We saw what happened with a boundary condition on one side of your system. Let us add another one, so that the system is finite. Our cable is now fixed at both ends, like here on the bridge. We could try to see if we can build something with the d'Alembert solution, with waves that are going to bounce from one side to another. But this is not very easy. Let us go directly to the harmonic solutions, but now with two boundary conditions, for instance a fixed displacement at the point of coordinate L, y(L,t)=0. We look for solutions that oscillate in time Y(x,t)=phi(x) sin(omega t). As before, to satisfy the left boundary condition we need to have Y(x,t)=sin(kx) sin(omega t) with k and omega related by the dispersion relation But now this needs to satisfy also the condition at the other boundary. Is that possible with the form above? Certainly, but in that case you need to have sin(kL)=0. This is very restrictive, it is only satisfied for wave numbers that are multiples pi/L, k_N=N pi/L. Let us summarize. The system with a finite length L can bear standing waves sin(kx) sin(omega t), but only for certain k values. For k_1=pi/L, here is the shape. FOr k_2=2 pi/L here is the shape, and so on. And each of these shapes will oscillate at frequencies that I call omega_1, omega_2.. omega_N. How do I know the frequencies ? Well the dispersion relation still holds, so that omega_n is given by the condition that D (omega_n, k_n)=0. So far we have not specified what the dispersion relation is. It can be that of a dispersive or non-dispersive medium. If the medium is non-dispersive, then the dispersion relation reads D(omega,k) = omega^2 - c^2k^2 = 0. This gives Omega=kc. So that with k_N=N pi /L, we have omega_N = N pi c/L. This is a nice and simple formula. Our tensioned cable can have standing wave motions of certain shapes and the corresponding frequencies are proportional - the second is twice the first, the third is three times the first, and so on. Look at the formula for the first frequency omega_1= pi c/L. The period of this oscillation is T_1 = 2pi/omega_1 = 2L/c. This is exactly the time that a wave takes to go back and forth along the length L Here we recover the fact that these standing waves are indeed the superposition of propagating waves, with their velocity c. The formula for frequencies is actually a fundamental result for musical instruments. Imagine a guitar string. By adjusting the string tension T, you can adjust the wave velocity c=sqrt(T/m), and so the frequencies. This is how you can tune your guitar! By changing the length of the string with you finger, you can adjust the frequency. This is how you change from A to B while playing. And when you play the string you will hear a nice sound because all the waves that will be heard will have many frequencies that are proportional - what we call harmonics. And you can do the same things in wind instruments, because acoustic waves are non dispersive too. In the slide trombone you can change the tone by changing the length In a recorder, with the same length I can play two different sounds, that have two different frequencies, although I don't change the length. Listen: first, second. How is that possible? Because they are several frequencies for the same pipe, here omega_1 and omega_2. And omega 2 is twice omega 1, this defines an octave. But what happens for dispersive media? The simplest one is the cable with an elastic foundation. Well, nothing much changes, we still have our selection of wave numbers that must be k_N=N pi/L to satisfy the boundary conditions. But now, the frequencies are given by the dispersion relation Omega_N= sqrt ((k_N c)^2 +b), which is Omega_N= sqrt ((N pi c/L)^2 +b). Everything is different - the frequencies are no more proportional. Music is going to be difficult. Note that what changes here is not the spatial shape, but the frequencies that the dispersion relation gives. You remember that there are many different kind of non-dispersive media: beams, deep water, and many others. As you will see soon, the spatial shape of these standing waves, and the corresponding frequencies may be quite different from what we had here. But something is common to all these systems, that are bounded. And even to systems in more complex geometries. In all these, there exist series of standing waves, freely oscillating solutions. They are called modes, because they are simply modes of vibrations, ways of vibrations. How does this work? In the space domain, Boundary conditions select acceptable wavenumbers. This gives the modal shapes In the time domain. These selected wave numbers give selected frequencies though the dispersion relation. These are the modal frequencies. You will see plenty of examples of this scheme in the forthcoming weeks. It is time to conclude. When the system was unbounded the dispersion relation was the only bounding relation, between space and time evolution. All wave numbers could exist. When we added a boundary condition to our system of dimension one, it just prescribed that some combinations of waves be made to satisfy the boundary condition. Still all wave numbers could exist. But when we had two boundary conditions, when the system was of finite length everything changed. Only some wave numbers were acceptable, and correspondingly through the dispersion relation, some frequencies. That defined the modes of the system. What can we do with modes? Well, actually a lot of things. Next, we shall see that they have plenty of useful properties.
