Wave

A wave is a spatially propagating change (fault) or vibration of a location - and time-dependent physical quantity. Differences are mechanical waves that are always linked to a medium, and waves that can propagate in vacuum ( for example, electromagnetic waves, matter waves, gravity waves ). In media, the propagation of a local perturbation by the coupling of adjacent oscillators ( oscillatory physical quantities ) conveys. A wave transports energy but not matter, ie the neighboring oscillators transport the disturbance through space without moving even when averaged over time. Direct perceptible waves include sound waves, water waves, and light.

• 2.1 wavefunction
• 2.2 amplitude
• 2.3 phase
• 2.4 Examples

Wave types

Waves are divided into several categories: the "classic" longitudinal and transverse waves (one of which can occur also mixed forms such as torsion ) and matter waves ( according to the theory of Louis de Broglie has a moving particle and a wave, which is also detected in the corresponding experimental setup may be ) and probability waves that describe the framework of quantum physics, the states of physical systems. Furthermore, it is assumed the existence of gravitational waves.

Mechanical longitudinal waves can be in any medium, whether solid, liquid or gaseous spread, whereas mechanical, pure transverse waves can propagate only in solids themselves. Electromagnetic waves in lossless media (e.g. in a vacuum) are transversal.

Longitudinal wave

Shafts oscillate parallel to the propagation direction is referred to as longitudinal or longitudinal shaft. An important example is the sound, which always propagates as a longitudinal wave in gases and liquids.

Mechanical longitudinal waves are pressure waves. This means that in a medium zones of increased pressure or compressive stress to propagate (or negative pressure or tension ) in the propagation direction or to move or spread. The individual particles in the dispersion medium, atoms or molecules that vibrate in this direction of propagation, by the amount of the amplitude of back and forth. After the passage of vibration to the particles to move back to its rest position, the position of equilibrium to return.

The power of a longitudinal wave is proportional to the square of the amplitude or of the compressive stress, see also the sound pressure and particle velocity. Have longitudinal waves in the same solid medium, a higher speed than the transverse waves of the same type with otherwise identical parameters.

Transverse wave

Waves that oscillate perpendicular to the direction of propagation are called transversal, transverse, shear or shear waves. Only transverse waves can be polarized.

Examples are electromagnetic waves, gravitational waves, bending waves and plasma waves. Sound in the solid state and seismic waves can propagate as a transverse wave at an appropriate material properties and water waves are a hybrid of longitudinal and transverse waves.

Mathematical Description

For the mathematical description of waves more sizes are needed. These include amplitude, phase, and propagation or phase velocity. The accompanying table gives an overview of the variables that are necessary for a complete description.

Wave function

Mathematically, one speaks of a wave when the wave function, ie the mathematical equation describing the wave, a solution of a wave equation. These functions depend in general of place and time.

The deflection is at the location of the time. Functions of this type correspond to the idea that waves are spatially extended oscillations. Specify a general function for each type of wave, it is not readily possible. Therefore, very simple solutions of the wave equation can be used and often considered the real wave as a superposition of many of these solutions. The most common elemental solutions are the plane wave and the spherical wave.

Amplitude

The amplitude is the maximum possible deflection of the shaft. She's with waves - as opposed to vibrations - a vector quantity, since in addition to the amount of deflection also their direction is crucial. Is the propagation direction parallel to the amplitude, is a longitudinal wave, is perpendicular to a transverse wave. In both cases the intensity of the wave is proportional to the square of the amplitude.

Phase

The phase of a wave indicates in which section the shaft is located at a reference time and place within a period. So you determines how large the deflection is. In the example of a plane wave, the phase at the time the place is. The phase thus depends on the two parameters wave vector and angular frequency.

Examples

The mathematical formulation for a harmonic (also: homogeneous, monochromatic ) plane wave in three-dimensional space in complex notation:

A spherical wave can be described by the following equation:

Generation of waves

Sources of waves can be pulsed excitations, vibrations or periodic oscillations. Periodic mechanical and electromagnetic waves can be generated by periodic oscillations. A simple example is a swinging pendulum: At one such shuttle, for example a pin, at which a sheet of paper is pulled at a constant speed. The pendulum is attached to the pin describes now on the paper strip, which represents the propagation medium, a sinusoidal wave. In this example the wavelength is dependent on the speed with which the paper strip is moved. The amplitude of the wave is determined by the maximum swing of the pendulum.

An electromagnetic wave can be generated by an antenna, which is connected to a vibration generator. As a vibration generator, a so-called resonance circuit is used, in which the electric current between a coil and a capacitor to flow to and fro. The electromagnetic energy is converted periodic electrical energy ( capacitor ) in magnetic energy (coil). This is done with a suitable frequency, the energy is emitted in the form of an electromagnetic wave from the antenna. This effect is in particular in wireless communication systems are of particular importance.

Superposition of waves

Naturally occurring waves are rarely pure monochromatic waves, but a superposition of many waves of different wavelengths. The superposition is carried out by the principle of superposition, which mathematically means that all wave functions of the individual waves are added. The shares of wavelengths are called spectrum. Examples:

• Sunlight is a superposition of the electromagnetic waves. The spectrum covers a wavelength range from infrared through visible light to ultraviolet. Such spectra are referred to as continuous.
• A musical sound of an instrument is composed of a root and several harmonics. The different proportions of harmonics are the reason why a trumpet sounds different than a flute. Such a spectrum is called discrete because it is composed only of individual, clearly separated wavelengths.

Many effects can occur:

• Interference - is superposed on waves, so there may be a constructive reinforcement, but also to a partial or even total annihilation of the shaft ( when both wavelengths and frequencies are equal and the waves exactly opposite swing ) can occur. This phenomenon plays in everyday life to radio a role - adjacent radio cells emit a station on different frequencies to avoid interference in the overlap region.
• Standing wave - When superimposing counterpropagating traveling waves of the same frequency and amplitude of two leads to the formation of standing waves. These do not spread, but form spatially constant vibration patterns: On the so-called movement bellies they resonate with the doubled amplitude and frequency of the original, on the intervening nodes movement is the amplitude of zero at all times. This phenomenon is a special case of interference. It occurs in particular in front of a reflective wall or between two appropriately tuned walls which together form a resonator.
• Beat - a superposition of two waves of adjacent frequency leads to a beat. The amplitude of such a wave will periodically increases and decreases - the closer the frequencies to each other, the more (in time), this process happens slowly. This effect is, for example, exploited for tuning musical instruments. One can find beats in older telephones that have an alarm clock with two bells shells. This beat feels man as pleasant.
• Wave packet - The superposition of waves at all frequencies of a frequency band generated a wave packet. Here, the envelope of the wave shows only a single mountain, in front of and behind the amplitude is negligible. Since the phase velocity of a wave in waveguides and dispersive media is frequency dependent, melt wave packets with time progresses. The messaging by means of shafts, the resulting broadening of wave packets must be considered.