Standing waves and the strain ellipsoid envelope. This is not an envelope but a polar coordinate plot of the main ellipsoid axes length against the angle of rotation. It provides a vector model of the displacement the strain causes. It is not a model of the actual motion of the strain ellipsoid, but gives a model of pressure acceleration , at a boundary.

Standing waves are a wave phenomenon usually modelled by a sine function combination in a Fournier series. This is based on constructive and destructive interference theory.
The sine wave is proposed as travelling through a medium in a given direction and opposed by another dine wave moving in a contra direction. The illustration shows the amplitude variation with respect to time as a function of position.

I noticed that watching a supposed wave transport, the sine wave used to explain the math was not consistent with the standing wave observable!
A wave has been identified with the sine curve, Fourier analysis has strengthened this identification. The identity with the exponentials has extended its reach. It is therefore difficult to think of any other concept of a wave.

The circle that underpins all common measures supports the sine curve and therefore the link between rotation and wave analyss.. However the circle is one of a number of forms which are cyclic, starting with a line segment!

A point moving along a line segment can describe an oscillation if a coordinate frame is set.
We can use an orthogonal frame and use the equal division of the lie as the scales for the axes. The wave we get is saw tooth.

Now if I use an angle as the shape and use an ordinate system on the angle I can plot a wave pattern from this. How it varies depends on how I traverse the anle and I get a disjoint saw tooth wave.

If I use a triangle I get a more boxlike shape to the curve. The curves vary but have the displacements either side of a horizontal axes . If an ellipse is used rather than a circle the wave shape is different. If a spiral is substituted the wave shape is different again. If I use a Brownian motion distribution I get a different wave shape once again.

So what is the best model for a wave in a fluid.?

The answer seems to be the envelope of the strain ellipsoids as it develops over time!



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