In "Constant power panning using square root of intensity" we used the square root of the desired intensity for each speaker to calculate the amplitude of each speaker. However, square root calculations are somewhat computationally intensive, and it would be nice if we could somehow avoid having to perform two such calculations for every single audio sample. As it happens, the sum of the squares of sine and cosine functions also equals 1.
The intensity of sound is proportional to the square of its amplitude. So if we want to have a linear change in intensity as we go from 0 to 1 or 1 to 0, we need to use the square root of that linear change to calculate the amplitude. This example patch is exactly like "Linear amplitude panning", except that we consider the linearly changing signal from line~ to be the intensity rather than the amplitude, and we take the square root of that value to obtain the actual amplitude for each speaker.
The simplest and most common way to localize a sound in a stereo field is to vary the relative intensity between the two speakers. To make a sound seem to move from one side to the other, for example, you can start with the level of one speaker set to 1 and the other speaker set to 0, then gradually turn one down to 0 as you bring the other up to 1. This patch demonstrates a direct linear pan from one speaker to the other.