Wavenumber–frequency diagram

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A wavenumber–frequency diagram is a plot displaying the relationship between the wavenumber (spatial frequency) and the frequency (temporal frequency) of certain phenomena. Usually frequencies are placed on the vertical axis, while wavenumbers are placed on the horizontal axis. [1] [2]

A plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. The plot can be drawn by hand or by a mechanical or electronic plotter. Graphs are a visual representation of the relationship between variables, which are very useful for humans who can then quickly derive an understanding which may not have come from lists of values. Graphs can also be used to read off the value of an unknown variable plotted as a function of a known one. Graphs of functions are used in mathematics, sciences, engineering, technology, finance, and other areas.

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

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In the atmospheric sciences, these plots are a common way to visualize atmospheric waves.

An atmospheric wave is a periodic disturbance in the fields of atmospheric variables which may either propagate or not. Atmospheric waves range in spatial and temporal scale from large-scale planetary waves to minute sound waves. Atmospheric waves with periods which are harmonics of 1 solar day are known as atmospheric tides.

In the geosciences, especially seismic data analysis, these plots also called fk plot, in which energy density within a given time interval is contoured on a frequency-versus-wavenumber basis. They are used to examine the direction and apparent velocity of seismic waves and in velocity filter design.

A velocity filter removes interfering signals by exploiting the difference between the travelling velocities of desired seismic waveform and undesired interfering signals.

Origins

In general, the relationship between wavelength ${\displaystyle \lambda }$, frequency ${\displaystyle \nu }$, and the phase velocity ${\displaystyle v_{p}}$ of a sinusoidal wave is:

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

${\displaystyle v_{\text{p}}=\lambda \nu }$

Using the wavenumber (${\displaystyle k=2\pi /\lambda }$) and angular frequency (${\displaystyle \omega =2\pi \nu }$) notation, the previous equation can be rewritten as

In physics, angular frequencyω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

${\displaystyle v_{\text{p}}={\frac {\omega }{k}}}$

On the other hand, the group velocity is equal to the slope of the wavenumber–frequency diagram:

The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

${\displaystyle v_{\text{g}}={\frac {\partial \omega }{\partial k}}}$

Analyzing such relationships in detail often yields information on the physical properties of the medium, such as density, composition, etc.

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The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, i.e., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

In electromagnetics and antenna theory, antenna aperture, effective area, or receiving cross section, is a measure of how effective an antenna is at receiving the power of electromagnetic radiation. The aperture is defined as the area, oriented perpendicular to the direction of an incoming electromagnetic wave, which would intercept the same amount of power from that wave as is produced by the antenna receiving it. At any point , a beam of electromagnetic radiation has an irradiance or power flux density which is the amount of power passing through a unit area of one square meter. If an antenna delivers watts to the load connected to its output terminals when irradiated by a uniform field of power density watts per square meter, the antenna's aperture in square meters is given by:

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

The four-frequency of a massless particle, such as a photon, is a four-vector defined by

In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. This is the emitted component of radiosity. The SI unit of radiant exitance is the watt per square metre, while that of spectral exitance in frequency is the watt per square metre per hertz (W·m−2·Hz−1) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m−3)—commonly the watt per square metre per nanometre. The CGS unit erg per square centimeter per second is often used in astronomy. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

Plasma parameters define various characteristics of a plasma, an electrically conductive collection of charged particles that responds collectively to electromagnetic forces. Plasma typically takes the form of neutral gas-like clouds or charged ion beams, but may also include dust and grains. The behaviour of such particle systems can be studied statistically.

Zero sound is the name given by Lev Landau to the unique quantum vibrations in quantum Fermi liquids.

The Frank–Tamm formula yields the amount of Cherenkov radiation emitted on a given frequency as a charged particle moves through a medium at superluminal velocity. It is named for Russian physicists Ilya Frank and Igor Tamm who developed the theory of the Cherenkov effect in 1937, for which they were awarded a Nobel Prize in Physics in 1958.

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

The Planck–Einstein relation is also referred to as the Einstein relation, Planck's energy–frequency relation, the Planck relation, and the Planck equation. Also the eponym Planck formula belongs on this list, but also often refers to Planck's law instead. These various eponyms are far from standard; they are used only sporadically, neither regularly nor very widely. They refer to a formula integral to quantum mechanics, which states that the energy of a photon, E, known as photon energy, is proportional to its frequency, ν:

In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations. In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.

In fluid dynamics, Beltrami flows are flows in which the vorticity vector and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.

In physics, sinusoidalplane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane.

References

1. R.G. Fleagle; J.A. Businger (1990). An introduction to atmospheric physics. Academic Press. pp. 183–198. ISBN   978-0-12-260355-6.
2. J. Pedlosky (1998). Geophysical fluid dynamics (2nd ed.). Springer. pp. 676–678. ISBN   978-0-387-96387-7.