly. (This observation is the basis for the method of stationary phase for evaluation of such integrals. [33]) The condition for f to vary slowly is that its rate of change with k
1 be small; this rate of variation is:][
]
where the evaluation is made at k
1 = k
because A(k
1)
is centered there. This result shows that the position x
where the phase changes slowly, the position where ? is appreciable, moves with time at a speed called the group velocity
:
The group velocity therefore depends upon the dispersion relation connecting ? and k
. For example, in quantum mechanics the energy E
= h? = (hk
)2/(2m
). Consequently,
showing that the velocity of a localized particle in quantum mechanics is its group velocity.[ Because the group velocity varies with k
, the shape of the wave packet broadens with time, and the particle becomes less localized. [34] 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 pattern as the wave propagates.
]
Standing wave
frame
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 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, longitudinal waves propagate out to where the string is held in place at the bridge and the "nut", whereupon the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and 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 average no net propagation of energy.
Also see: 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 of the string (T
) over the linear mass density (µ
):
v=\sqrt{\frac{T}{\mu}}, \,
where the linear density µ
is the mass per unit length of the string.
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 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 isotropic medium
if its physical properties are the same
in different directions
WKB method
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.
Notes
- Gravitational waves have never been directly detected but are widely believed by the scientific community to exist.
- Modulated waves: theory and application
- Lighthill, M.J., Whitham, G.B., 1955. On kinematic waves. II. A theory of traffic flow on long crowded roads. Procedings of Royal Society A 229, 281-345] and Richards [Richards, P.I., 1956. Shockwaves on the highway. Operations Research 4, 42-51
- op. cit.
- Communication Systems and Techniques
- The physics and chemistry of the interstellar medium
- The versatile soliton
- An introduction to seismology, earthquakes, and earth structure
- ''op. cit.''
- Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators
- Principles of physics
- Seismic waves and rays in elastic media
- Geometric wave equations
- Wave motion in elastic solids
- Introduction to Macromolecular Crystallography
- All you wanted to know about mathematics but were afraid to ask
- Introduction to Macromolecular Crystallography
- An introduction to numerical methods in C++
- Quantum Mechanics for Applied Physics and Engineering
- Advances in Electronics and Electron Physics
- Quantum Mechanics
- The applied dynamics of ocean surface waves
- Quantum Mechanics
- The picture book of quantum mechanics
- Modern mathematical methods for physicists and engineers
- FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection
- Oscillations and waves
- Fundamentals of carrier transport
- Foundations for guided-wave optics
- Localized Waves
- Quantum Mechanics
- Quantum Mechanics: An introduction
- Quantum many-particle systems
- Principles of quantum mechanics: as applied to chemistry and chemical physics
Bibliography
{{#if:March 2009{{#ifexist:Category:Articles to be expanded since March 2009
See also
- Audience wave
- Beat waves
- Capillary waves
- Cymatics
- Doppler effect
- Envelope detector
- Group velocity
- Harmonic
- Inertial wave
- List of wave topics
- List of waves named after people
- Ocean surface wave
- Phase velocity
- Reaction-diffusion equation
- Resonance
- Ripple tank
- Rogue wave (oceanography)
- Shallow water equations
- Shive wave machine
- Standing wave
- Transmission medium
- Wave turbulence