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.

• 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.

• Blood Pressure

  • The systolic pressure is defined as the peak pressure in the arteries during the cardiac cycle; the diastolic pressure is the lowest pressure at the resting phase of the cardiac cycle.
  • The blood pressure of the systole phase and the diastole phase gives the two readings for blood pressure .
  • Blood pressure is related to the blood velocity in the arteries and arterioles.
  • In the capillaries and veins, the blood pressure continues to decease, but velocity increases.

• 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.

• Velocity and Duration of Muscle Contraction

  • The shortening velocity affects the amount of force generated by a muscle.
  • The force-velocity relationship in muscle relates the speed at which a muscle changes length to the force of this contraction and the resultant power output (force x velocity = power).
  • Though they have high velocity, they begin resting before reaching peak force.
  • As velocity increases force and power produced is reduced.
  • Maximum power is generated at one-third of maximum shortening velocity.