We’ve previously talked about **standing waves**. But *what is it ?*

Visually, it looks like a *dancing wave*: the wave is going up and down, but does not seem to travel, because each point has a constant amplitude.

Standing wave generated from a wave maker.

From the video above, you can see that the wave is not moving the same way everywhere: some parts of it go up and down with a maximum amplitude (these points are called *antinodes*) while some other points seem static (these are called *nodes*). This is characteristic of a standing wave.

Well, that sounds cool, but* how does it occur* ?

A standing wave is formed when two identical waves (same frequency, amplitude, speed…) travel in opposite directions and meet: the sum of these two waves will create a standing wave. This is how we generated it in the movie above: by setting an appropriate wave maker frequency, we can generate a wave which propagates along the basin. When it reaches the end wall, it is reflected and comes back in the opposite direction. When meeting the incoming waves, standing waves appear.

Formation of the standing wave: the wave travels along the basin, is reflected, and meets the incoming wave to give a standing wave.

The number of standing waves in the basin depends on the frequency of the wave maker. This one must satisfy an equation involving the length of the basin, the number of waves that we wish (the “*wave number*“) and the wave length (basically the distance between two crests). In the video below you can count 7 waves, but we could have generated 1, 2 or 13 waves just by changing the frequency of the wave maker.

**Can this occur in daily life ?**

Of course it can ! Just take a string, hold it in both ends, and move it up and down. Give it a try with different speeds: one of them (corresponding to the *resonance frequency*), will create a standing wave.

You can also observe standing waves in shallow water, in music (harmonics), light, and a lot of other applications.

Ok, but *how is Maths involved in this* ?

Well, you can use Maths to simulate these standing waves: by solving waves equations analytically or numerically and choosing an initial condition (that is, the solution at initial time t=0) that satisfies the characteristics of a standing wave, you can end up with the kind of simulations below. These simulations can then be used to set testbeds, which are a primordial step in the production process of any object. For instance, we can predict the behaviour of a boat in such waves, in order to improve its design.

Simulations of standing waves.

Nice, right ? 😉

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