# Proof that wave packet moves with the group velocity

1. Jan 10, 2007

### mbs

I’m currently taking a class in fluids in which we are studying different types of wave propagation. We discussed how for certain types of waves (such as deep-water ocean waves), the frequency (and phase speed) of each sinusoidal component is a functions of the wave number. This makes the waves dispersive as different components move at different speeds. This part is clear to me.

What I am having trouble with is the claim that an initial packet of the form

$f(x,y,z)sin(ax+by+cz)$

will move with the group velocity, defined as

$\frac{\partial \omega}{\partial k}\mathbf{\vec{i}}+\frac{\partial \omega}{\partial l}\mathbf{\vec{j}}+\frac{\partial \omega}{\partial m}\mathbf{\vec{k}}$

where $\omega$ is a function of k, l, and m (the wavenumbers in the x, y, and z directions) in the dispersion relation.

All the texts I’ve seen give a sort of hand waving explanation involving the superposition of two sine waves with slightly different wavenumbers and frequencies. In that case the waves travel through each other and set up an interference pattern where they add and cancel at different locations. I don’t see how this argument applies to an isolated packet where the waves are not periodic and don’t extend indefinitely. If there were an isolated packet of waves my intuition tells me that in the absence of a periodic domain the disturbance would simply spread apart and disintegrate as the components separate.

Maybe someone could post a more careful/rigorous derivation here (assuming the mathematics aren’t too difficult or tedious to type out). If not, maybe someone has a good reference. Thanks.

Marshall

Last edited: Jan 10, 2007
2. Jan 10, 2007

### Staff: Mentor

This might help you get started, although it probably has gaps:

http://physics.oregonstate.edu/~tate/COURSES/ph424/lectures/L14.ppt [Broken]

The heart of what you're looking for starts on slide 17 of this presentation. This is actually from quantum mechanics, not fluid mechanics, but waves are waves.

You can probably find more of this sort of thing in intermediate and advanced quantum-mechanics books, or in books that discuss Fourier analysis. All of my books are at the office so I can't check them at the moment.

Last edited by a moderator: May 2, 2017
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