The end of this post concludes with a visualization of a certain kind of three dimensional standing wave produced by a point source. The rest of this post will explain some of the intuition behind the modelling.

A standing wave is what results when a traveling wave such as this

and its time-reversed analog traveling in the opposite direction

are considered together and allowed to interfere with each other constructively and de-constructively. The positive or ‘upward waving’ parts of one of the waves are cancelled out by the negative or ‘downward waving’ parts of other wave.

This is example of a 1 dimensional wave, the curve, which is ‘waving’ into a higher 2nd dimension represented by the vertical extent on the plane that the curve lies in.

In this way, a 2 dimensional wave can be visualized as a 2 dimensional surface which ‘waves’ into a higher 3rd dimension. Here is a certain kind of 2D traveling wave produced by a point source which emits symmetrical in 2 directions creating these circular shaped waves:

And a standing wave produced from the interference of an outward and inward moving wave:

Trying to continue this way in order to visualize a 3 dimensional wave results in some issues, because the 3 dimensional thing that is doing the ‘waving’ needs to wave in a higher 4th dimension which we can’t really visualize by simply drawing in the 4th dimension.

Thus far, we have been associating waviness to movement in a higher spatial dimension. Instead, assign a color to the thing that is doing the waving to represent its waviness. Doing this allows us to eliminate that higher spatial dimension needed in the previous wave visualizations, and alternatively associates this to a color dimension.

Using this method to visualize the 2 dimensional traveling wave resulting from a point source looks like this, where lighter regions can represent positive values and darker regions negative values:

Now we can visualize a 3 dimensional wave by assigning a 3 dimensional region colors according to the wave structure.

Below is a traveling wave resulting from a point source emitting symmetrically in 3 dimensions producing spherical wavefronts. One of the upper quadrants of the region has been removed in order to expose the internal structure of the wave, and the bounding box is present to allow for a better sense of depth.

The standing wave resulting from an outward and inward traveling wave would then look like this:

Increasing the size of the waving region so that it fills the bounding box looks a little more interesting. The faces of the cube show 2 dimensional projections of the 3 dimensional wave which are identical to the 2D analog.

A traveling wave:

And a standing wave:

What was exemplified here only considered a certain kind of wave, which is the most symmetric of waves. In general, a standing wave can be produced by taking any waveform and adding it to the waveform produced when the time direction is reversed.

Interactive Mathematica code: notebook, CDF