Wave

• 1. Introduction

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  A wave is a propagating dynamic disturbance, a change from equilibrium, in one or more quantities. That single sentence hides an enormous range of phenomena. The same word covers the ripple on a pond, the light from a distant star, and a tremor that travels through the Earth's deep interior. It also covers things you might not expect to call a wave at all. The heights of spectators standing and sitting at a sporting event can behave as one. So can the level of anxiety spreading through a crowd of political protestors. What links a violin string, a beam of X-rays, and a ripple in spacetime itself? Why does no single definition manage to describe all of them at once? And how can something carry energy across a room while the material it moves through barely shifts from where it started? These are the questions that follow.

• 2. The Trouble With Defining It

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  No single definition adequately describes a wave. Scientists fall back on examples and shared characteristics instead, because the abstract view misses too much. One way to see a wave is as the dynamic manifestation of time-dependent field theory, an analogue to ballistics in particle mechanics. That framing is precise, but it loses the visual appeal of a vibrating stringed instrument or a fluid ripple. Viewed microscopically, a wave is a change in some physical property at a point in space, produced by a delayed response to changes in adjacent regions. The property might be pressure, temperature, height, or gravitational force. The delay matters. Because each region only responds to its neighbors after a finite time, a disturbance cannot leap across space instantly. A wave is the dynamic response of a field caused by effects that can only propagate at a finite speed. The rotation of an electric dipole produces electromagnetic waves, and the mutual rotation of binary stars produces gravitational waves. Both of these travel at the speed of light, set by that same finite limit on how fast a field can answer its surroundings.

• 3. Mechanical And Electromagnetic

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  Sound waves are variations in local pressure and particle motion that travel through a medium. They belong to the family classical physics studies most: mechanical waves. In a mechanical wave, stress and strain fields oscillate about a mechanical equilibrium. A local deformation creates stresses that strain neighboring particles, and the disturbance hands itself from particle to particle. Seismic waves, gravity waves, surface waves, and string vibrations all work this way. Electromagnetic waves work without any particles to pass through. Coupling between the electric and magnetic fields sustains them according to Maxwell's equations, and they can travel through a vacuum. James Clerk Maxwell showed in the 19th century that in a vacuum the electric and magnetic fields satisfy the wave equation, each moving at the speed of light. From that came the idea that light itself is an electromagnetic wave. Heinrich Hertz confirmed the unification of light and electromagnetic waves experimentally at the end of the 1880s. The whole family sorts by frequency into wavebands. Radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays form one continuous range, and the boundaries between bands are mostly arbitrary. Visible light is the exception, since it must be visible to the normal human eye.

• 4. Traveling, Standing, Solitary

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  When an entire waveform moves in one direction, it is a traveling wave. Send two identical periodic waves toward each other in opposite directions and you get something different. Their sum is a standing wave, whose envelope stays in a constant position. Standing waves are fundamental to music. When a violin string is displaced, transverse waves run out to the bridge and the nut where the string is fixed, and there they reflect back. At the bridge and nut the two opposed waves arrive in antiphase and cancel, producing a node. Halfway between two nodes sits an antinode, where the counter-propagating waves reinforce each other maximally. No net energy travels along the string over time. A soliton, or solitary wave, breaks from both patterns. It is a self-reinforcing wave packet that holds its shape while propagating at constant velocity. Solitons exist because nonlinear and dispersive effects in the medium cancel each other out. Dispersion, here, means the speed of a wave depends on its frequency. Solitons solve a widespread class of weakly nonlinear dispersive partial differential equations, the kind that describe many physical systems at once.

