### Can You Believe Your Eyes?

Scene from John Ford’s Stagecoach (1939).

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained?

The illusion is known as the stroboscopic effect. We don’t see it in real life, where time flows smoothly and continuously. But, in the cinema, films are projected at a rate of 24 frames per second: time is chopped into discrete moments separated by gaps. We only see the position of the wheels 24 times each second, with no indication of what is going on between the frames.

When the coach is moving fast, the wheels whirl rapidly and the spokes are a blur. But as the speed drops, the stroboscopic effect comes into play. Suppose the wheel has a one metre radius or a circumference of 6 m. If the coach is travelling at 6 m/s, the wheel rotates once every second, which means ${\frac{1}{24}}$ of a rotation every frame. With 24 spokes in the wheel, each spoke moves to the position of the one ahead between frames, so that every frame appears identical: we see a wheel that is not spinning.

If the speed were twice 6 m/s, each spoke would move to the position of second one in advance and, again, the wheel would appear stationary. Indeed, for any integral multiple of 6 m/s, the wheel would appear not to spin.

Frame by Frame

When the stage-coach is standing still, every frame is identical. We see the spokes in the same position each frame, as shown in Fig. 1.

Fig 1. Three successive frames for a stationary coach. Red and green lines both represent physical spokes.

Suppose now that the coach starts to pull away. Both the green and red spokes advance through a small angle between frames, and the clockwise rotation is clear (Fig. 2).

Fig 2. Three successive frames for low coach speed. Red lines represent a physical spoke. Green lines also represent a single spoke.

When the coach speed reaches 3 m/s, each spoke moves to the mid-point between spokes. From frame to frame, the spokes flicker between two positions, and we cannot discern the direction of spin.

As the speed increases further, approaching 6 m/s, each spoke moves to a position slightly behind that of the one in front of it. This is shown by the red spokes in Fig. 3. But what we perceive is that the green spoke (actually, a different spoke in each frame) appears to move slightly counter-clockwise, and our brains interpret this as a backward rotation of the wheel.

Fig 3. Three successive frames for higher coach speed. Red lines represent a physical spoke. Green lines represent three different spokes that appear to be a single spoke moving counter-clockwise.

When the coach reaches 6 m/s, each spoke moves to the position of the one ahead of it between frames. But all we see is that the spokes are in the same position on each frame, so the wheels appear to be static.

Fig 4. Three successive frames for a coach moving at critical speed. Red lines represent a physical spoke. Green lines represent three different spokes that appear to be a single stationary spoke.

The cycle is repeated with alternate forward and reverse rotations as the speed increases.

Aliasing

The stroboscopic effect is a result of the discretisation of time. There is an analogous effect for discrete space. In solving partial differential equations, we divide the spatial domain into grid boxes. So, a line segment is split into small sections of length ${\Delta}$. The shortest wave that can be represented on the grid has wavelength ${2\Delta}$.

Shortwave signal (blue), sampled on a coarse grid (black dots). is aliased to a long wave (red).

The Figure shows a shortwave signal (blue) represented on a coarse grid (black dots). It is indistinguishable from the long wave (red).

If the governing equations are nonlinear, long waves multiplied together can produce waves shorter than ${2\Delta}$. Because the wave is specified only on the grid, these short waves appear as long waves, just as high-frequency rotation of the wheel appears as low frequency. This phenomenon is called aliasing: the short wave `goes under the name’ of a long wave.

Numerical modellers must take precautions to avoid aliasing so that the solution is not corrupted by spurious projection of the short waves onto the resolved spectrum.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
Full details and links to suppliers at
http://logicpress.ie/2020-3/

>>  Review in The Irish Times  <<

* * * * *