The Manley–Rowe relations are

mathematical expression In mathematics, an expression or mathematical expression is a finite combination of Glossary of mathematical symbols, symbols that is well-formed formula, well-formed according to rules that depend on the context. Mathematical symbols can desi ...

s developed originally for

electrical engineer Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

s to predict the amount of

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...

in a wave that has multiple frequencies. They have since been found to describe systems in

non-linear optics Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typica ...

fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...

and the theory of non-linear dynamical systems, as they provide a pair of invariants or

conserved quantities In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant (mathematics), constant along each trajectory of the system. Not all systems have conserved quantities, and c ...

for the

three-wave equation In nonlinear systems, the three-wave equations, sometimes called the three-wave resonant interaction equations or triad resonances, describe small-amplitude waves in a variety of non-linear media, including electrical circuits and non-linear optics. ...

. For example, in a

resonant interaction In nonlinear systems, a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria coupling wave-vectors and the dispersion equation are met ...

in non-linear optics, the Manley–Rowe relations can be interpreted as saying one photon is created as two more are destroyed (or conversely, two are created when one is destroyed.) For the three-wave equation, the Manley–Rowe invariants can be related to the modular invariants

g_2

and

g_3

of the Weierstrass ℘-function. This essentially follows because the three-wave interaction has exact solutions that are given by

elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...

History

The original papers, written by two researchers at

Bell Labs Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial research and scientific development company owned by mult ...

, J. M. Manley and H. E. Rowe between 1956 and 1960 was for an

electrical circuit An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, ...

containing

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...

s and

inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...

s. One or more

oscillator Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

s, operating at specified frequencies, are connected to the input of this circuit. The Manley–Rowe relations predict the energy present in waves at various frequencies, including new frequencies (such as

harmonics A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...

and

sideband In radio communications, a sideband is a band of frequencies higher than or lower than the carrier frequency, that are the result of the modulation process. The sidebands carry the information transmitted by the radio signal. The sidebands co ...

s) that arise in the circuit due to

nonlinearity In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

. The theory is based partly on the principle of conservation of energy. It requires that

energy storage Energy storage is the capture of energy produced at one time for use at a later time to reduce imbalances between energy demand and energy production. A device that stores energy is generally called an accumulator or battery. Energy comes in ...

in the circuit is a

stationary process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...

that varies with time only due to the oscillations and not due to some steady increase or decrease with time. More precisely, the theory describes a

between waves at various different frequencies; the resonant interaction describes which frequencies can mix and interact, and the strengths by which they couple. Because the Manley–Rowe relations are based on general concepts like nonlinear

wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...

s and

conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...

, their use is not limited to the original application in radio-frequency electrical circuits. They have also found use in other scientific fields, for example

nonlinear optics Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typica ...

. In the electrical circuit for the original derivation of Manley–Rowe relations, capacitors and inductors store energy from a wave and then release it. Other physical systems that involve energy storage for waves, and nonlinear generation of new waves, can make use of the same relations. John Manley and Harrison Rowe were protégés of

Ralph Hartley Ralph Vinton Lyon Hartley (November 30, 1888 – May 1, 1970) was an American electronics researcher. He invented the Hartley oscillator and the Hartley transform, and contributed to the foundations of information theory. Biography Hartley was ...

at Bell Laboratories. The work with nonlinear reactances (inductors and capacitors) was started back in 1917 by John Burton and Eugene Peterson.Peterson E., “Atomic Physics and Circuit Theory”; Bell Laboratories Record, Volume 7, February 1929, pp 231–233 When Hartley joined Bell Laboratories after being part of Western Electric, he started a research group on nonlinear oscillations. This group was later joined by Peterson, Manley, and Rowe.

Notes

{{DEFAULTSORT:Manley-Rowe relations Applied mathematics Electrical engineering