group velocity

Examples of group velocity in the following topics:

• Wavelength, Freqency in Relation to Speed

  • They also have two kinds of velocity: phase and group velocity.
  • This is the velocity at which the phase of any one frequency component of the wave travels.
  • Finally, the group velocity of a wave is the velocity with which the overall shape of the waves' amplitudes — known as the modulation or envelope of the wave — propagates through space.
  • This shows a wave with the group velocity and phase velocity going in different directions.
  • (The group velocity is positive and the phase velocity is negative. )

• Water Waves

  • The deep-water group velocity is half the phase velocity.
  • In shallow water for wavelengths larger than about twenty times the water depth (as often found near the coast), the group velocity is equal to the phase velocity.
  • We see a wave propagating in the direction of the phase velocity.
  • The wave can be thought to be made up of planes orthogonal to the direction of the phase velocity.

• Classification of Nerves

  • Peripheral nerve fibers are grouped based on the diameter, signal conduction velocity, and myelination state of the axons.
  • Fibers of the A group have a large diameter, high conduction velocity, and are myelinated.
  • Fibers of the B group are myelinated with a small diameter and have a low conduction velocity.
  • Fibers of the C group are unmyelinated, have a small diameter, and low conduction velocity.
  • The lack of myelination in the C group is the primary cause of their slow conduction velocity.

• Root-Mean-Square Speed

  • Although the velocity of gaseous particles is constantly changing, the distribution of velocities does not change.
  • Particles moving in opposite directions have velocities of opposite signs.
  • By squaring the velocities and taking the square root, we overcome the "directional" component of velocity and simultaneously acquire the particles' average velocity.
  • This equation determines the average speed of a given group of gaseous particles.
  • Recall the mathematical formulation of the root-mean-square velocity for a gas.

• Distribution of Molecular Speeds and Collision Frequency

  • (Velocity is a vector quantity, equal to the speed and direction of a particle) To properly assess the average velocity, average the squares of the velocities and take the square root of that value.
  • This is known as the root-mean-square (RMS) velocity, and it is represented as follows:
  • If we assume that all velocity states are equally probable, higher velocity states are favorable because there are greater in quantity.
  • Using the above logic, we can hypothesize the velocity distribution for a given group of particles by plotting the number of molecules whose velocities fall within a series of narrow ranges.
  • Velocity distributions are dependent on the temperature and mass of the particles.

• Defining Kinematics

  • Kinematics is the study of the motion of points, objects, and groups of objects without considering the causes of its motion.
  • Kinematics is the branch of classical mechanics that describes the motion of points, objects and systems of groups of objects, without reference to the causes of motion (i.e., forces).
  • To describe motion, kinematics studies the trajectories of points, lines and other geometric objects, as well as their differential properties (such as velocity and acceleration).
  • The study of kinematics can be abstracted into purely mathematical expressions, which can be used to calculate various aspects of motion such as velocity, acceleration, displacement, time, and trajectory.
  • The physical quantities relevant to the motion of a particle include: mass m, position r, velocity v, acceleration a.

• Viscosity

  • The force required to move the plate at a constant speed is directly proportional to the area of the plate and to the fluid's velocity gradient as we move at a greater perpendicular distance from the plate (meaning how fast the velocity of the layers is changing as we move away from the plate).
  • Liquids such as syrups and honey are much more viscous because the sugars they contain are studded with hydroxyl groups (–OH).
  • Velocity of a fluid's layers, or lamina, during smooth flow.
  • The velocity is greatest at the center of the tube.
  • Note the magnitude of the velocity vectors for layers increasingly away from the moving plate.

• Relative Velocity

  • Relative velocity is the velocity of an object B measured with respect to the velocity of another object A, denoted as $v_{BA}$.
  • Relative velocity is the velocity of an object B, in the rest frame of another object A.
  • Is the velocity of the fly, $u$, the actual velocity of the fly?
  • No, because what you measured was the velocity of the fly relative to the velocity of the boat.
  • The velocity that you observe the man walking in will be the same velocity that he would be walking in if you both were on land.

• Tangent and Velocity Problems

  • Velocity is defined as rate of change of displacement.
  • The average velocity becomes instantaneous velocity at time t.
  • Instantaneous velocity is always tangential to trajectory.
  • Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.
  • Its slope is the velocity at that point.

• Instananeous Velocity: A Graphical Interpretation

  • Instantaneous velocity is the velocity of an object at a single point in time and space as calculated by the slope of the tangent line.
  • Typically, motion is not with constant velocity nor speed.
  • However, changing velocity it is not as straightforward.
  • Since our velocity is constantly changing, we can estimate velocity in different ways.
  • Motion is often observed with changing velocity.