# Longitudinal wave

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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.

Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves, rarefaction waves also exist in nature. A common rarefaction wave is the area of low relative pressure following a shock wave.

## Contents

The other main type of wave is the transverse wave, in which the displacements of the medium are at right angles to the direction of propagation. Some transverse waves are mechanical, meaning that the wave needs an elastic medium to travel through. Transverse mechanical waves are also called "shear waves".

A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer.

By acronym, "longitudinal waves" and "transverse waves" were occasionally abbreviated by some authors as "L-waves" and "T-waves" respectively for their own convenience. [1] While these two acronyms have specific meanings in seismology (L-wave for Love wave [2] or long wave [3] ) and electrocardiography (see T wave), some authors chose to use "l-waves" (lowercase 'L') and "t-waves" instead, although they are not commonly found in physics writings except for some popular science books. [4]

An acronym is a word or name formed as an abbreviation from the initial components of a phrase or a word, usually individual letters and sometimes syllables.

Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake environmental effects such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic, atmospheric, and artificial processes such as explosions. A related field that uses geology to infer information regarding past earthquakes is paleoseismology. A recording of earth motion as a function of time is called a seismogram. A seismologist is a scientist who does research in seismology.

In elastodynamics, Love waves, named after Augustus Edward Hough Love, are horizontally polarized surface waves. The Love wave is a result of the interference of many shear waves (S–waves) guided by an elastic layer, which is welded to an elastic half space on one side while bordering a vacuum on the other side. In seismology, Love waves are surface seismic waves that cause horizontal shifting of the Earth during an earthquake. Augustus Edward Hough Love predicted the existence of Love waves mathematically in 1911. They form a distinct class, different from other types of seismic waves, such as P-waves and S-waves, or Rayleigh waves. Love waves travel with a lower velocity than P- or S- waves, but faster than Rayleigh waves. These waves are observed only when there is a low velocity layer overlying a high velocity layer/ sub–layers.

Longitudinal waves include sound waves (vibrations in pressure, particle of displacement, and particle velocity propagated in an elastic medium) and seismic P-waves (created by earthquakes and explosions). In longitudinal waves, the displacement of the medium is parallel to the propagation of the wave, and waves can be either straight or round. A wave along the length of a stretched Slinky toy, where the distance between coils increases and decreases, is a good visualization.

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road.

In physics, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate forces are applied to them. If the material is elastic, the object will return to its initial shape and size when these forces are removed.

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 gases, liquids, or solids.

## Sound waves

In the case of longitudinal harmonic sound waves, the frequency and wavelength can be described by the formula

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.

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.

${\displaystyle y(x,t)=y_{0}\cos {\Bigg (}\omega \left(t-{\frac {x}{c}}\right){\Bigg )}}$

where:

• y is the displacement of the point on the traveling sound wave;
• x is the distance the point has traveled from the wave's source;
• t is the time elapsed;
• y0 is the amplitude of the oscillations,
• c is the speed of the wave; and
• ω is the angular frequency of the wave.

The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.

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.

The quantity x/c is the time that the wave takes to travel the distance x.

The ordinary frequency (f) of the wave is given by

${\displaystyle f={\frac {\omega }{2\pi }}.}$

The wavelength can be calculated as the relation between a wave's speed and ordinary frequency.

${\displaystyle \lambda ={\frac {c}{f}}.}$

For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave.

Sound's propagation speed depends on the type, temperature, and composition of the medium through which it propagates.

## Pressure waves

In an elastic medium with rigidity, a harmonic pressure wave oscillation has the form,

${\displaystyle y(x,t)\,=y_{0}\cos(kx-\omega t)}$
${\displaystyle y(x,t)\,=y_{0}\cos(kx-\omega t+\varphi )}$

where:

• y0 is the amplitude of displacement,
• k is the angular wavenumber,
• x is the distance along the axis of propagation,
• ω is the angular frequency,
• t is the time, and
• φ is the phase difference.

