# Wave propagation

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Wave propagation is any of the ways in which waves travel.

## Contents

With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.

Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The term vibration is precisely used to describe mechanical oscillation. Familiar examples of oscillation include a swinging pendulum and alternating current.

Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure.

In physics, a transverse wave is a moving wave whose oscillations are perpendicular to the direction of the wave.

For electromagnetic waves, propagation may occur in a vacuum as well as in a material medium. Other wave types cannot propagate through a vacuum and need a transmission medium to exist [ citation needed ].

Vacuum is space devoid of matter. The word stems from the Latin adjective vacuus for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a perfect vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum.

A transmission medium is something that can mediate the propagation of signals for the purposes of telecommunication.

## Reflection of plane waves in a half-space

The propagation and reflection of plane waves-- e.g. Pressure waves (P-wave) or Shear waves (SH or SV-waves) are phenomena that were first characterized within the field of classical seismology, and are now considered fundamental concepts in modern seismic tomography. The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding the Helmholtz decomposition of the displacement field, which is then substituted into the wave equation. From here, the plane wave eigenmodes can be calculated.

A P-wave is one of the two main types of elastic body waves, called seismic waves in seismology. P-waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any affected location or at a seismograph. P-waves may be transmitted through Solids and liquids, but not gasses.

In seismology, S-waves, secondary waves, or shear waves are a type of elastic wave, and are one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves.

Seismic tomography is a technique for imaging the subsurface of the Earth with seismic waves produced by earthquakes or explosions. P-, S-, and surface waves can be used for tomographic models of different resolutions based on seismic wavelength, wave source distance, and the seismograph array coverage. The data received at seismometers are used to solve an inverse problem, wherein the locations of reflection and refraction of the wave paths are determined. This solution can be used to create 3D images of velocity anomalies which may be interpreted as structural, thermal, or compositional variations. Geoscientists use these images to better understand core, mantle, and plate tectonic processes.

### SV wave propagation

The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of reflected SV wave is identical to the incidence wave, while the angle of reflected P wave is greater than the SV wave. Note also that for the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture. [1]

### P wave propagation

Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different.

## Wave velocity

Wave velocity is a general concept, of various kinds of wave velocities, for a wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as:

In physics and mathematics, the phase of a periodic function of some real variable is the relative value of that variable within the span of each full period.

In everyday use and in kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero.

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

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

where:

The phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency. The angular frequency ω cannot be chosen independently from the wavenumber k, but both are related through the dispersion relationship:

${\displaystyle \omega =\Omega (k).\,}$

In the special case Ω(k) = ck, with c a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed c. For instance electromagnetic waves in vacuum are non-dispersive. In case of other forms of the dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instance electromagnetic, sound or water waves).

The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation:

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

In almost all cases, a wave is mainly a movement of energy through a medium. Most often, the group velocity is the velocity at which the energy moves through this medium.

## Related Research Articles

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

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.

In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

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.

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.

A wavenumber–frequency diagram is a plot displaying the relationship between the wavenumber and the frequency of certain phenomena. Usually frequencies are placed on the vertical axis, while wavenumbers are placed on the horizontal axis.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.

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.

An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it. The medium has an intrinsic impedance, 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 electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods. The key property of a PML that distinguishes it from an ordinary absorbing material is that it is designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML to strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior.

Equatorial Rossby waves, often called planetary waves, are very long, low frequency waves found near the equator and are derived using the equatorial beta plane approximation.

Precursors are characteristic wave patterns caused by dispersion of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of electromagnetic radiation as well as in fluid surface waves and seismic waves.

In the fields of nonlinear optics and fluid dynamics, modulational instability or sideband instability is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.

Seismic inversion involves the set of methods which seismologists use to infer properties through physical measurements. Surface-wave inversion is the method by which elastic properties, density, and thickness of layers in the subsurface are obtained through analysis of surface-wave dispersion. The entire inversion process requires the gathering of seismic data, the creation of dispersion curves, and finally the inference of subsurface properties.

Mathematical Q models provide a model of the earth's response to seismic waves. In reflection seismology, the anelastic attenuation factor, often expressed as seismic quality factor or Q, which is inversely proportional to attenuation factor, quantifies the effects of anelastic attenuation on the seismic wavelet caused by fluid movement and grain boundary friction. When a plane wave propagates through a homogeneous viscoelastic medium, the effects of amplitude attenuation and velocity dispersion may be combined conveniently into the single dimensionless parameter, Q. As a seismic wave propagates through a medium, the elastic energy associated with the wave is gradually absorbed by the medium, eventually ending up as heat energy. This is known as absorption and will eventually cause the total disappearance of the seismic wave.

## References

1. The animations are taken from Poursartip, Babak (2015). "Topographic amplification of seismic waves". UT Austin.