Wave packet

A wave packet, a wave group or wave train is a limited space or time wave. Mathematically, a wave packet be considered as a composite system of simpler waves. In particular, a wave packet can be represented by multiple plane waves superposition (addition). This decomposition of the wave packet on the frequency components is motivated by the Fourier transform and can be experimentally determined with a spectrometer.

Mathematical formulation

A wave packet can be defined as a sum of plane waves

The amplitude of each plane wave are arbitrary and determine the particular structure of the wave packet. Each plane wave is monochromatic and oscillates at the angular frequency. In contrast, the wave packet has no single frequency but a frequency distribution. Is given by the wave number. In this case [Note 1] is the phase velocity of the plane wave, which can be frequency-dependent, depending on the medium. Actual frequency independent as the medium is free of dispersion and the wave packet does not change its shape with time. In general, this is not the case and one speaks of the run-out of the wave packet.

A wave packet is, just as a plane wave, a solution of the general wave equation

This follows from the linearity of the wave equation and the superposition principle.

Physically meaningful is only the real part or imaginary part (also) of the wave packet.

It continues to have a solution of the wave equation, when one passes from the sum to the integral. The amplitude distribution sets, which now depends on the wave number:

Example: Gaussian wave packet

A commonly used example of a wave packet is called the Gaussian wave packet. This is a wave whose amplitude distribution is a Gaussian distribution. A special feature of the Gaussian wave packet is that the Fourier transform of a Gaussian is again a Gaussian. Thus, the specification of a Gaussian distribution amplitude distribution leads to a Gaussian wave in the spatial domain. Are you reversed a wave packet in position space the Gaussian shape, the frequency distribution of this wave packet is Gaussian distributed automatically.

In addition, the Gaussian wave packet is that the wave packet with the least blur. That no other wave packet the product of the width of the wave in the spatial domain and its width in frequency space is less.

Mathematically

Substituting in the above equation (1) for the amplitude distribution of a Gaussian function

A, we obtain after integration at the time:

The figure shows the result. It now has just one area in which the amplitude is significantly different from 0.

Dispersion

In most cases, the propagation velocity of the wave is frequency dependent (eg light into matter ), so that the wave packet " bleeds ", ie its width with time is larger ( or smaller) and the spatial certainty always accurate. Wave packets, which do not exhibit dispersion thus maintain its shape and width are referred to as solitons.

With the following experiment can prove that electromagnetic waves up to several kilometers (frequency range 20 kHz to about 2 GHz) spread over an extremely wide range of wavelengths of a few centimeters at the same speed, ie, that occurs no dispersion of electromagnetic waves in a coaxial manner: An pulse generator generates a short voltage pulse of about 10 ns duration at a repetition frequency of about 20 kHz. If you send this by an approximately 20 m long coaxial cable, they are reflected at the open end and running back. Depending on cable loss can be observed one hundred pulses, the shape does not change. The unavoidable ohmic losses in the cable and the connection resistance between the generator and cables introduce decrease in amplitude but no change in shape of the envelope of the wave packets.

With a Fourier analysis one can determine the frequency content of the very short voltage pulses:

• The lowest frequency is the repetition frequency of the pulses, ie 20 kHz.
• The highest frequency is about 100 times the reciprocal of the pulse width in the above assumed case, at 10 GHz.

Would the duration of the pulses due to dispersion differ markedly, and the decay of the waveform would vary according to the laws of Fourier synthesis. Since this is not observed, it follows the constancy of the velocity of propagation in the cable described in the frequency domain.

Application in various fields

• Water waves: Wave packets are used as surface waves in water for application, for example, the transfer functions (English: RAO = response amplitude operator) of ships and offshore structures in the model test to measure. The fact that all waves emanating from the wave maker, meet at the same time and same place only succeeds because according to the dispersion relation is short ( high frequency ) waves on the water surface spread more slowly than long ( low frequency ) waves. As a provider of such model tests occur (few ) shipbuilding research institutions.
• Matter waves: In quantum mechanics using wave packets to represent particles in the wave picture. The width of a wave packet in position and momentum space are on the Heisenberg uncertainty principle linked. A locally well given particle therefore has a very broad momentum distribution and vice versa. The same is true for energy ( frequency ) and time.

Multidimensional wave packet

Equation (1) is also in vector expressible:

One can at the time the room an initial pattern [Note 2] impress (generator) the means of the Huygens' principle then for all subsequent time steps is further propagated spatially ( iterator ).

Wave

Under a wave train is understood wave of a frequency a limited time ( duration). Although all the vibrations of the wave train of the same period, the spectrum of the wave train is not only from the frequency component. From Fourier theory follows with the time limitations of a minimum width of the frequency spectrum: ( Küpfmüllersche uncertainty relation ).