wave

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Wave refers to the propagation of vibration. The propagation of electromagnetic vibration is electromagnetic wave. For the sake of intuition, take the simplest example of rope shaking. There is a vibration source of up and down vibration at one end of the rope, and the vibration propagates forward along the rope. As a whole, the wave crest and trough move forward continuously, while the mass point of the rope only moves up and down and does not move forward.^[1]

Chinese name: wave
Foreign name: wave

Research category: physics
Definition: Physical quantities propagating in a specific form in space

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Physical definition

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wave

We will physical quantity Of Disturbance or Vibration The motion formed during point by point transmission in space is called wave. Although there are great differences between different forms of waves in terms of generation mechanism, propagation mode and interaction with matter, they have many commonalities in propagation, and can be used to Mathematical method Description and handling.

Generation and category

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wave

Volatility is material The important form of movement exists widely in nature. The transmitted physical quantity disturbance or vibration has many forms, mechanical vibration The transmission of constitutes a mechanical wave, electromagnetic field Transmission composition of vibration electromagnetic wave (including light wave ）, transmission composition of temperature change Temperature wave (See Liquid helium ）， Crystal lattice The transmission of vibration forms a lattice wave (see Lattice dynamics ）， Spin magnetic moment The disturbance of Ferromagnet Formed during internal propagation Spin wave (See Solid State Physics ）In fact, the disturbance of any macroscopic or microscopic physical quantity can form waves when it is transmitted in space. The most common mechanical wave is the mechanical motion of the particles constituting the medium (causing displacement , density pressure The propagation process in space, such as waves Surface wave , air or solid acoustic wave Etc. The prerequisite for generating these waves is the existence of adjacent particles in the medium Elastic force Or quasi elastic force Interaction force So that the vibration of a certain point can be transmitted to adjacent particles, so these waves are also called elastic wave 。 The physical quantity of vibration can be scalar , the corresponding wave is called Scalar wave (such as sound waves in the air), can also be vectors, and the corresponding waves are called vector waves (such as electromagnetic waves). A scale whose vibration direction is consistent with the wave propagation direction longitudinal wave , vertical is called transverse wave 。

Common characteristics

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The common feature of all forms of waves is their periodicity. Disturbed animals have time when their physical capacity changes Periodicity That is, the physical quantity at the same point will completely return to the original value after a cycle; It also has spatial periodicity when transmitting in space, that is, it passes through a certain Spatial distance The same vibration state (such as displacement And speed). Therefore, the physical capacity of disturbed animals u Is time t , again spatial location r Of Periodic function , function u （ t ， r ）Known as wave function or wave expression, it is a mathematical expression that quantitatively describes the wave process. In a broad sense, all descriptions motion state The functions with temporal periodicity and spatial periodicity can be called waves, such as Gravitational wave , microparticle Probability wave (See Wave particle duality ）Etc.

The common characteristics of various waves are as follows: ① reflection and refraction can be generated on the interface of different media Isotropic medium And comply with the laws of reflection and refraction (see Reflection law 、 Law of refraction ）；② General linear wave superposition Superposition principle of wave (See Independent propagation principle of light ）；③ When two or more waves are superimposed under certain conditions, interference can occur (see Interference of light ）；④ Waves can be generated when encountering obstacles on the propagation path Diffraction phenomenon (See diffraction of light); ⑤ The transverse wave can produce polarization (see Optical polarization phenomenon ）。

Simple harmonic

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Simple harmonic vibration The wave formed during space transmission is called Simple harmonic Its wave function is in the form of sine or cosine function. The vibration of each point has the same frequency f , called wave frequency , the reciprocal of the frequency is cycle , i.e T =1/ f 。 The distance between two adjacent points with identical vibration status in the wave propagation direction is called the wavelength λ Indicates that the reciprocal of the wavelength is called the wave number. Disturbance propagation distance per unit time u be called wave velocity 。 The relationship among wave speed, frequency and wavelength is u = fλ 。 The wave velocity is related to the type of wave and the nature of the propagation medium. Wavelike amplitude and phase Generally spatial location r Function of. Curved surface scale formed by points with equal phase in space Wavefront The surface formed by the front points of the wave must be an isophase surface, called wavefront or Wavefront 。 Waves are often divided into plane wave 、 spherical wave And cylindrical waves, whose wave surfaces are plane, spherical and cylindrical in turn. The vibration transmitted by the actual wave is not necessarily Simple harmonic vibration , but more complex periodic motion, called non simple harmonic. Any anharmonic wave can be regarded as the superposition of many simple harmonic waves with different frequencies.

