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The phase velocity c p (blue) and group velocity c g (red) as a function of water depth h for surface gravity waves of constant frequency, according to Airy wave theory. Quantities have been made dimensionless using the gravitational acceleration g and period T , with the deep-water wavelength given by L 0 = gT 2 /(2π) and the deep-water phase ...
Frequency dispersion in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the green circles propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square overtakes two green circles when moving from the left to the right of the figure.
Frequency dispersion of surface gravity waves on deep water. The red square moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square traverses the figure in the time it takes the green dot to traverse half.
A galaxy's recessional velocity is typically determined by measuring its redshift, a shift in the frequency of light emitted by the galaxy. The discovery of Hubble's law is attributed to work published by Edwin Hubble in 1929, [ 2 ] [ 3 ] [ 4 ] but the notion of the universe expanding at a calculable rate was first derived from general ...
The Doppler effect (also Doppler shift) is the change in the frequency of a wave in relation to an observer who is moving relative to the source of the wave. [1] [2] [3] The Doppler effect is named after the physicist Christian Doppler, who described the phenomenon in 1842.
The group velocity is positive (i.e., the envelope of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward). The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.
= is the escape velocity, and β e = v e / c {\displaystyle \beta _{e}=v_{e}/c} is the escape velocity, expressed as a fraction of the speed of light c. To illustrate then, without accounting for the effects of rotation, proximity to Earth's gravitational well will cause a clock on the planet's surface to accumulate around 0.0219 fewer seconds ...
The meaning of the constants and can be easily found: setting = on the equation above we see that () =, so that is the initial position of the particle, =; taking the derivative of that equation and evaluating at zero we get that ˙ =, so that is the initial speed of the particle divided by the angular frequency, =.