Plane wave
A plane wave is a wave in three-dimensional space whose wavefronts (ie surfaces of the same phase angle ) are planes that are extended perpendicular to the propagation direction. Synonymous with this is that the direction of propagation of the wave is constant in space.
The term is used almost exclusively for waves that are homogeneous and balanced, i.e., have a spatially constant amplitude and exhibit a sinusoidal time course with a constant frequency. Such waves are among the simplest solutions of the wave equation, which plays an important role in classical mechanics, electrodynamics and in quantum mechanics.
Other solutions of the wave equation are the spherical wave (concentric about a point ) and the cylinder shaft ( concentrically around a straight line). These can be approximated in the distance from the center in small areas well by a plane wave.
The two-dimensional analogue of the plane wave is a wave whose wavefronts are straight lines that move on a flat surface. An illustrative, but only approximately [note 1] suitable example is the tapering on the beach ocean waves.
Homogeneous harmonic plane wave
The picture to the local course of a harmonic plane wave is shown propagating in the x direction and the magnitude of A (x, t) in the y- direction oscillates ( A snapshot at time t = 0). The maximum deflection (amplitude ) of the shaft is designated by their wavelength and their phase relationship at that time with. The wavelength indicates the periodicity of A in place.
In the following picture of the time course at a fixed location is shown as an animation. The frequency is a measure of the periodicity of A in the time. The phase velocity c is the ratio of time period T and spatial period:
A plane wave is most easily described if the coordinate system is chosen such that an axis corresponding to their direction of propagation. In the directions perpendicular to the propagation does not occur vibration. Thus can a harmonic homogeneous plane wave as
Represent. At this point, the constant phase move with the phase velocity c in the positive x-direction. In the inner bracket, the increase of x / c compensated precisely the time t, so that
Is a plane equation of the wave front.
For a change of direction, as it (eg refractive index or characteristic acoustic impedance change ) occurs about by reflection at an inhomogeneity in the medium, the sign of x or the x - axis is to turn itself.
The physics of periodically varying quantity A is not important for the concept of the plane wave. It can be a mechanical displacement, a pressure change, a field strength or as a probability amplitude. If it is a vector quantity, the direction of their amplitude is relative to the propagation direction of its polarization to.
General form of a plane wave
A plane wave is the simplest form of a wave
With an arbitrary ( scalar or vector-valued ) function and: . 1 In this case, the plane wave propagates in the direction of the speed c. If one observes the wave at the location, the size considered A changes over time in any way. The levels are the same oscillation phase
The plane wave is a solution of the wave equation
In practice, however, only plane harmonic waves are used because each common plane wave can be represented as a sum of harmonic plane waves. This is because one can represent the general form of plane wave A as the Fourier integral:
This corresponds to a sum of harmonic plane waves with frequency-dependent amplitudes. Here only the physically meaningful real part of the Fourier transform is considered and presented in the last part of the equation using the identity with the complex conjugation *. Because now the validity of the superposition principle for the wave equation, it is sufficient for further consideration only the spectral component of the angular frequency
Look at. g harmonic plane wave is called. Usually, this form will be expressed with the help of the wave vector. It applies and thus
The real part of the harmonic plane wave corresponds to and imported in the previous section sinusoidal plane wave.
Inhomogeneous plane wave
A plane wave is always a solution of the Helmholtz equation ( temporal Fourier transform of the wave equation )
With real dispersion relation. The Helmholtz equation is also solved if one allows complex components of the wave vector:
Thus, the Helmholtz equation is fulfilled, but the wave number must be real, reflecting the condition
Leads and imposes a restriction on the choice of the complex wave vector. This condition clearly means that the real part () of the wave vector must be perpendicular to its imaginary part ().
A wave of the form
Is called inhomogeneous plane wave or non-uniform plane wave. It spreads in the direction and its amplitude drops vertically from the direction of propagation. In contrast to the homogeneous plane wave the planes of constant amplitude are perpendicular to the planes of constant phase here. Furthermore, the phase velocity is always less than that of a homogenous plane wave at the same frequency.
Absorption
Selects one real and imaginary part of the complex wave vector parallel as vectors, the imaginary part of the wavenumber is not as in the previous section is zero and the wave number is complex
Is called the absorption coefficient or damping constant and referred to as phase constant. This leads to a damped harmonic plane wave. Taking the x-axis in the propagation direction, as follows
The planes of constant phase and a constant amplitude are identical, only the amplitude decreases in the propagation direction decrease exponentially. It is a homogeneous plane wave.
Idealization
A plane wave always fills an infinitely extended space and is therefore an idealization of the real wave. On the one hand can not a plane wave emitted from a last extended channel, and on the other hand, the power of a plane wave is infinite. Both are unphysical.