Wave and energy

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Wave propagation is always accompanied by energy transmission, mechanical wave transmits mechanical energy, and electromagnetic wave transmits electromagnetic energy. The energy passing through the unit area perpendicular to the propagation direction in unit time is called wave Energy flux density , is often used to describe the intensity of waves. The energy flow density is proportional to the square of the amplitude. Generally, the phase propagation direction and energy propagation direction of the wave must be distinguished. identical phase (i.e Wavefront ）The propagation direction of normal The propagation velocity of direction and phase (or wave surface) is called Phase velocity Or normal speed. For isotropic media, the normal direction of wave is energy transfer When the direction is combined, the phase velocity and energy propagation velocity are the same. yes Anisotropic medium The normal direction of the wave is generally not coincident with the energy propagation direction, and the phase velocity is not equal to the energy propagation velocity.

In the process of wave motion, each particle of the medium only vibrates near the equilibrium position, and does not move along the direction of vibration propagation. Therefore, wave is the propagation of vibration state, not the propagation of matter itself.

Physical classification:

According to the nature, there are mainly four kinds - mechanical wave, electromagnetic wave, gravitational wave and material wave. Mechanical waves are caused by the propagation of disturbances in matter momentum And energy transmission. General objects are affected by a large number of interactions particle When one part of the object vibrates, the other parts will also vibrate one after another due to the interaction of particles, and the material itself has no corresponding large block movement. For example, waves, sound waves, water waves that travel along strings or springs. We call the substances that propagate waves medium, and they are deformation Of or elastic and continuous extension. about electromagnetic wave or Gravitational wave The medium is not necessary, and the transmitted disturbance is not the movement of the medium but the field - the former is the electromagnetic field moving in the form of waves in space, and the gravitational wave is space-time Bending moves in space in the form of waves. quantum mechanics Think that any material It has both particle property and wave property, that is, any substance has Wave particle duality , so there is the so-called Matter wave (Also called De Broglie wave ）, such as Electronic wave , neutron wave, etc.

According to the relationship between the vibration direction and the propagation direction, there are mainly three kinds - shear wave, longitudinal wave and spherical wave. particle The wave whose vibration direction is perpendicular to the propagation direction of the wave is called transverse wave The wave whose particle vibration direction is parallel to the wave propagation direction is called longitudinal wave 。

According to the shape of the wave: indeterminate, what the shape of the wave looks like is called a wave. as square wave (Some are also called rectangular waves) Sawtooth wave 、 Pulse wave , sine wave, cosine wave, etc.

By wavelength: long wave, medium wave, medium short wave and microwave.

By strength: Ordinary wave (general wave), shock wave.

There are also ultrasonic waves and infrasound waves in the sound wave, etc. There is no unified requirement, and what classification method is generally used under what conditions.

Waves in Nature

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wave

There are many forms of waves. The space on which it spreads can be filled with material or vacuum (Yes electromagnetic wave For). Some forms of waves can be felt by people's senses, while others cannot. People are most familiar with Surface wave , it has several types. For example, on the surface of deep water, there are surface wave , typical wavelength is 1m to 100m; There are mainly surface Tension is The wave length of the restoring force ripple is about 0.07m. These two waves often have a sinusoidal shape. In deep water Gravity wave , appears in areas with density stratification in the ocean. Not only in the ocean atmosphere Internal gravity waves can also occur. More widely encountered in the air, of course acoustic wave 。 Sound waves propagate disturbances of physical quantities such as pressure and density in the air No sound wave The deviation from the original value.

