Wave Propagation part 1

A while ago I created a Java applet/application to play with propagation of waves in an ideal medium. The theoretical background necessary to compute this kind of simulation is documented in this series of post.

The first step is to find an algorithm that reproduces the behavior of propagation. Propagation is a very common phenomenon that can take many forms: water waves, sound, electromagnetism, vibrating string, waves on jelly, etc. Whatever the form, propagation is always based on the local change of a property of the medium (e.g. water or air) over which the propagation occur. Even if there is an apparent movement, matter is not displaced, only the local perturbation moves.

In the case of water waves, the modified property is the local level of the water surface. For sound waves, it is air pressure. For electromagnetism it is not a scalar value but a vector. In this case, each component of the vector is assumed independent of the others and behaves like a scalar property showing propagation.

A propagation is only possible if the medium is such that the local property is in a state of stable equilibrium, i.e. the induced deformation will tend to be corrected by the surrounding environment, like a spring comes back to its initial position.

A network of springs and masses is a simple, non-continuous propagation medium that can be modeled easily.

For example, in one dimension, let’s imagine a long chain of alternating masses and springs, attached to each others. Let’s assume that each mass can move only up and down, and that gravity plays no role. In this system, if a mass is moved from its initial position, the surrounding springs will tend to pull it back. But at the same time, because both ends of the springs can move, the other masses will be pulled as well, and so on.

Now let’s consider one of the masses (at position 2 on the picture), all masses have a mass of m, the springs have a stiffness of k:

Because the masses move only up and down, only the vertical components of the forces applied by the springs need to be considered. Let’s write the sum of the forces applied to the mass at position 2, where a2 is the acceleration of the mass, and Ui the vertical position of the masses:

m a2 = ∑ forces applied to mass at 2

m a2 = F12 + F32 = k ( U1 – U2 ) + k ( U3 – U2 )

m a2 = k ( U1 – 2 U2 + U3 )

Now, different things can be noted about this relation. To begin with, no real surprise, it is a differential equation (the acceleration a2 is the second-order derivative of the position U2). If the masses at positions 1 and 3 would not move (e.g. U1=U3=0), we would obtain a perfect oscillator (the mass at 2 would go up and down following a sinusoidal movement), typically represented by an equation of the form y” + K y = 0.

Back to the considered system where the masses at 1 and 3 move and all masses of the chain are interconnected via springs. Obviously the relation shows that the vertical position U2 cannot be computed without knowing U3 and U1, which in turn depend on the positions of the next masses in the chain, and so on. It turns out that any mass in the chain can be influenced by the movement of any other mass. As a result, for each step of the computation, it is not possible to solve only one local equation for one mass, but all positions must be computed.

Stay tuned for the next part.

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