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#appliedmathematics

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Here we see three identical pendulums, oscillating independently. The red and purple ones are vibrating with small amplitudes and so their periods are nearly the same. But the blue one is undergoing what would be considered to be very large amplitude oscillations and has a significantly longer period. In fact, as the amplitude approaches π radians, the period increases without bound and approaches infinity.

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A subtlety probably difficult to spot in the animation is that the interaction of the two waves leaves them phase shifted, with the taller wave gaining position, while the shorter one loses it. This further animation shows the interactions again (purple) but I’ve also shown what would happen if each wave moved without interaction with the other (red and blue).

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When you start looking at #NonlinearWaves, some of these principles no longer apply. For example, in the Korteweg-de Vries equation, which has #Soliton or solutions, you can no longer simply add two solutions together as the resulting function would not be a solution of the governing equation. Waves of different heights travel at different velocities, with the taller waves moving faster than the shorter ones. Instead, they interact #nonlinearly.

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“My PhD was all about understanding what happens to blood flow in collapsible blood vessels like the giraffe jugular vein. In my postdoc I was investigating how to optimise ventilator settings for patients in ICU and then how to deliver inhaled therapies into the lungs. Since then, my focus has been in trying to understand how diseases like Asthma and other respiratory diseases originate and then progress. This involves incorporating biology and physics into mathematical and computational models, using approaches from different areas of applied maths. More recently I have started to look into the mechanisms that could lead to a rare lung disease called lymphangioleiomyomatosis (LAM) and Long Covid.” - Bindi Brook

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(Applied?) maths terminology question:

In the context of signal analysis (doing things like Fourier transforms) you might have some variable (say, t) that you're treating as being equivalent to time and you're interested in some periodic function of t (say, sin(t)) with wavelength 2pi on the t scale.

It's common to (informally?) use the term "phase" to refer to the value of t within the first cycle of the periodic function. In that usage: phase = t modulo 2pi

By extension, I have seen people refer to location anywhere on t (not just the first cycle) as "phase", but this seems to me to be ambiguous, and not necessarily helpful, terminology.

My question is: Given that the variable t is possibly something much more abstract and not necessarily "time", is there some accepted terminology for referring to the variable t in this context?

(This is possibly a non-issue for pure mathematicians because they might never talk about phase and use only symbolic representations without using descriptive words. If so, I would be seeking terminology from some applied discipline where this kind of generic descriptive labelling is used.)

In brief, if I wanted to say:
phase = x modulo 2pi
because I am talking about periodic functions of x, is there some generic descriptive term for variables like x (e.g. time-like)?