Waves are constantly occurring in solids. In terms of large objects, such as those often appearing on the earth seismic wave ； In terms of small objects, such as crystal atom lattice Dot matrix wave that is constantly driving. Special for physical property It can also excite some special waves: for example, there may be electroacoustic in piezoelectric materials surface wave ； stay Ferromagnetic material There are Spin wave , magnetoelastic wave, etc. stay Plasma It can also excite some different types of waves. Earthly the ionosphere Because the magnetic induction line moving with the fluid exerts magnetic pressure on the fluid, and because the fluid pressure can automatically adjust to balance the changing magnetic pressure, it can be excited to propagate along the magnetic induction line Magnetoacoustic wave 。 This is just one of many types of waves that can be generated in a plasma. Plasma gas can also have, for example, plasma- Electronic wave Plasma ion wave and so on. The solid can also be filled with streamers to form plasma, which can excite some characteristic waves, such as

wave

Including light waves electromagnetic wave Is one of the most closely related waves with human life. It can propagate not only in fluid, solid and plasma, but also in vacuum. The universe is filled with all kinds of light and radio.

General relativity And predicted existence Gravitational wave , is now experiment And Astronomical observation Confirmed. On February 11, 2016, LIGO Scientific cooperation organizations and Virgo The cooperative team announced that they have used the advanced LIGO detector to detect the gravitational wave signal from the merger of two black holes for the first time. It is believed that a strong gravitational wave source is a binary system.

quantum mechanics Think that any Particle It has both particle property and wave property, that is, any matter has wave particle duality, so there is a so-called Matter wave , such as Electronic wave , neutron wave (see Wave particle duality ）。 Wave is a very extensive phenomenon in the universe. The concept of wave is one of the most important unified concepts in physics; In practice, wave is the carrier of information.

Longitudinal and transverse waves

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wave

The physical quantity (or its disturbance) represented by the wave function can be scalar or vector, so F It can be a vector. The relevant physical quantities of electromagnetic waves are electric field or magnetic field And these are vectors. The particle displacement of sound waves in solids is also a vector. If this physical quantity of wave is the same as the propagation direction of wave（ Wave vector ）It is parallel, and waves are called longitudinal waves, such as fluid Sound waves in; If it is vertical, the wave is called shear wave, such as light wave 。 Sometimes the corresponding physical quantity has both components parallel to the propagation direction and vertical components, such as the electric field or magnetic field of the electromagnetic wave in the waveguide.

Main properties

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Waves have some unique properties, which are obviously different from particles from the perspective of classical physics. These properties mainly include wave superposition Interference phenomenon 、 diffraction Phenomenon, etc.

wave

be careful: The following properties completely ignore the deflection of wave propagation path caused by space-time bending in general relativity.

Superimposition

This is a very important attribute of waves (linear waves to be exact, see below). If more than two columns of similar waves meet in space, in the coexistence space, the total wave is the vector sum of each component wave (that is, when adding, not only the amplitude but also the phase is considered), while each component wave does not affect each other, and their properties remain unchanged after separation. The basis of superimposition is that（ linear ）The sum of several solutions of the wave equation is still this Solution of the equation ； This principle is called superposition principle.

interference

Interference pattern

Due to superposition, when two trains of the same kind of waves with the same frequency and fixed phase difference coexist in space, the amplitude will strengthen or weaken each other, which is called interference. When mutually reinforcing, it is called Constructive interference , which is called Destructive interference 。

diffraction

Diffraction phenomenon

When a wave encounters a large obstacle or a hole in a large obstacle in its propagation, it will bypass the edge of the obstacle or the edge of the hole, showing a curved path and spreading behind the edge of the obstacle or hole. This phenomenon is called wave diffraction. The larger the wavelength is relative to the barrier or pore, Diffraction effect The stronger. Figure 2 shows light wave Diffraction produced when encountering a circular aperture. Diffraction is an important example of wave superposition. Near the edge Wavefront It is decomposed into many point wave sources, which emit separately wavelet These wavelets are superimposed on each other to generate diffraction patterns in the geometric shadow area of the obstacle. The concept of wavelet here is more universal Huygens principle An application of.

coherence

Huygens principle The points on the wave front can be regarded as the wave source of many wavelets, and the envelope surface of these wavelets is the wave front at the next moment. The schematic diagram of the principle is shown in Figure 3 (see Huygens Fresnel principle ）。

