Making more connections

Four interwoven strands in an approach to understanding waves

So far there have been four parallel, but interwoven strands in the explanation of waves. Here we'll show how they are interconnected, and point out how they continue to grow together as the explanation of radiations and radiating deepens beyond study at this level.

Firstly there has been the modelling of phenomena as vibrations that travel. So a source vibrates with a particular frequency and amplitude. After a delay set by the trip time (distance from sourcepropagation speed), the detector mimics that vibration. This passage of information without matter we call waves.

Secondly the idea that beams can be predicted by drawing rays has been revisited. These rays follow seemingly arbitrary rules; but it turns out that these rules can in turn be predicted if you select, from a whole range of possible paths from source to detector, the path with the minimum trip time.

Thirdly the importance of the minimum trip time itself has begun to be explored. Minima of time occur where the differences in trip time between adjacent paths don't vary by much. These are paths for which the contributions add up. As the contributions rely on the source vibration this begins to draw together the first two strands.

Finally, the world at the smallest scale is a quantum world. All the radiating – starting with emission at the source, and absorption at the detector – is grainy. So the power emitted or absorbed is measured in the number of photons per second. The resultant amplitudes from the contributions predict where, on average, these photons will be detected.

Multiple paths for reflection, refraction and propagation and least time paths

For reflection, refraction and propagation, the least time path predicts the ray. Here we consider a triplet of least paths, and the difference in time between these paths. If the difference in trip time is small, then the contributions from the three paths will superpose constructively, and there will be a large chance of detecting a photon there. This triplet will always be centred on the actual path of least time. So the least time path is the path that, if blocked, would make the greatest difference to the number of photons arriving at the detector, as it is the path that makes the greatest contribution to the resultant amplitude. It's all about considering possibilities, and these are constrained by geometry.

How to engineer mirrors and lenses with photons and paths

Here are found more complex systems, probably most easily thought of as being optical, but there is nothing in the model that demands this. Any travelling vibration, or waves, is predicted to behave in the same way.

The power detected – the activity – is predicted by the sum of the contributions. The sum of the contributions – the resultant amplitude – depends on the trip times.

So, finally, it all comes together:

• frequency
• amplitude
• trip times
• photons
• power

Waves and radiating are not best thought of as a list of independent rules about different phenomena. Then reason that they're connected into a single topic is a theoretical reason, and here you have reached one level of explanation as to why this unity is so compelling.