Undulations can be used for many cool and easy to set up effects. Basically, undulations are the incremented loop of sound or movement. If you have played music you may noticed the use of effects like Chorus or Flange in your keyboard or guitar pedal, these use Undulations to create the sound. If you live near a lake or other body of water, the water is using Undulations to create wave movement. All undulations really are is this simple math function f = sin, cos or tan ( ( n * speed ) ) * depth

Pixels3D | studio | v3.6 and above
ShaderMakerPro

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First we need to create our shader or at least demonstrate how we can use Math Functions to create Undulations. Below is how we stack our Nodes to control both the speed and depth of the undulation. Notice that there is another Math Function at the beginning, and Add function, this is to keep our value being inputed above 1, a value below 1 will cause unecessary probelms when animating.

Create a default Sphere and add the above formula to the Ambient Input inside ShaderMaker ( cmd-w ). Lower the Diffuse and Specular to 0.0 and also add an Edge Blend Node to the Opacity Input.

Animate it by setting the Render Setup ( cmd-u ) to render a range of frames and do a final render ( cmd-f ). We achieved a blinking light.

This time, add the downloaded shader to the Sphere. This shader has added the Formula to each Index input of the Turbulence Node located in the Opacity Input of the Main Shader. Create a Shapes>Cube and then Controls>Scale the cube larger than the camera from the centre. Change the Cube's shader to black.

Now render it. We have achieved a blinking light, and the turbulence is expanding/contracting.

We can apply the same formula as in my tutorial on Faking Caustics, to the spots node, and create an animated caustic for our underwater scene.

Add the formula to the Spot Scaling Input. Final Render

Now that we have learned something about this wonderful process, experiment and create your own animated scene, have fun!