|phase velocity (m/s):||group velocity (m/s):|
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So what do we mean by the velocity of a wave? This may seem like a simple question. When we talk about a wave on a string (or a sound wave) we can talk about the velocity as v = λ f. We can rewrite this expression in terms of the wave's wave number, k, and angular frequency, ω, given that λ = 2π/k and that f = 2π/ω. We therefore find that v = ω/k. We note here that the velocity of the wave is also fundamentally related to the medium in which the wave propagates.
But what happens when we want to add several traveling waves together? In this case we are interested in several waves traveling in the same direction. We can change the wave number and angular frequency for each wave, but we must ensure that the wave speeds are identical. In this animation we add the red wave to the green wave to form the resulting blue wave (position is given in meters and time is given in seconds). Restart.
Consider what happens when we change k1 to 8 rad/m and ω1 to 8 rad/s. Note the interesting pattern that develops in the superposition. Notice that there is an overall wave pattern that modulates a finer-detailed wave pattern. The overall wave pattern is defined by the propagation of a wave envelope with what is called the group velocity. The wave envelope has a wave inside it that has a much shorter wavelength that propagates at what is called the phase velocity. For these values (of k and ω), the phase and group velocities are the same.
Now consider k1 = 8 rad/m and ω1 = 8.4 rad/s. What happens to the wave envelope now? It does not move! This is reflected in the calculation of the group velocity. The finer-detailed wave has a phase velocity of 1.02 m/s. Now consider k1 = 8 rad/m and ω1 = 8.2 rad/s. The group velocity is now about half that of the phase velocity (certain water waves have this property). Now consider k1 = 8 rad/m and ω1 = 7.6 rad/s. The group velocity is now about twice that of the phase velocity.
For a superposition of two waves the group velocity is defined as vgroup = Δ;ω/Δk and the phase velocity as vphase = ωavg/kavg. In general, the group velocity is defined as vgroup = ∂ω/∂k and the phase velocity as vphase = ω/k.
So what velocity do we want? The physical velocity is that of the wave envelope, the group velocity. For waves on strings we got lucky: the phase and group velocities are the same (these are harmonic waves).
Illustration by Wolfgang Christian and Mario Belloni.