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mbs
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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
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
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