Related to interference is wave coherence. This is a concept that attracted people's attention before and after the emergence of laser, especially after. Not all two waves can interfere, but they need to meet certain conditions, which are called coherence conditions, mainly the same frequency and fixed phase difference. Generated by two common light sources light wave It is difficult to interfere. Because the light source has a certain area, including many Luminous center For ordinary light sources, these luminous centers are uncoordinated and have no connection with each other. For this reason, in the classic Young's interference experiment It is necessary to separate two light waves from the same light source to obtain interference. Laser Otherwise, its multiple luminous centers are phase related, and the waves it emits are not single frequency, but frequency band Very narrow. In this way, it is said that the wave coherence radiated by ordinary light sources is poor, while that radiated by lasers is good. The coherence of a wave is actually how accurate the wave can be Simple harmonic To represent the description of. This is a qualitative formulation. To quantitatively describe the coherence (strictly speaking, the degree of coherence), we need to use the statistical point of view, and use the time average of the disturbances at different moments on two points. Coherence can be divided into two parts to a certain extent: one is Spatial coherence , because the light source occupies a limited space; One is Time coherence , due to Radiation wave Limited of bandwidth 。

reflex

Fixed end reflection of wave

There are two kinds of wave reflection: free end reflection (the reflection end is not constrained) and fixed end reflection (the reflection end is constrained).

Free end reflection of wave

When the free end reflects, the reflected wave is in phase with the incident wave (the vibration state remains unchanged);

When the fixed end reflects, the reflected wave is out of phase with the incident wave (the vibration state is opposite).

The reflection angle is equal to the incidence angle.

Circular wave

The wave extends outwards in a circle with the wave source (the position where the wave is generated) as the center to generate a circular wave.

Circular wave

The direction of the wave must be perpendicular to the wave front (the line between the peaks or troughs). Parallel waves can be considered to be generated by overlapping circular waves.

refraction

Refraction of light

When the wave slants into different media, the velocity changes, and the change of velocity when passing through the interface changes the direction of the wave.

standing wave

When continuous waves are continuously reflected at both ends, interference will occur between them, and left and right movement will stop to produce standing waves. The strongly vibrating part is called the antinode, and the completely static part is called the wave node.

standing wave

decompose

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wave

Several waves can be superimposed into a total wave, on the contrary, a wave can also be decomposed into the sum of several waves. according to Fourier series Representation, any function can be represented as a series of different Frequency sine and cosine function So any wave can be summed up as a series of simple harmonic waves with different frequencies. This analysis method is called spectrum analysis method, which provides a powerful tool for understanding some complex fluctuation phenomena.

energy

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All waves carry energy. Surface wave The up and down vibrations of the water surface are transmitted to the static water surface in front of the wave front, which means that the wave has kinetic energy and potential energy. The energy carried by a wave is usually measured by the energy per unit volume in the wave, which is called the energy density of the wave. Through vertical in unit time Yu Bo The energy transmitted by the unit area of a vector is called the wave strength or Energy flux density , which is the product of wave energy density and wave propagation speed.

When the elastic wave propagates to a certain place in the medium, the original motionless particle at that place starts to vibrate, so it has kinetic energy. At the same time, the medium at that place will also have deformation, so it also has potential energy

strength

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light

Energy flow P - The energy that passes through a section vertically in unit time is called the energy flow of wave passing through the section, or the energy flux. Set the wave speed as u , at Δ t Passing through the section Δ perpendicular to the wave velocity in time S Energy of:

w - energy density, so the energy flow is:

Energy flow over time Periodic change , which is always positive. The average value of the energy flow in a cycle is called the average energy flow through the unit area of the average energy flow perpendicular to the wave propagation direction is called the average energy flow density, which is usually called the energy flow density or wave intensity. In acoustics sound intensity This is an example of the above definition.

The energy flow density is the average energy passing through the unit section perpendicular to the wave speed direction in unit time.

The energy flow density is a vector whose direction is the same as the wave velocity.

Linear and nonlinear waves

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Some properties of waves, including wave superposition, are conditional. The main condition is that the wave is linear. The reason why this condition is not emphasized above and generally is that the waves commonly discussed, such as general sound waves and light waves, are almost all linear. But in practice, there are also many nonlinear waves. Then the situation changed.

Linear is a mathematical term. The dependent variable is proportional to the independent variable linear relationship ； Otherwise, it is a nonlinear relationship. If the physical quantity (or its disturbance) of the wave is small enough Equation of motion The quadratic term of the physical quantity in the and the term higher than the quadratic term can be ignored compared with the primary term, so the primary term plays a decisive role in the properties and behavior of waves. according to linear This kind of wave is called linear wave. The nature is not so simple, limiting some phenomena to the linear range. But as it happens, among many kinds of waves, it is this kind of wave that can be expressed in linear relation that is most commonly encountered in human life, such as general (not all) water waves, sound waves, light waves, etc. In this way, linear waves are highlighted first in the process of understanding waves.