• 5. Plane Waves And Polarization

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  A plane wave is a mathematical idealization where the disturbance is identical along any infinite plane drawn perpendicular to a single direction of travel. The simplest of all is a sinusoidal plane wave, where every point undergoes simple harmonic motion at one frequency. In linear media this idealization becomes a tool of decomposition. Complicated waves can generally be written as a sum of many sinusoidal plane waves, each with its own direction and frequency. The orientation of the oscillation sorts plane waves into two kinds. A transverse wave has its field disturbance described by a vector perpendicular to the direction of travel. A longitudinal wave has those vectors aligned with the direction of travel instead. Mechanical waves include both, but electromagnetic plane waves are strictly transverse, and sound waves in fluids like air can only be longitudinal. The direction of the oscillating field relative to propagation is called polarization. Polarization arises when wave motion can occur in two orthogonal directions, so transverse waves can be polarized while longitudinal waves like sound cannot. A transverse wave that oscillates in only one plane is linearly polarized. Electromagnetic waves in free space can be polarized by passing them through a polarizing filter.

• 6. What Waves Do When They Travel

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  White light passed through a prism fans out into the colors of the rainbow. Isaac Newton was the first to recognize that this meant white light is a mixture of light of different colors. That spreading is dispersion, the frequency dependence of the refractive index, rooted in the atomic nature of materials. It is one of several behaviors waves show in standard situations. Reflection turns a wave back from a surface, with the angle of incidence equal to the angle of reflection about the normal line. Refraction changes a wave's speed, usually as it passes from one medium into another, with the two directions related through Snell's law and the refractive indices involved. Diffraction appears when a wave bends around an obstacle or spreads after passing through an opening, and the effect grows when the obstacle or opening is comparable in size to the wavelength. Absorption is the loss of energy as a material converts a wave into heat, which is why objects appear colored. Interference comes from the superposition principle. Where waves of the same frequency meet in a fixed phase relationship, some positions find them in phase and adding, others out of phase and canceling. The Doppler effect, the change in a wave's frequency for an observer moving relative to its source, is named after the Austrian physicist Christian Doppler, who described it in 1842.

• 7. Through Earth, Air, And String

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  Seismic waves are waves of energy traveling through the Earth's layers, set off by earthquakes, volcanic eruptions, magma movement, large landslides, and large man-made explosions. They split into body waves and surface waves. Body waves travel through the interior along paths controlled by density and modulus, which vary with temperature, composition, and material phase. The primary waves, or P waves, and the secondary waves, or S waves, are the two body types. Surface waves include Rayleigh waves, Love waves, and Stoneley waves. The speed of a transverse wave on a vibrating string is directly proportional to the square root of the tension divided by the linear mass density, the mass per unit length. Acoustic waves are compression waves moving through gases, liquids, solids, and plasmas, traveling at the square root of the adiabatic bulk modulus divided by the ambient density. A shock wave is a special case. When a disturbance moves faster than the local speed of sound in a fluid, it becomes one, marked by an abrupt, nearly discontinuous change in pressure, temperature, and density. Gravity waves restore equilibrium through gravity or buoyancy, with ripples on a pond being a familiar mix of transverse and longitudinal motion, so surface points trace orbital paths.

• 8. Waves As Particles

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  Louis de Broglie postulated that every particle with momentum has a wavelength, given by the Planck constant divided by the magnitude of the particle's momentum. This hypothesis sat at the foundation of quantum mechanics. The wavelength now carries his name. The electrons in a CRT display, for instance, have a de Broglie wavelength of about ten to the minus thirteen meters. A wave with a single definite wavelength is not localized in space, so it cannot stand for a particle pinned to a location. De Broglie's answer was to superpose many wavelengths around a central value into a wave packet, often given a Gaussian shape. The mathematics carries a strict trade. A narrow range of wavelengths is needed to produce a localized packet, and the more localized the envelope, the larger the spread in wavelengths required, an expression of the Heisenberg uncertainty principle. Two equations govern this quantum behavior. The Schrodinger equation describes the wave-like behavior of particles, with solutions called wave functions that give a particle's probability density. The Dirac equation is a relativistic wave equation for electromagnetic interactions. It accounted for the fine details of the hydrogen spectrum and implied a new form of matter, antimatter, previously unsuspected and unobserved, and later confirmed by experiment. The most recent of these confirmations sits at cosmic scale. The first observation of gravitational waves, disturbances in the curvature of spacetime predicted by Einstein's general relativity, was announced on the 11th of February 2016.