Таласна дужина

The wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests (on top), or troughs (on bottom), or corresponding zero crossings as shown.

Објашњење таласне дужине

Таласна дужина је карактеристика сваког таласа, као и фреквенција. Сваки талас има и своју амплитуду која означава интензитет таласа. Путовање таласа описује синусна функција, а таласна дужина код трансверзалних таласа је дужина између два суседна врха таласа (или два удубљења). Мерна јединица за таласну дужину у Међународном систему јединица СИ је метар. Таласна дужина је такође и најкраћа раздаљина између две честице које осцилују у истој фази. Таласна дужина означава се са грчким словом lambda - $\lambda$

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. 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.

Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.

Wavelength depends on the medium (for example, vacuum, air, or water) that a wave travels through. Examples of waves are sound waves, light, water waves and periodic electrical signals in a conductor. A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary.

The range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum.

Sinusoidal waves

In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by

$\lambda ={\frac {v}{f}}\,\,,$

where v is called the phase speed (magnitude of the phase velocity) of the wave and f is the wave's frequency. In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear.

In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×10⁸ m/s. Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about: 3×10⁸ m/s divided by 10⁸ Hz = 3 metres. The wavelength of visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm (for other examples, see electromagnetic spectrum).

For sound waves in air, the speed of sound is 343 m/s (at room temperature and atmospheric pressure). The wavelengths of sound frequencies audible to the human ear (20 Hz–20 kHz) are thus between approximately 17 m and 17 mm, respectively. Somewhat higher frequencies are used by bats so they can resolve targets smaller than 17 mm. Wavelengths in audible sound are much longer than those in visible light.

Sinusoidal standing waves in a box that constrains the end points to be nodes will have an integer number of half wavelengths fitting in the box.

A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue)

Standing waves

A standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, and the wavelength is twice the distance between nodes.

The upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box (an example of boundary conditions) determining which wavelengths are allowed. For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall.

The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Consequently, wavelength, period, and wave velocity are related just as for a traveling wave. For example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum.

Mathematical representation

Traveling sinusoidal waves are often represented mathematically in terms of their velocity v (in the x direction), frequency f and wavelength λ as:

$y(x,\ t)=A\cos \left(2\pi \left({\frac {x}{\lambda }}-ft\right)\right)=A\cos \left({\frac {2\pi }{\lambda }}(x-vt)\right)$

where y is the value of the wave at any position x and time t, and A is the amplitude of the wave. They are also commonly expressed in terms of wavenumber k (2π times the reciprocal of wavelength) and angular frequency ω (2π times the frequency) as:

$y(x,\ t)=A\cos \left(kx-\omega t\right)=A\cos \left(k(x-vt)\right)$

in which wavelength and wavenumber are related to velocity and frequency as:

$k={\frac {2\pi }{\lambda }}={\frac {2\pi f}{v}}={\frac {\omega }{v}},$

or

$\lambda ={\frac {2\pi }{k}}={\frac {2\pi v}{\omega }}={\frac {v}{f}}.$

In the second form given above, the phase (kx − ωt) is often generalized to (k•r − ωt), by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r. In that case, the wavenumber k, the magnitude of k, is still in the same relationship with wavelength as shown above, with v being interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction.

Generalizations to sinusoids of other phases, and to complex exponentials, are also common; see plane wave. The typical convention of using the cosine phase instead of the sine phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave

$Ae^{i\left(kx-\omega t\right)}.$

Види још

• Фреквенција
• Таласна једначина
• Електромагнетни спектар
• Спектар

Литература

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• Feynman, R. P.; Hibbs, A. R. (1965), Quantum Mechanics and Path Integrals, New York: McGraw-Hill, ISBN 978-0-07-020650-2  (Dover, 2010, ISBN 0-486-47722-3.)
• Wheeler, Nicholas (2004), Energetics of a Gaussian wavepacket 
• Jones, Peter Ward (2001). Calculation of Small-Angle Scattering Patterns. Oxford Music Online. Oxford University Press. 
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• Patel, R.; Achamfuo-Yeboah, S.; Light R.; Clark M. (2014). „Widefield two laser interferometry”. Optics Express. 22 (22): 27094—27101. Bibcode:2014OExpr..2227094P. PMID 25401860. doi:10.1364/OE.22.027094 . 
• Born, Max; Wolf, Emil (1999). Principles of Optics. Cambridge: Cambridge University Press. ISBN 0-521-64222-1. 
• Levitin, Daniel J. (2006). This is Your Brain on Music: The Science of a Human Obsession. Dutton. ISBN 978-0525949695. 
• Greene, Brian (1999). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: W.W. Norton. стр. 97–109. ISBN 978-0-393-04688-5. 
• Halliday, David; Resnick, Robert; Walker, Jearl (2005). Fundamentals of Physics (7th изд.). John Wiley & Sons. ISBN 0-471-42959-7. 
• Serway, Raymond A.; Faughn, Jerry S. (1992). College Physics (3rd изд.). Saunders College Publishing. ISBN 0-03-076377-0. 
• Streets, J. (2010). „Chapter 16 - Superposition and Standing Waves” (PDF). Department of Physics. PHYS122 Fundamentals of Physics II. University of Maryland. Приступљено 23. 8. 2020. CS1 одржавање: Формат датума (веза)

Спољашње везе

• Conversion: Wavelength to Frequency and vice versa – Sound waves and radio waves
• Teaching resource for 14–16 years on sound including wavelength
• The visible electromagnetic spectrum displayed in web colors with according wavelengths