After a certain understanding of common waves, people began to explore other practical waves and found that many are nonlinear waves. Even some common waves are sometimes nonlinear. The sound waves people use to talk are linear waves. The aircraft Supersonic velocity The shock wave or boom generated by operation is nonlinear. For another example, large amplitude electromagnetic waves will generate frequency doubling, parametric oscillation, parametric amplification, etc. in some crystals, which is not possible for ordinary linear electromagnetic waves. In recent years, the soliton that attracts attention is an extension of the nonlinear water wave that was noticed as early as the 19th century. Compared with the previous discussion on linear waves, one prominent property of nonlinear waves is that the superposition principle is not tenable.

In order to avoid too wide coverage, this item focuses on linear waves. This is what we did before, and we will continue to do so.

Probability wave

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Probability wave includes material wave, light wave, etc. It refers to the probability that a point in space may appear at a certain time. For example, if an electron is a free electron, its wave function is a traveling wave, which means that it may appear at any point in space with equal probability. If you are bound to hydrogen atom And in the ground state, it is possible to appear at any point in space, but Bohr radius The probability is the highest. The same is true for yourself. You may also appear on the moon, but compared with the chance of sitting in front of a computer, it is very, very small, so it is impossible to see this situation. These are the basic concepts of quantum mechanics, which are very interesting.

That is to say, quantum mechanics believes that there is no fixed position of matter, and the position it shows in macro view is actually the average value of the probability wave function. When it is not measured, it can appear anywhere. Once it is measured, its average value and determined position can be obtained.

In quantum mechanics, noncommutative mechanical quantities, such as position and momentum, cannot be measured at the same time, so the accurate position and momentum of an object cannot be obtained. The more accurate the position measurement is, the more accurate the momentum is. This is called Uncertainty principle Of course, it exists even without measurement.

Matter wave

The matter wave is neither Mechanical wave , nor Classical electrodynamics Meaningful electromagnetic wave (Mechanical wave is the propagation of periodic vibration in the medium, and electromagnetic wave is the propagation of periodic electromagnetic field). After de Broglie put forward the matter wave, people had put forward various explanations for it. By 1926, the German physicist Bohn (1882-1970) had put forward a statistical explanation that was later recognized as consistent with the experimental facts: the strength of the matter wave in a certain place was proportional to the probability of finding the particle it represented. According to Bohn's explanation, Matter wave is a kind of probability wave De Broglie wave The probability of particles appearing nearby is directly proportional to the intensity of the wave.

light wave

Light dependent Wave particle duality The light wave is the probability wave of photons appearing in space, which can be determined by the wave rule.

Propagation law

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Reflection, refraction and scattering

In a uniform medium, waves travel in straight lines. Wave may encounter new environment during propagation. A simple case is that the wave consists of a uniform medium It shoots at another homogeneous medium, and the interface between the two mediums is plane. Wave incident on the interface（ incident wave ）, part of which is reflected back to the first medium (called reflected wave ）The other part is folded into the second medium (called Refracted wave ）。 As we all know, the reflection angle is equal to Angle of incidence And the size of the refraction angle depends on the ratio of the relevant physical quantities of the two media. For electromagnetic waves, this physical quantity is Dielectric constant with Permeability The square root of the product of. For other waves, sometimes the situation is more complicated. For example, when the sound wave in a solid is projected from one solid medium to another, in the first medium, the incident wave will be reflected into two waves instead of one, one of which is longitudinal wave One is shear wave. Two waves will also be refracted when entering the second medium (Fig. 4). The reflection angle of the two reflected waves and the refraction angle of the two refracted waves have certain rules.

When the wave encounters a real object in its propagation, not only will there be simple reflection and refraction, but also other waves with complex distribution, including diffracted waves. This phenomenon is generally called scattering (in some literatures, the concepts of scattering and diffraction are not strictly distinguished). Tracking aircraft with radar sonar This is the case with the search for submarines.

Traveling wave and standing wave

Generally speaking, the wave refers to a wave that is moving forward. However, in special cases, a wave that seems to be trapped in a certain space can also be established. In order to distinguish, the former is called traveling wave and the latter is called standing wave.

