Construct · 3 fields · 3 citations

standing wave: one construct, many systems

One name. One construct. Different systems. One map.

● invariant · one construct, three systems Nearest construct by meaning·not indexed
same wave eigenmodes, now in fieldsstationary states are standing matter wavesstandingwave3 SYSTEMSStanding waveRAYLEIGH · THE MODESEM standing waveBORN & WOLF · CAVITYStationary stateGRIFFITHS · EIGENSTATE

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1877Standing wave1999EM standing wave2018Stationary state
2026

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Acoustics

A fixed pattern of nodes in a vibrating medium

A standing wave is the stationary pattern formed when two waves of the same frequency travel in opposite directions and interfere, giving fixed points of no motion (nodes) and points of maximum motion (antinodes) rather than a travelling disturbance. In acoustics this is how a stretched string or an air column resonates, selecting a discrete set of frequencies (the harmonic series) that give a musical instrument its pitch and timbre. Rayleigh's The Theory of Sound laid out the mathematics of these vibrating systems. The referent is the set of resonant modes of a bounded medium, and it is the same construct that recurs in electromagnetism and quantum mechanics.[1]

Unit
The resonant-mode pattern of a bounded vibrating medium, with its nodes and allowed frequencies
Recovery means
Not applicable
Measured by
Found from the system's boundary conditions as a discrete set of mode frequencies and node/antinode positions

Electromagnetism

electromagnetic waves trapped between reflectors that interfere to form fixed nodes and antinodes, sustaining only certain wavelengths.

The same construct governs electromagnetic fields confined between reflectors or within a cavity: counter-propagating waves interfere to form a standing wave with fixed nodes and antinodes of the field, so only certain wavelengths (the cavity modes) are sustained. Born and Wolf's Principles of Optics develops the interference and resonator theory behind it. It is the identical eigenmode idea as a vibrating string, realised in the electromagnetic field and read from cavity mode wavelengths and the standing-wave ratio rather than from the modes of a string.[2]

Unit
The resonant field pattern of an optical or microwave cavity, with its mode wavelengths
Recovery means
Not applicable
Measured by
Characterised by cavity mode spacing, the standing-wave ratio, and the location of field nodes and antinodes

Quantum mechanics

A particle's stationary state as a standing wave

In quantum mechanics the same mathematics describes a bound particle: a stationary state is a standing wave of the particle's wavefunction, so confining a particle (as in a particle in a box or an atom) admits only discrete energy levels, the system's quantised eigenstates. Griffiths's Introduction to Quantum Mechanics derives these standing-wave solutions of the Schrödinger equation. It is the same bounded-eigenmode construct as a resonating string or an optical cavity, here realised as matter waves and read from energy levels rather than acoustic frequencies.[3]

Unit
The stationary states and quantised energy levels of a bound quantum system
Recovery means
Not applicable
Measured by
Obtained as the eigenfunctions and discrete energy eigenvalues of the Schrödinger equation under the system's boundary conditions

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References

  1. Rayleigh, J. W. S. (1945). The theory of sound (Vol. 1). Dover Publications. (Original work published 1877) ISBN 9780486602929
  2. Born, M., & Wolf, E. (1999). Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (7th ed.). Cambridge University Press. ISBN 9780521642224
  3. Griffiths, D. J., & Schroeter, D. F. (2018). Introduction to quantum mechanics (3rd ed.). Cambridge University Press. ISBN 9781107189638

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