Standing Wave

standing wave [′stand·iŋ ′wāv]

(physics)

A wave in which the ratio of an instantaneous value at one point to that at any other point does not vary with time. Also known as stationary wave.

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Standing Wave 

(or stationary wave), a wave arising as a result of the interference of two waves propagating in opposite directions. In practice, a standing wave is formed, owing to the superposition of the incident wave and the reflected wave, when a wave is reflected from an obstacle or inhomogeneity. Different parts of a standing wave oscillate in the same phase but with different amplitudes (see Figure 1). In contrast to a traveling wave, there is no net energy flow in a standing wave.

Figure 1. Pressure and velocity distribution in a standing wave in a tube with a closed end and a tube with an open end

Standing waves can arise in, for example, an elastic system, such as a bar or a column of air located inside a tube that is closed at one end and in which traveling waves are produced by the oscillation of a piston in the tube. The traveling waves are reflected from the boundaries of the system, and standing waves are formed in the system as a result of the superposition of the incident and reflected waves. In this case, displacement (velocity) nodes and antinodes are formed along the length of the column. The displacement nodes are planes perpendicular to the axis of the column; in these planes, there are no displacements of air particles but the pressure amplitudes are at a maximum. The antinodes are planes in which the displacements are at a maximum and the pressures are equal to zero. The displacement nodes and antinodes are located in the tube at quarter-wavelength distances. Moreover, a displacement node and a pressure antinode are always formed at the solid wall at the end of the tube. A similar situation is observed if this solid wall is removed, but the velocity antinode and pressure node are then located approximately in the plane of the opening. Standing waves can be formed in any volume with definite boundaries and a sound source, but in this case the standing waves have a more complicated structure.

Any wave process associated with the propagation of disturbances can be accompanied by the formation of standing waves. Standing waves can arise not only in gaseous, liquid, and solid media but also in a vacuum where electromagnetic disturbances are propagated and reflected. Such waves may arise, for example, in electric transmission lines. The antenna of a radio transmitter is often made in the form of a linear dipole or system of di-poles, along which a standing wave is formed. Specific types of standing waves are established in sections of wave guides and in closed, variously shaped volumes used as resonators in microwave equipment. The electric and magnetic fields in electromagnetic standing waves are separated in much the same way as the displacement and pressure in elastic standing waves.

Strictly speaking, pure standing waves can be established only in the absence of damping in the medium and only when there is total reflection of waves from the boundary. Traveling waves, which feed energy to points of energy absorption or emission, are usually present in addition to standing waves.

In optics, standing waves with visible maxima and minima of the electric field may also be formed. If the light is nonmonochro-matic, then the electric field antinodes of different wavelengths are located at different points in the standing waves, and separation of colors is often observed.

REFERENCE

Gorelik, G. S. Kolebaniia i volny. Moscow-Leningrad, 1950.