Hello! So far we have considered unbounded continuum media. We have seen that these media could bear waves, propagating waves. But at some point these waves in real systems are going to reach a boundary. What happens then? And what happens if waves are confined between two boundaries? That is the topic of the week we start now. Let us see first what happens when a wave reflects on a boundary. To make things simple, I will stay with waves on dimension one continuum media, as we did before. We did that with cables, beams, sound and surface waves. Remember that we could make a very important distinction between non-dispersive waves and dispersive waves. Non-dispersive waves existed on the tensioned cable but also for sound in pipes, or on the surface of shallow water. The model system for all this was the tensioned cable. Dispersive waves could be found in many different cases: bending beams, deep water and the model system was the tensioned cable resting on elastic foundations. Let us work with the tensioned cable, with or without elastic foundation What kind of boundary condition can we have on a cable? The simplest would be that the cable is fixed and cannot move. That would mean that at this point, we have y=0 for all times. Here is a semi-infinite cable which is fixed at x=0, so that y(0,t)=0. You can imagine other boundary conditions. For instance, a free end (with tension) would be dy/dx=0. How does this condition interact with waves on the cable? What happens when a wave hits the boundary? We can take first the simple non-dispersive case. You remember the d'Alembert solution that states that waves are always the sum of two functions, one of x-ct and one of x+ct, y=F(x-ct)+G(x+ct). Can we have any form of F and G? Well, this satisfies the equation for the tension cable in the medium. But now we have a condition that the displacement should be equal to zero at x=0. This means that for any time t F(-ct)+G(ct)=0. These two functions F and G are now related. If I take an argument z, then F(-z) = -G(z). They cannot be chosen independantly. The displacement that satisfies both the cable equation and the boundary equation must therefore be y(x,t) = -G(-x+ct)+G(x+ct). What does this look like? Well, we have a left going part and the right going part is just the opposite. Why is that? Because the sum of the two will exactly cancel at x=0. Let us take a simple function G like here. A pulse, G is zero everywhere except on a small zone. At early times, on the cable, I will see the pulse G moving left: The second part of the solution does not materialize on the cable When the pulse reaches the boundary, the second part of the solution becomes non-zero and exactly compensates the first part, at the point x=0. This is how the condition is satisfied. And then, for later time I will only see the second part. What happened? the wave reflected, and at reflection it changed sign. This is a very common phenomenon. And now you know why the sign changes. It is because we need a wave opposite in sign and direction to exactly balance the wave so that the boundary condition is satisfied. Of course, you can do the same for other boundary conditions. If the cable is free, dy/dx=0 you will get y(x,t) = G(-x+ct) + G(x+ct). The wave will reflect but not change sign. Of course, this is true for any form of waves, any G function. So, let us have a harmonic function G(x+ct) = sin[k(x+ct)]. The total wave is y(x,t) = sin[k(x+ct)] - sin[k(-x+ct)]. This reads also y = 2sin(kx) sin(kct). What is that? this is just a harmonic standing wave oscillating at the frequency omega=kc. We had this solution before. But what is new here is that because it must satisfy the condition at x=0 it cannot be anywhere on the x axis. And this solution is the sum of a left going wave and the reflected inverted right going wave. All this was for non dispersive waves. But what happens if the waves are dispersive? Well, as you know, the d'Alembert solution does not exist. But we solved the case of harmonic waves, remember. We had exactly these same standing waves. We found that wavenumbers k and frequencies omega were related by the dispersion equation D(omega, k) = 0. For the tensioned cable on elastic foundation that was D(omega,k) = omega^2 - c^2k^2 - b = 0. Now, all we have to check is that the sum of harmonic waves is zero at origin. The solution takes the same form y(x,t)=sin(kx) sin(omega t), where k and omega satify the dispersion relation. And this is the sum of a left going wave sin(kx+omegat) and a right going wave sin (kx -omega t) that combine to give you a zero displacement at origin. Pressure waves reflect - this is sound echo Surface waves do reflect. All waves reflect. To conclude, the boundary condition does not interfere with the dispersion relation, that connects space and time evolution. It just selects the right combination of waves so that the condition is satisfied. But what if the system is now bounded on both sides? Let us do that next.