# Oscillations, waves and mathematical models

**Mechanical oscillators**

Every mechanical oscillator, isochronous or not, has these features:

- it is displaced successively to one side then the other of an equilibrium position;
- it is accelerated towards the equilibrium position by a force; the force is related to its displacement in some way;
- it has inertia, which means that it continues through the equilibrium position, rather than coming to rest there;
- it possesses kinetic energy as it passes through the equilibrium position, potential energy at the extreme ends of its motion, and usually a combination of both at points in between;
- there are resistive forces against which it must do work; as a result the oscillator loses energy.

**Oscillations can make waves**

Every mechanical wave has its origin in an oscillating object. The particles in the medium themselves oscillate, in a similar way to the originating object, but a certain time later.

Electrons oscillating in a transmitting aerial give rise to fluctuating electric and magnetic fields which travel and cause electrons to oscillate in a receiving aerial.

**Isochronous oscillators **

There are simple arguments for relating the steady time-keeping of an isochronous oscillator to the relationship between displacement and restoring force.

1. The time is constant. If the amplitude is doubled, the average speed must be doubled. The double speed is acquired in the same time, so the average acceleration is doubled. Double the acceleration means double the force. So the average acceleration is proportional to the amplitude. This can be achieved by having a restoring force proportional to the displacement.

2. If the restoring force is proportional to the displacement, twice the displacement gives twice as much force, giving twice as much acceleration. Thus the velocity gained in any given short time is doubled, and twice the distance is covered in this same time. But the displacement was doubled to start with. There is just twice as far to go, and double the distance will be covered in the same time as the original motion.

The motion of an oscillator for which the restoring force is inversely proportional to displacement from its equilibrium position would be called *harmonic*. Simple harmonic motion is an idealized kind of motion. Real oscillators - atoms in a crystal or a molecule, car bodies on springs, buildings, and bridges - are likely only to approximate to this motion.

**Simple harmonic motion as a mathematical model**

You might consider whether restoring force is likely to be accurately proportional to displacement, in one or two practical cases. Mathematical models can be built up, to be used in describing phenomena. Such models are often strictly limited to ideal cases, and represent real events more or less well. But such seeming inadequacy can be a strength, for the ideal model can be quite simple and can apply to many things. If necessary, more complicated mathematical models can be devised, to cope with damping, for example.