Longitudinal wave

Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also referred to as compressional waves or compression waves.

Plane pressure pulse wave

Representation of the propagation of an omnidirectional pulse wave on a 2d grid (empirical shape)

Non-electromagnetic

Longitudinal waves include sound waves (alternation in pressure, particle displacement, or particle velocity propagated in an elastic material) and seismic P-waves (created by earthquakes and explosions).

Sound waves

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

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

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 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

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

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 pressure of the medium through which it propagates.

Pressure waves

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

$y(x,t)\, = y_0 \cos(k x - \omega t +\varphi)$

where:

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

The force acting to return the medium to its original position is provided by the medium's bulk modulus.^[1]

Electromagnetic

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).^[2] However, waves can exist in plasma or confined spaces. These are called plasma waves and can be longitudinal, transverse, or a mixture of both.^[2][3] Plasma waves can also occur in force-free magnetic fields.

In the early development of electromagnetism there was some suggesting that longitudinal electromagnetic waves existed in a 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.^[4] But it should be stated that Maxwell's equations do lead to the appearance of longitudinal waves under some circumstances in either 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.^[5] The longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in a cavity. The longitudinal modes correspond to the 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.^[6]

See also

• Longitudinal mode
• Transverse wave

References

1. ^ Weisstein, Eric W., "P-Wave". Eric Weisstein's World of Science.
2. ^ ^{a b} David J. Griffiths, Introduction to Electrodynamics, ISBN 0-13-805326-X
3. ^ John D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X.
4. ^ Heaviside, Oliver, "Electromagnetic theory". Appendices: D. On compressional electric or magnetic waves. Chelsea Pub Co; 3rd edition (1971) 082840237X
5. ^ Corum, K. L., and J. F. Corum, "The Zenneck surface wave", Nikola Tesla, Lightning observations, and stationary waves, Appendix II. 1994.
6. ^ Haifeng Wang, Luping Shi, Boris Luk'yanchuk, Colin Sheppard and Chong Tow Chong, "Creation of a needle of longitudinally polarized light in vacuum using binary optics," Nature Photonics, Vol.2, pp 501-505, 2008

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 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, Kettering University Applied Physics.
• Longitudinal Waves, with animations "The Physics Classroom"

External links

Websites

• An interactive simulation of longitudinal travelling wave
• Wave types explained using high speed film and animations