The standing wave is formed by superposition of similar waves with the same amplitude and frequency but opposite propagation direction. The common establishment method is to make one included radio wave reflected by the medium boundary, so as to generate a reverse wave that meets the conditions, and let the two forms a superposition standing wave 。 For example, simple harmonic waves reflect back and forth in the standing wave cavity (Figure 5). The length of the standing wave cavity is an integral multiple of the half wavelength, and each interface at the cavity end generates π phase difference when reflecting. The point in standing wave whose amplitude is always zero is called Nodal , the adjacent wave nodes are half a wavelength apart, and the amplitude between two wave nodes is sine Shape distribution. The point with the largest amplitude is called Antinode 。

Standing waves are also widely used, for example, in orchestral instruments. In addition, it also derives an important concept, that is, the separation of frequencies. The distance between two interfaces is required d Is an integral multiple of half wavelength, which can be understood as that only those frequencies are n （ v /2 d ）The standing wave can only be established by the wave of. This concept of frequency separation has inspired the creation of quantum mechanics.

Dispersion and group velocity

In ordinary media, the phase velocity of a simple harmonic wave is a constant. For example, the phase velocity of light of any color in vacuum is always constant, equal to 2.99792458 × 10 eight m/s。 But in some cases medium The phase velocity varies with frequency (or wavelength). This phenomenon is called dispersion or Dispersion 。 For Nonlinear Wave, phase velocity or amplitude function. The dispersion of waves is determined by the characteristics of media, so media are often classified as dispersive or non dispersive. One of the reasons why media can cause wave dispersion is that its size is limited. This dispersion is called configuration dispersion. For example, in a solid block much larger than a wave, elastic wave The phase velocity of is constant. However, for elastic waves propagating along a rod surface with the same diameter and wavelength, rods of the same material are dispersive.

Another reason for the dispersion of media is its internal microstructure. Some media, regardless of their shape, are always dispersive for certain kinds of waves in certain frequency ranges. For example, charged particles (such as electrons) in some media are excited by the electric field of the incident visible light and vibrate, thus reacting to the light and causing its dispersion (see Electronic theory ）。 Because of the dispersion of water, it is possible to show a rainbow after rain.

Dynamic diagram of group velocity (green) and phase velocity (red)

For a wave with a single frequency, its propagation velocity is its phase velocity. The actual wave is not single frequency, if medium If the wave is dispersive, the propagating wave will "diffuse" due to the inconsistent movement speed of components at different frequencies. However, if the wave is composed of a group of simple harmonic waves with little difference in frequency, then the total wave will remain as a whole and run at a fixed speed during a quite long propagation distance. This special Wave group call wave packet This speed is called group velocity 。 Different from the phase velocity, the group velocity is smaller than the central phase velocity of the wave packet, and the difference between the two is proportional to the average rate of the central phase velocity changing with the wavelength. Group velocity is the energy of wave packet Propagation rate Is also the propagation rate of the signal represented by the wave packet.

Propagation in complex media

Uniform (macro view) and isotropic media are simple propagation media, and many media are more complex. Some media are isotropic but inhomogeneous. A simple example is the water in the ocean salinity , increasing with depth pressure The sea water zone is stratified. The propagation speed of sound waves is a function of these factors, so it varies with layers, and the result is that the propagation path of sound waves is far from a straight line. It is possible that sound source There is no sound wave area in the ocean ahead. Media that are more heterogeneous than stratification are also common in the ocean and in other environments.

The medium may be homogeneous but anisotropic. Single crystals are such media. When a beam of light shines into a uniaxial crystal like calcite, it will split into two beams, one of which follows the ordinary Law of refraction , said Ordinary light The other beam is not observed, called extraordinary light. Ordinary light and unusual Polarization of light The faces are perpendicular to each other. This phenomenon is called birefringence. Similar to it, there is a so-called cone Refraction phenomenon , which occurs when light shoots into aragonite along the optical axis of the crystal Biaxial crystal When. When a thin beam of light vertically shoots into such a flat crystal, it will emit a circle of light at the back of the crystal due to conical refraction. It can be pointed out that such images can also be observed for sound waves.

