Standing wave

The title of this article is ambiguous. For information about the flow over obstacles, see standing wave (flow) .

A standing wave (black) as a superposition of two opposing traveling waves ( red and blue). The nodes of the standing wave are at the red points.

A standing wave , also called a standing wave , is a wave whose deflection always remains at zero at certain points. It can be understood as the superposition of two oppositely advancing waves of the same frequency and the same amplitude . The opposing waves can originate from two different pathogens or can be caused by the reflection of a wave on an obstacle. For water waves, see Clapotis .

A mechanical example of a one-dimensional standing wave is a rope wave, in which one end of the rope is moved up and down, creating a progressive wave in the rope. If the other end of the rope is attached, the wave is reflected there and runs back on the rope. As a result, you no longer see a progressive wave, but the rope performs an oscillation in which certain points remain at rest ( wave nodes or vibration nodes , also fast nodes ), while others with a large oscillation range ( amplitude ) oscillate back and forth ( wave antinodes or antinodes , too Quick belly ).

background

The distance between two wave nodes or two wave antinodes is half the wavelength of the original advancing waves.

With some standing waves a second quantity is important, the nodes and bellies of which are shifted by a quarter of a wavelength compared to those of the first quantity. In the case of a standing electromagnetic wave , the two quantities are the electric and magnetic field strength ; in the case of a standing sound wave in a wind instrument, the air pressure and the speed of sound . In these cases the terms belly and knot are therefore ambiguous; Designations such as pressure belly, pressure knot, fast belly (amplitude belly), fast knot (amplitude knot) are clear .

If the standing wave is generated by means of two in-phase (synchronously vibrating) exciters, there is an (amplitude) belly exactly in the middle between them, since the waves arrive here at the same time and always reinforce each other. A quarter of a wavelength away from this, the time difference of arrival is half an oscillation period. The waves are out of phase here and cancel each other out, creating a (amplitude) node. (Node) By generalizing this consideration one finds the conditions:

Belly : The distance d of an antinode from the center is a multiple of half the wavelength.

${\displaystyle d=n\cdot {\frac {\lambda }{2}}\qquad {\text{mit}}\qquad n=0,1,2,\dots }$

Node : The distance d of a vibration node from the center is a multiple of half the wavelength plus a quarter.

${\displaystyle d=\left(n+{\frac {1}{2}}\right)\cdot {\frac {\lambda }{2}}\qquad {\text{mit}}\qquad n=0,1,2,\dots }$

The energy transported by the wave is thrown back by the reflection. No energy transport therefore takes place on a waveguide with a standing wave created by complete reflection. If the wave is only partially reflected, there is a superposition of standing and advancing waves. In this case, energy is transported.

Standing waves between two reflectors

Only standing waves with certain wavelengths can form between two reflectors. The frequencies for these wavelengths are called natural frequencies or natural resonances.

Which boundary condition leads to the fact that the wavelengths cannot be arbitrary depends on the type of wave under consideration. For example, if the ends of a vibrating string are firmly clamped, there must be a vibration node at both ends, as shown in the figure below.

In the case of a standing electromagnetic wave , the electrical field strength on the reflective conductor must be zero, whereas the magnetic field strength there always has an antinode. In the resulting electromagnetic wave, the electric field and magnetic field are now phase-shifted by 90 °, with the E and H fields of the outgoing and returning waves being in phase.

With a standing (acoustic) longitudinal wave, there is always a sound pressure belly on every reflecting wall in a room; see room modes . In acoustics, it is primarily the size of the sound field that is of interest as sound pressure .

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  Electric field and magnetic field when reflected to a standing wave

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  The three largest wavelengths that result from a tightly clamped string.

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  Fundamental oscillation, including the first and second overtone

Standing wave ratio

A measure of the proportion of standing waves on an electrical conductor is the standing wave ratio (English: s tanding w ave r atio = SWR).

Applications

• Electromagnetic waves in the cavity , e.g. B. in particle accelerators
• Quantum mechanical explanation of the hydrogen atom
• Musical instruments (see e.g. string instruments , woodwind instruments , also Chladnian sound figure ).
• In concert halls, resonance caused by standing waves is avoided as far as possible . Here emphasis is placed on a uniformly high damping for all frequencies .
• An optical resonator stabilizes the wavelength (“color”) of the laser light .
• Standing waves (rhythmic oscillation) of the water in lakes, bays or docks are called Seiche (see also the water miracle of Constance ).

More pictures

The movement of the molecules can also move larger particles such as water droplets. The drops collect in the vibration nodes of a standing wave, which is formed due to a sound reflector arranged below the drops (reflector was not also photographed). The distance between the transducer face and the reflector must be selected to match the wavelength in air.

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  If short trains of waves meet, a standing wave only occurs during the time of the encounter.

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  For a circular standing wave there must be an integer ratio of circumference / wavelength.

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  Visualization of a standing acoustic wave in a sonicated glass tube filled with styrofoam balls in the dynamic

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  Standing wave forms (1–5) in a conical tube open on one side

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  Three-dimensional standing wave in the afterburner flame of a J58 engine on the test bench

literature

• W. Demtröder: Experimental Physics 1 . 5th edition, Springer 2008, ISBN 978-3-540-79294-9
• Andreas Friesecke: The audio encyclopedia. A reference book for sound engineers. Saur, Munich 2007, ISBN 978-3-598-11774-9 .
• Philipp Bohr: Physics. Textbook for the upper level, Norderstedt 2004, ISBN 3-833-45041-X .
• Peter Kaltenbach, Heinrich Meldau: Physics and radio technology for seafarers. Friedrich Vieweg & Sohn Verlag, Braunschweig 1938.
• FW Gundlach: Fundamentals of high frequency technology. Springer Verlag, Berlin / Heidelberg 1950.

See also

• Interference (physics)
• Modes of vibration
• Node surface
• Mach knot

Web links

Commons : Standing Waves  - collection of images, videos and audio files

• Dynamic illustration of the formation of standing waves and beats
• Effect of standing wave on room acoustics (sample calculation & graphic)
• The difference between standing waves in strings and the room modes as sound pressure distribution
• Standing waves (accessed September 25, 2017)