Let us look at the other problem, that of

the shape of harmonic evolutions. What is the shape phi(x) that oscillates at a frequency

omega? Again, I can insert this in the equation of

motion and I get an equation on phi. For a given frequency of oscillation, omega, the

corresponding shape will be defined by this differential equation.

And we get something new here : we only have a solution for our unbound cable when omega

is large enough, larger that sqrt of b. In that case, the shape in space is harmonic,

with a wavenumber k=sqrt(omega^2-b) / c. Below that limit frequency or cut-off frequency

omega_c = sqrt(b), there just no solution.

Well, there is a solution, following the same

idea as above we can define a phase velocity which will depend on omega. It is simply

c_phi = c omega/sqrt (omega^2-b).

Here is what it looks like : the phase velocity becomes infinite at the cut-off frequency.

And there is none below. When you think of it, this is a bit puzzling.

We do have wave propagations, but the only velocity we have is sometimes infinite.

And sometimes, there is just no velocity because there is no propagation.

To understand this better, we can look at something a bit different, the case of two

waves going in the same direction. More precisely, of two waves with almost the same wavenumbers

and frequencies. The first one woud be

cos[(k-delta k)x-(omega-delta omega)t]. The second one cos[(k+delta k) x-(omega+delta omega)t].

Both waves are solutions of the equation of

motion if their wavenumber and frequency satisfy the dispersion relation. They have slightly

different parameters, so they propagate with slightly different phase velocities.

Because the equation of motion is linear, I can add them and form a motion as the sum

of two waves.

The result of the sum, using simple cosine combinatiosn formulas, reads

2 cos (delta k x - delta omega t) cos (kx - omega t).

Here is what this looks like, schematically.

What we have here is simply our classical propagating wave cos(kx -omega t),

but modulated with an amplitude that propagates

also.

So, The carrier wave has a phase velocity of omega/k,

as before. But the modulating wave has a different phase

velocity of delta omega/delta k which is about d omega/dk. You can see that on the animation.

This means that, for a frequency omega, or for a wavelength k, there are two velocities

involved: that of the phase of the carrier and that of the phase of the modulations.

This second velocity is called the group velocity, c_g. It can be shown that this is also the

velocity of propagation of the energy in the waves.

What is the value of this group velocity for

our cable with elastic foundation? Here it is expressed as a function of omega

c_g=d omega/dk= c sqrt (omeha^2-b) / omega. The evolution with omega is totally different

from the phase velocity. As omega approaches the cut-off frequency, the group velocity

goes to zero. At the cutoff frequency, where the cable oscillates in phase everywhere,

there is just no propagation of energy and so a zero group velocity.

And below the cut off frequency there is no propagation, of course.

To summarize, in the case of an equation of

motion that differs from the tensioned cable, we have something new: there is not just one

wave velocity for everything. Actually, for each wavenumber there are even two velocities,

one for the phase of harmonic motions, and one for the modulations of these motions.

Because all these depend on the wavenumber,

you can easily imagine that a perturbation that contains several wavenumbers is going

to disperse in the medium, because all components will go at different velocities. We have what

are called dispersive waves. And in that case, the velocity that becomes

of interest is actually the group velocity c_g.

In the non-dispersive case time and space had similar roles.

Frequencies and wavenumbers were proportional. There was a time-space symmetry.

phase and group velocities were identical and constant.

But in our dispersive case here, all this is lost. Each harmonic component has a different velocity

and the time-space symmetry is lost. Phase and group velocities are different and

depend on the wavenumber or the frequency. There might even be a range of frequencies

with no propagation. And all this appeared by just

adding the elastic foundation!

We have seen this on the case of the tensioned cable on elastic foundation. But how does

this apply to beams and to a fluid surface, where the equations also differ from the cable

equation? Certainly you can describe the waves on these

systems using the concept we just derived. Let us see that next.