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I know that the velocity of a wave is given by $v=\lambda f$ but what does this velocity represent in the physical sense. For instance, if I am told a car moves at a velocity of 5 $m/s$ I know that the car itself will cover 5 meters in displacement every second. What part of the wave is moving at velocity $v$?

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The car is just an example of the concept of velocity applied elsewhere where I can understand the meaning of velocity easily it has nothing to do with the wave in this case. –  guest101 Nov 13 '12 at 15:50
    
For more on phase velocity vs. group velocity, see e.g. Phys.SE posts here, here and here. –  Qmechanic Nov 13 '12 at 17:55

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The velocity of a wave can be though of as the rate of change of displacement of any single peak. In more generic terms, It's the speed of a surfer who is riding the wave with no relative displacement.

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I think this response highlights my confusion. From my understanding: a standing wave also has velocity, but if a surfer were positioned on an antinode they would simply bob up and down and not move forward or backward at all. –  guest101 Nov 13 '12 at 15:52
    
A standing wave isn't moving by definition, hence it has zero velocity. But a non-zero phase velocity doesn't necessarily mean the particles are moving at this velocity e.g. if they are oscillating in a transverse direction. –  Jason Davies Nov 13 '12 at 17:23
    
Ok, so are you saying this formula doesn't apply to standing waves? $v=\lambda f$ –  guest101 Nov 13 '12 at 19:18
    
Correct; it only applies to a pure travelling sine wave. To create a standing wave you add two travelling sine waves going in opposite directions (and synchronised). So the two opposite phase velocities cancel each other out. A standing wave has no longitudinal velocity, hence the name! –  Jason Davies Nov 13 '12 at 19:46
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Of course v=λf applies to standing waves. –  David Jun 30 at 23:17

The phase velocity of a wave tells us how fast any given phase of the wave is moving e.g. the crest. If we examine some fixed position, then we can count how frequently a full wavelength passes this position as the wave moves with time. This leads us directly to $v = λf$.

Note that this doesn't necessarily mean that the particles in the medium are moving at this velocity. For example, the wave could consist of particles oscillating in a transverse direction, in which case no particle has any lateral motion at all.

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