We have seen that the tensioned cable equation is somewhat generic of all the systems carrying non-dispersive waves. But you remember that some of our continuous systems have a very different equation of motion. That was the case of beams or of surface of a liquid. There was actually a very simple model that we built earlier, with just a slight modification of the cable equation. We called it the cable resting on elastic foundations. The equation of motion was identical to that of the tensioned cable, except for the additional term proportional to the displacement ay, so, m d^2y/dt^2 - T d^2y/dx^2 + ay = 0. Dividing by m again and using the simplified notations for the derivatives, this reads yddot-c^2y''+by=0, where c is the same quantity as before and b is just a/m. We are going to see how waves propagate here. Can you imagine what is the influence of this additional term ? There is going to be a huge influence and this is not so easy to guess! First of all, do we have something like the very general and powerful D'Alembert solution? No, we don't, unfortunately. Can we build some interesting solutions of the waves ? We can certainly. Let us look at the some of the cases we solved for the non-dispersive waves. First, the dispersion relation. If we take y of the form of real part of e^i(kx-omegat) and insert this in the equation of motion, we have D(omega,k) = omega^2- c^2 k^2 -b = 0. What is the difference? Well, now omega and k are not just proportional, we have the -b term. What does this change? Take now the case of the response to a harmonic deformation as initial condition, Ycos(kx). We have the solution as cos(kx) cos(omega t) but now with omega = sqrt (c^2k^2+b). Here it is. We can write this, in some analogy with the D'Alembert form, as y(x,t)= 1/2[Y cosk(x-Ct) + Y cos k(x+Ct)] where C=omega/k=c sqrt (1 + b/c^2k^2). What do we have here? We have, as before, a solution that oscillates in space and time. Because it is harmonic I could write it as the sum of two propagating waves. And this waves propagate with a velocity C which I will call the phase velocity. Why? Because it is the velocity of the phase of the harmonic functions. But this velocity which I write c_phi is not a constant, it is proportional to the velocity c, and has a dependence on k. Let us plot the phase velocity schematically. As you can see it is very much dependent on the wavenumber. The harmonic deformations with a short wavelength, or large wavenumber, go at about the same velocity as in the free tensioned cable, c. But for long wavelength, or small wave numbers, the velocity increases a lot. At k=0 we have an infinite phase velocity. Why? Because then the whole cable oscillates simultaneously as a mass on the springs. Everything is in phase. So, the phase velocity is infinite. To summarize, this is very different from the case of the pure tensioned cable. We are talking here of wave velocities that are not constant at all. 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.