The restoring force, which acts to return the medium to its original position, is provided by the medium's bulk modulus. [5]

## Electromagnetic waves

Maxwell's equations lead to the prediction of electromagnetic waves in a vacuum, which are transverse (in that the electric fields and magnetic fields vary perpendicularly to the direction of propagation). [6] However, waves can exist in plasmas or confined spaces, called plasma waves, which can be longitudinal, transverse, or a mixture of both. [6] [7] Plasma waves can also occur in force-free magnetic fields. [8]

In the early development of electromagnetism, there were some like Alexandru Proca (1897-1955) known for developing relativistic quantum field equations bearing his name (Proca's equations) for the massive, vector spin-1 mesons. In recent decades, some extended electromagnetic theorists, such as Jean-Pierre Vigier and Bo Lehnert of the Swedish Royal Society, have used the Proca equation in an attempt to demonstrate photon mass [9] as a longitudinal electromagnetic component of Maxwell's equations, suggesting that longitudinal electromagnetic waves could exist in a Dirac polarized vacuum.

After Heaviside's attempts to generalize Maxwell's equations, Heaviside came to the conclusion that electromagnetic waves were not to be found as longitudinal waves in " free space " or homogeneous media. [10] But Maxwell's equations do lead to the appearance of longitudinal waves under some circumstances, for example, in plasma waves or guided waves. Basically distinct from the "free-space" waves, such as those studied by Hertz in his UHF experiments, are Zenneck waves. [11] The longitudinal modes of a resonant cavity are the particular standing wave patterns formed by waves confined in a cavity. The longitudinal modes correspond to those wavelengths of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. Recently, Haifeng Wang et al. proposed a method that can generate a longitudinal electromagnetic (light) wave in free space, and this wave can propagate without divergence for a few wavelengths. [12]

## Related Research Articles

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

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

In the physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes. Mathematically a plane wave takes the form

Polarization is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves in solids. In some types of transverse waves, the wave displacement is limited to a single direction, so these also do not exhibit polarization; for example, in surface waves in liquids, the wave displacement of the particles is always in a vertical plane.

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 radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields. For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. When relating to music, normal modes of vibrating instruments are called "harmonics" or "overtones".

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 plasma physics, an upper hybrid oscillation is a mode of oscillation of a magnetized plasma. It consists of a longitudinal motion of the electrons perpendicular to the magnetic field with the dispersion relation

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:

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

Acoustic waves are a type of longitudinal waves that propagate by means of adiabatic compression and decompression. Longitudinal waves are waves that have the same direction of vibration as their direction of travel. Important quantities for describing acoustic waves are sound pressure, particle velocity, particle displacement and sound intensity. Acoustic waves travel with the speed of sound which depends on the medium they're passing through.

Sinusoidal plane-wave solutions are particular solutions to the electromagnetic wave equation.

The Appleton–Hartree equation, sometimes also referred to as the Appleton–Lassen equation is a mathematical expression that describes the refractive index for electromagnetic wave propagation in a cold magnetized plasma. The Appleton–Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartree and German radio physicist H. K. Lassen. Lassen's work, completed two years prior to Appleton and five years prior to Hartree, included a more thorough treatment of collisional plasma; but, published only in German, it has not been widely read in the English speaking world of radio physics. Further, regarding the derivation by Appleton, it was noted in the historical study by Gilmore that Wilhelm Altar first calculated the dispersion relation in 1926.

Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plate normal. In 1917, the English mathematician Horace Lamb published his classic analysis and description of acoustic waves of this type. Their properties turned out to be quite complex. An infinite medium supports just two wave modes traveling at unique velocities; but plates support two infinite sets of Lamb wave modes, whose velocities depend on the relationship between wavelength and plate thickness.

Brillouin spectroscopy is an empirical spectroscopy technique which allows for the determination of elastic moduli for a given material. The technique uses inelastic scattering of light when it encounters acoustic phonons in a crystal, a process known as Brillouin scattering, to determine phonon energies and therefore interatomic potentials of a material. The scattering occurs when an electromagnetic wave interacts with a density wave, photon-phonon scattering.

In the physics of continuous media, spatial dispersion is a phenomenon where material parameters such as permittivity or conductivity have dependence on wavevector. Normally, such a dependence is assumed to be absent for simplicity, however spatial dispersion exists to varying degrees in all materials.

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