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A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line). The group velocity of a collection of waves is defined as =.
In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency. The phase velocity is the rate at which the phase of the wave propagates in space. The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the ...
1D Gaussian wave packet, shown in the complex plane, for =, =, =, =. The overall group velocity is positive, and the wave packet moves as it disperses. The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor.
If the group velocity (see below) is wavelength-independent, this equation can be simplified as: [14] (,) = (+), showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation .
The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small wave packet will move, i.e. the direction of the group velocity. For light waves in vacuum, this is also the direction of the Poynting vector. On the other hand, the wave vector points in the direction of phase velocity.
Aircraft Agency Velocity-Mach- Altitude - m - Comments X-1D #1: July 24, 1951 Jean Ziegler 48-1386 Bell 1 ? ? Pilot familiarization. Glide flight. Nose gear broken on landing. X-1D #2: August 22, 1951 Frank Everest 48-1386 USAF 1 ? ? Launch aborted. Small explosion during propellant jettison process. Pilot evacuated to B-50. X-1D jettisoned and ...
The group velocity ∂Ω / ∂k of capillary waves – dominated by surface tension effects – is greater than the phase velocity Ω / k . This is opposite to the situation of surface gravity waves (with surface tension negligible compared to the effects of gravity) where the phase velocity exceeds the group velocity. [13]
It is often convenient to formulate the trajectory of a particle r(t) = (x(t), y(t), z(t)) using polar coordinates in the X–Y plane. In this case, its velocity and acceleration take a convenient form. Recall that the trajectory of a particle P is defined by its coordinate vector r measured in a fixed reference frame F.