phase velocity

Examples of phase velocity in the following topics:

• Wavelength, Freqency in Relation to Speed

  • They also have two kinds of velocity: phase and group velocity.
  • where v is called the wave speed, or more commonly,the phase velocity, the rate at which the phase of the wave propagates in space.
  • For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity.
  • 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. )

• Spherical and Plane Waves

  • Spherical waves come from point source in a spherical pattern; plane waves are infinite parallel planes normal to the phase velocity vector.
  • Constructive interference occurs when waves are completely in phase with each other and amplifies the waves.
  • Destructive interference occurs when waves are exactly out of phase with either other, and if waves are perfectly out of phase with each other, the wave will be canceled out completely.
  • A plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector .

• 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.

• Refraction

  • Due to change of medium, the phase velocity of the wave is changed but its frequency remains constant (most commonly observed when a wave passes from one medium to another at any angle other than 90° or 0°).
  • Refraction is described by Snell's law, which states that for a given pair of media and a wave with a single frequency, the ratio of the sines of the angle of incidence θ1 and angle of refraction θ2 is equivalent to the ratio of phase velocities (v1/v2) in the two media, or equivalently, to the opposite ratio of the indices of refraction (n2/n1):

• Thermal Bremsstrahlung Emission

  • The most important case astrophysically is thermal bremsstrahlung where the electrons have a thermal distribution so the probablility of a particle having a particular velocity is
  • We would like to integrate the emission over all the velocities of the electrons to get the total emission per unit volume,
  • If we look at the emission for a particular velocity, the emisision rate diverges as $v \rightarrow 0$, but the phase space vanishes faster; however, it is stll reasonable to cut off the integral at some minimum velocity.

• Position, Velocity, and Acceleration as a Function of Time

  • By taking derivatives, it is evident that the wave equation given above holds for $c = \frac{\omega}{k}$, which is also called the phase speed of the wave.
  • To find the velocity of a particle in the medium at x and t, we take the temporal derivative of the waveform to get $\frac{\partial y(x,t)}{\partial t} = -A \omega cos(kx - \omega t + \phi)$.
  • Note the phase relationship among the trigonometric functions in y(x,t), y'(x,t), y''(x,t).
  • When the particle displacement is maximum or minimum, the velocity is 0.
  • When the displacement is 0, particle velocity is either maximum or minimum.

• 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.

• 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.

• Relativistic Addition of Velocities

  • A velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.
  • A velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.
  • As Galileo Galilei observed in 17th century, if a ship is moving relative to the shore at velocity $v$, and a fly is moving with velocity $u$ as measured on the ship, calculating the velocity of the fly as measured on the shore is what is meant by the addition of the velocities $v$ and $u$.
  • Since this is counter to what Galileo used to add velocities, there needs to be a new velocity addition law.
  • This change isn't noticeable at low velocities but as the velocity increases towards the speed of light it becomes important.

• Addition of Velocities

  • Relative velocities can be found by adding the velocity of the observed object to the velocity of the frame of reference it was measured in.
  • As learned in a previous atom, relative velocity is the velocity of an object as observed from a certain frame of reference.
  • demonstrates the concept of relative velocity.
  • When she throws the snowball forward at a speed of 1.5 m/s, relative to the sled, the velocity of the snowball to the observer is the sum of the velocity of the sled and the velocity of the snowball relative to the sled:
  • The magnitude of the observed velocity from the shore is the square root sum of the squared velocity of the boat and the squared velocity of the river.