Phase velocity question

In summary, the phase velocity v=w/k can be derived by rewriting a typical harmonic wave in the form f(x,t)=a\sin(k(x-\frac{\omega}{k}t)). This shows that the signal value can be seen as moving along the positive x-direction with velocity \frac{\omega}{k}. This is represented by the trajectory x=s+\frac{\omega}{k}t, where s is a constant. This relationship explains the propagation velocity of the signal.
  • #1
asdf1
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why is phase velocity v=w/k ?
 
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  • #2
Let us look at a typical harmonic wave, [tex]f(x,t)=a\sin(kx-\omega{t})[/tex]
Here, a,k,[tex]\omega[/tex] are constants, where a is the amplitude, k the wave number and [tex]\omega[/tex] the frequency.

Rewrite this in the following form:
[tex]f(x,t)=a\sin(k(x-\frac{\omega}{k}t))[/tex]
Do you see that if [tex]x-\frac{\omega}{k}t=s[/tex] where s is a CONSTANT, makes the value of f constant as well (equal to [tex]a\sin(ks)[/tex])?
But that means, that the signal value [tex]a\sin(ks)[/tex] can be regarded as MOVING ALONG THE POSITIVE X-DIRECTION WITH VELOCITY [tex]\frac{\omega}{k}[/tex]!
For, (remembering that s is constant) we have the trajectory for our signal:
[tex]x=s+\frac{\omega}{k}t[/tex]
and this simply shows what the propagation velocity of our signal is..
 
Last edited:
  • #3
@@a
"But that means, that the signal value can be regarded as MOVING ALONG THE POSITIVE X-DIRECTION WITH VELOCITYw/k !"
can you explain that a little clearer?
 

1. What is phase velocity?

Phase velocity is the speed at which the phase of a wave propagates through a medium. It is the rate at which the wave's crests or troughs move in space, and is measured in meters per second (m/s).

2. How is phase velocity different from group velocity?

Phase velocity and group velocity are two different ways of measuring the speed of a wave. While phase velocity describes the speed at which the wave's phase changes, group velocity describes the speed at which the energy of the wave propagates. In some cases, the two may be the same, but in other cases, such as in dispersive media, they can be different.

3. What factors affect the phase velocity of a wave?

The phase velocity of a wave is influenced by the properties of the medium through which it travels, such as its density, elasticity, and temperature. It is also affected by the frequency and wavelength of the wave.

4. How is phase velocity calculated?

The phase velocity of a wave can be calculated by dividing the wave's frequency by its wavelength. This calculation is based on the wave equation v = λf, where v is the phase velocity, λ is the wavelength, and f is the frequency.

5. Why is phase velocity important in physics?

Phase velocity is an important concept in physics because it helps us understand how waves behave and propagate through different media. It is also crucial in many practical applications, such as in telecommunications, where the speed of a wave can affect the quality of communication signals.

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