For some kinds of media, external fields can sometimes be applied to affect and control the wave propagation in the media. M. Faraday It was discovered more than 100 years ago that Isotropy When the material is applied with a strengthened magnetic field, the polarization plane of light propagating in the material can be rotated. There can also be other media conditions, such as

The propagation of different kinds of waves under different conditions is ever-changing in details, but there are often similarities in large aspects. Among them, the electromagnetic waves often encountered in daily life are especially similar to the sound waves (in the air), and the mathematical processing of several problems is also interconnected or mutually inspired. In fact, in the 19th century, there was a period when light was regarded as ether Elastic waves.

attenuation

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In the process of wave propagation vacuum It's impossible to maintain it amplitude Constant. stay medium In the process of propagation, the energy carried by the wave will always be converted into heat energy or other forms of energy quickly or slowly due to some mechanism, thus weakening and eventually disappearing. Conversely, other forms of energy can sometimes be artificially supplied continuously to the propagating waves, such as slow electromagnetic waves or Piezoelectric semiconductor The internal ultrasonic wave not only does not weaken, but also strengthens. However, if the energy is not replenished, the wave propagating in the medium will always gradually decay. The attenuation mechanism of different kinds of waves in different kinds of media is very different. Even if the same wave propagates in the same medium, the attenuation mechanism may vary with frequency. The interaction between wave and some microstructure in the medium causes wave attenuation, and this interaction also causes dispersion. In this case, attenuation and dispersion are related. As for this kind of interaction, we can mention a quite common law, called Relaxation Phenomenon. Relaxation means that the balance of two states needs a limited time, rather than a moment to complete. Not all wave attenuation is due to microscopic factors. The attenuation of waves can also be attributed to macroscopic reasons. For example, the attenuation of sound waves in viscous fluid is partly due to friction generates heat (See Acoustic absorption ）。

Another time, the wave is scattered rather than really weak. For example, the plane wave is refracted by many small obstacles on the way, and part of it turns. From the original motion direction of the plane wave, the wave energy becomes smaller.

Particle property

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With its superposition, interference, diffraction, energy Space and time It is different from particles with concentrated mass, such as raindrops and bullets. However, at the beginning of the 20th century, some experiments and theories showed that light, which has been determined as a wave, behaves as a particle when interacting with matter. stay Blackbody radiation 、 Photoelectric effect X-ray Of Free electron scattering （ Compton effect ）Such experimental phenomena cannot be explained unless light is regarded as a particle. For example, in the photoelectric effect, the concept of wave cannot explain why photoelectron is the largest kinetic energy and incident light It has nothing to do with the intensity of light, but it has something to do with the frequency of light. Why photoelectron It will shoot from the metal surface at the moment of light incidence. Under the above experimental conditions, the energy of light is discontinuous quantization Of. In other words, light is a quantum, called photon , its energy is hv ， h yes Planck constant ， v Is the frequency of light.

Similar to light, sound, which is generally called sound wave, is also obviously shown as particles, called phonons, when the wavelength is very short. However, phonons only exist in matter, which is the collective effect of material vibration, and are different from photons.

Therefore, waves are also particle like, and comply with energy and Law of conservation of momentum 。 This generally occurs when waves interact with matter. On the other hand, the static mass is not zero Microparticle When propagating, it will also have wave characteristics. This expands the range of waves.

The wave equation expresses the characteristics of waves in mathematical language, and it gives the relationship between the wave function and space coordinates and time. By using the wave equation The solution can deeply describe the propagation law of wave and understand the essence of wave. Wave equations can be divided into classical and quantum mechanics.

wave equation

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formula

Is the second order Linear partial differential equation , its general form is,

here v Is a parameter with velocity dimension, F （ r , t ）Is an observable physical quantity, namely wave function, r Is the space coordinate, t It's time Laplace Operator, which can be expressed in different coordinates as required. For specific problems, the wave equation may be simplified. For example, for a point wave source in a homogeneous isotropic medium, the wave function is only related to the vector diameter, and then the wave equation can be simplified as

The wave equation on the string is the simplest one

FC = CA

ξ （ x , t ）Is the particle displacement. ξ The wave equation of plane acoustic wave propagating in fluid also has the same form.

FC = CA

wave equation

The wave equation of electromagnetic wave can be written as

G = CB

E and B Are electric field strength and Magnetic induction ， v Is the phase velocity, in vacuum v = с , is 2.99792458 × 10 eight Constant of m/s, in medium v = с / n ， n Is the refractive index of the medium.

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