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Section 5.9: Phase and Group Velocity
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A classical wave obeys a classical wave equation. In one dimension the classical wave equation is:
[v−2 d2/dt2 − d2/dx2] y(x,t) = 0, (5.8)
where v is the wave velocity and y(x, t) is the wave function, the solution to the wave equation. In general, the solution to this equation is y(x,t) = f(x −/+ vt). For harmonic waves, the solutions to the wave equation are
y(x, t) = A cos (kx −/+ ωt) + B sin (kx −/+ ωwt), (5.9)
for the right-moving (upper signs) and left-moving (lower signs) solutions.
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 = ω/2π. 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. At first, we will ensure that the wave speeds are identical. In this animation we add the red wave (wave 1) to the green wave (wave 2) to form the resulting blue wave (position is given in meters and time is given in seconds). Restart.
Consider what happens when we leave the second wave unchanged, but we change k1 to 8 rad/m and ω1 to 8 rad/s for the first wave. 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 (now wave one and wave two have different speeds). 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 (solutions to the free, V = 0, Schrödinger equation have this property).
For a superposition of two waves the group velocity is defined as vgroup = Δω/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 get lucky: the phase and group velocities are the same (these are harmonic waves).