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Terms and Formulae

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Terms

**
Principle of superposition
** -
When any two waves occupy the same point or region in space, the resulting disturbance of the medium is
the sum of the disturbances of the individual waves (in other words, just add the amplitudes, paying attention
to the sign). This is the same as saying that the wave equation is linear: if *μ*_{1} and *μ*_{2} are
solutions, then *aμ*_{1} + *bμ*_{2} are also solutions, for some constants *a* and *b*. One consequence of
this is that two or more waves can pass through each other with each being unaffected by the other.

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Fermat's principle
** -
The path taken by a light ray will be the one that minimizes the time taken in passing between any two
points. This is equivalent to saying that the duration of the path taken by light is stationary with respect to
small variations in the path.

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Scattering
** -
This occurs when light is incident on an atom. The oscillating electric and magnetic fields of the light wave
cause the electrons in the atom to vibrate at the same frequency as the incident wave, causing a re-radiation
of light in all directions (a spherical wave) about the atom. The light is said to be scattered by the atom.
Such scattering is always elastic.

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Longitudinal wave
** -
An oscillation in which the displacement of the particles of the medium about their equilibrium position is in
the direction parallel to the direction of propagation. Longitudinal waves exhibit many of the opposite
behaviors to transverse waves (for example, they speed up in denser media). Sound is a longitudinal wave.

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Transverse wave
** -
An oscillation in which the displacement of the particles of the medium about their equilibrium position is in
a direction perpendicular to the direction of propagation. Light is a transverse wave.

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Harmonic
** -
Waves that take on the shape determined by the harmonic functions, sine and cosine. They are also called
sinusoidal waves or simple harmonic waves. Not only are these functions simple to deal with, but Fourier
analysis tells us that any wave can by synthesized by the superposition of harmonic waves.

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Phase
** -
In a harmonic function, the phase the argument of the sine or cosine function. In general it is given by: *ψ*(*x*, *t*) = (*kx* - *σt* + *ε*), where *ε* is called the initial phase. The phase determines
whether the wave is at a peak or trough or somewhere in between at a particular point in space and time.

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Amplitude
** -
The maximum disturbance, or the maximum displacement of the particles of the medium from their
equilibrium position. This is given by the constant term preceding the sine or cosine in a harmonic wave.

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Wavelength
** -
The wavelength of a wave is denoted *λ* and is the distance in space from one peak to any adjacent
peak, one trough to any adjacent trough, or indeed from any one point to a similar point on an adjacent
cycle. In other words, it is the number of units of length per complete wave cycle.

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Wavenumber
** -
Denoted *k*, the wavenumber is the constant that appears in the expression for the phase (usually the
coefficient of *x*). It is defined as *k* = 2*Π*/*λ*, and as such as units of inverse length.

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Frequency
** -
Denoted *ν*, the frequency is the number of complete wave cycles which pass a given point in space in
one unit of time (one second). It is the inverse of the period of the wave (and has units of inverse time, or 1
Hertz = 1 second^{-1}), and is given by *ν* = *v*/*λ*.

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Angular frequency
** -
Denoted *σ*, the angular frequency is the number of radians of a harmonic wave which pass a given
point per unit time (second). One complete wave cycle has 2*Π* radians, so the angular frequency is given
by *σ* = 2*Πν*. It also has units of inverse time (or radians per second, but radians are not proper
units and are dimensionless).

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Period
** -
The amount of time *T* taken for a complete wave cycle to pass a particular point. In other words, the
number of units of time per wave. It has units of time and is the inverse of the frequency.

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Phase velocity
** -
Is the speed of propagation of the condition of constant phase. What this means is that the phase velocity is
the speed at which you would have to be traveling alongside the wave in order to observe to change in the
phase of the wave next to you. In other words, it is the speed of propagation of a particular crest or trough.
It is not hard to deduce from the wave equation that *v* = *σ*/*k* = *λν*.

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Photon
** -
A quanta of light. Photons are particles that have no mass or charge and travel only at speed *c*, regardless
of medium or reference frame. They have an energy given by *E* = *hν* where *ν* is the frequency of
the light to which they correspond, and *h* = 6.626×10^{-34} J.s (Planck's constant). We can
account for the behavior of light by considering it as consisting of a very large number of photons. In this
regime the electromagnetic field appears continuous and the granularity of the light beam is negligible.

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Poynting vector
** -
Named after John Henry Poynting (1852-1914), this is given by:

This is the unit power per area crossing a surface with normal

. The direction of

is
parallel to the direction of the propagation of the light ray.

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Spherical wave
** -
The linear wave described in

Waves is not the only solution to the wave equation. In
three-dimensions plane and spherical waves can also exist. In spherical waves, the disturbance of the
medium is a function of

*r*, isotropic in all directions (think of two-dimensional circular waves generated by
dropping a stone into a pond). The wavefronts are spheres. The symmetry of spherical waves makes them
very important when optics is treated in three-dimensions.

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Absorb
** -
When light is incident on an atom, if its frequency corresponds to a possible quantum jump between energy
levels for the electrons in that atom, it may be absorbed, and the atom excited into a higher energy state.
Usually this excitation energy is very rapidly transferred, via collisions into thermal motion (for this reason it
is sometimes called dissipative absorption).

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Resonant frequency
** -
The resonance frequencies of an atom are those frequencies which correspond via

*E* = *hν* to energies at
which the electron can make jumps between quantized energy states. At these frequencies, light is likely to
be absorbed by atoms. Confusingly, the natural frequency at which electrons in an atom can vibrate as
atomic dipoles, given by

*σ*_{0} = is also sometimes called the resonant frequency. A
forced oscillation will be most effective when it is close to the resonant frequency.

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Formulae