Group velocity and dispersion
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The document discusses group velocity, defined as the velocity of the wave envelope's propagation. It explains the relationship between group velocity and phase velocity, detailing how variations in wave frequency can lead to distortion of the wave envelope. Additionally, it highlights the significance of the dispersion relation and illustrates how wave packets behave under different frequency ranges.
Group velocity and dispersion
- 1. GROUP VELOCITY AND DISPERSION
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- 3. GROUP VELOCITY AND DISPERSION • GROUP VELOCITY:- 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. For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The shorter waves travel faster than the group as a whole, but their amplitudes diminish as they approach the leading edge of the group. The longer waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group.
- 4. • GRAPICALL EXPLANATION The group velocity vg is defined by the equation where, ω =2πf (angular frequency) k =2π radians/ λ (wave number) And, vp = ω/k (phase velocity) ∂ω/∂k vg =
- 5. DISPERSION RELATION � The function ω(k), which gives ω as a function of k, is known as the dispersion relation. � If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity. � If ω is a linear function of k, but not directly proportional (ω = ak + b), then the group velocity and phase velocity are different. The envelope of a wave packet will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity.
- 6. If ω is not a linear function of k, the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies (and hence different values of k), the group velocity ∂ω/∂k will be different for different values of k. Therefore, the envelope does not move at a single velocity, but its wavenumber components (k) move at different velocities, distorting the envelope. If the wave packet has a narrow range of frequencies, and ω(k) is approximately linear over that narrow range, the pulse distortion will be small, in relation to the small nonlinearity. For example, for deep water gravity waves, ω=√ gk , and hence vg = vp/2. This underlies the Kelvin wake pattern for the bow wave of all ships and swimming objects. Regardless of how fast they are moving, as long as their velocity is constant, on each side the wake forms an angle of 19.47° = arcsin(1/3) with the line of travel.