Longitudinal wave

The longitudinal waves are waves in which the displacement through the medium is in the same direction of displacement of the wave, independent of direction.

Introduction

Representation of the propagation of a wave (not longitudinal) in a 2-dimensional reticulate.

Pressure wave.

Longitudinal waves are also called compression waves or compressibility waves, since they produce compression and rarefaction when they travel through a medium, and pressure waves produce increases and decreases in pressure.

The first figure illustrates the case of a sound wave. If the center of the figure is a point source generating the wave, the wave fronts move away from the focus, transmitting the sound through the propagation medium, such as air.

On the other hand, each particle of any wavefront oscillates in the direction of propagation, initially it is pushed in the direction of propagation due to the increase in pressure caused by the focus, returning to its previous position due to the effect of the pressure drop caused by its displacement. In this way, the consecutive layers of air (fronts) push each other transmitting sound, and for this reason sound waves are longitudinal waves, and they need a material medium to move (solid, liquid or gas).

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

Example of longitudinal waves

Included in the concept of longitudinal wave: sound waves (vibrations in pressure, particle displacement and velocity of particles propagated in an elastic medium) and seismic waves type P (created by earthquakes and explosions).

In longitudinal waves, the displacement of the medium is parallel to the propagation of the wave, which means that a wave that propagates the length of a spring, where the distance between the loops increases and decreases, is a good display. Sound waves in air are longitudinal pressure waves.

Sound wave

In the case of harmonic longitudinal sound waves, the relationships between displacement, time, and frequency can be described by the formula

and(x,t)=A0cors(ω ω (t− − xc)){displaystyle y(x,t)=A_{0}cos(omega (t-{frac {x}{c}{c})}}

where:

• "and"represents the shift of the point in the motion sound wave;
• "x"represents the distance that this point has traveled from the source of the wave;
• "t"represents the time elapsed;
• "A₀"represents the amplitude of the oscillations,
• "c"represents the speed of the wave;
• "ω " represents the angular frequency of the wave.

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

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

f=ω ω 2π π {displaystyle f={frac {omega }{2pi }}}}

The wavelength, which can be calculated as the ratio between speed and ordinary frequency:

λ λ =cf{displaystyle lambda ={frac {c}{f}}}}

It is the distance between two consecutive points along the propagation axis that have the same pressure.

For sound waves, the amplitude of the wave is the difference between the pressure of the air that has not been disturbed and the maximum pressure caused by the wave.

The speed of sound propagation depends on the type, temperature and composition of the medium through which it propagates.

Pressure Waves

In an elastic medium with a certain stiffness, a harmonic oscillation of a pressure wave has the form:

u(x,t)=u0cors(kx− − ω ω t+φ φ ){displaystyle u(x,t)=u_{0}cos(kx-omega t+varphi)}

where:

• u₀ is the breadth of displacement,
• k It's the wave number,
• x is the distance along the axis of propagation,
• ω is the angular frequency,
• t It's time.
• φ It's the phase difference.

The restoring force, which acts to return the medium to its original position, is given by the modulus of compressibility.

Electromagnetism

Maxwell's equations lead to the prediction of electromagnetic waves in a vacuum, which are transverse (where electric fields and magnetic fields vary perpendicularly in the direction of propagation). However, waves can exist in plasmas or confined spaces, so called plasma waves, which can be longitudinal, transverse, or a mixture of the two. In addition, plasma waves can also exist in magnetic fields that are free of forces.

At the beginning of the development of electromagnetism, there were some scientists like Alexandru Proca (1897-1955), known for developing the equations of relative quantum fields that bear his name (Proca's equations) to massive ones.

In recent decades, some widespread electromagnetic theorists, such as Jean-Pierre Vigier and Bo Lehnert of the Royal Swedish Society, have used the Proca equation in an attempt to demonstrate the photon mass as a longitudinal electromagnetic component of the Maxwell's equations, suggesting that longitudinal electromagnetic waves could exist in a Dirac polarized vacuum.

After several attempts by Heaviside to generalize Maxwell's equations, Heaviside came to the conclusion that electromagnetic waves do not exist in the form of longitudinal waves in "free space" or in homogeneous media. But Maxwell's equations lead to the appearance of longitudinal waves in some circumstances, for example, in plasma waves or in guided waves. Basically different from "free space" waves, as studied by Hertz in his UHF experiments, are Zenneck waves. The longitudinal modes of a resonant cavity are the particular standing wave patterns formed by the waves confined within of a cavity. Then, the longitudinal modes correspond to the wavelengths of the wave that are reinforced by constructive interference after many reflections from the reflective surfaces of the cavity.

Recently, Haifeng Wang together with other scientists have proposed a method that can generate longitudinal electromagnetic (light) wave in free space, and this wave can propagate without divergence.

Further reading

• Varadan, V. K., and Vasundara V. Varadan, "Elastic wave scattering and propagation". Attenuation due to scattering of ultrasonic compressional waves in granular media - A.J. Devaney, H. Levine, and T. Plona. Ann Arbor, Mich., Ann Arbor Science, 1982.
• Schaaf, John van der, Jaap C. Schouten, and Cor M. van den Bleek, "Experimental Observation of Pressure Waves in Gas-Solids Fluidized Beds". American Institute of Chemical Engineers. New York, N.Y., 1997.
• Krishan, S, and A Selim, "Generation of transverse waves by non-linear wave-wave interaction". Department of Physics, University of Alberta, Edmonton, Canada.
• Barrow, W. L., "Transmission of electromagnetic waves in hollow tubes of metal", Proc. IRE, vol. 24, pp. 1298–1398, October 1936.
• Russell, Dan, "Longitudinal and Transverse Wave Motion". Acoustics Animations, Pennsylvania State University, Graduate Program in Acoustics.
• Longitudinal Waves, with animations "The Physics Classroom"