Defining a Wave

It just dawned on me you can construct a clean definition of a total wave, as a collection of individual waves, by simply stating their frequencies and their offsets from some initial position. For example, we can define a total wave T as a set of frequencies \{f_1, f_2, \ldots, f_k\}, and a set of positional offsets \{\delta_1, \delta_2, \ldots, \delta_k \}, where each f_i is a proper frequency, and each \delta_i is the distance from the starting point of the wave to where frequency f_i first appears in the total wave. This would create a juxtaposition of waves, just like you find in an audio file. Then, you just need a device that translates this representation into the relevant sensory phenomena, such as a speaker that takes the frequencies and articulates them as an actual sound. The thing is, this is even cleaner than an uncompressed audio file, because there’s no averaging of the underlying frequencies –

You would instead define the pure, underlying tones individually, and then express them, physically on some device.

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