# How are complex sound waves combined?

Audio is often explained by single frequencies. Typically this is a sound wave:

plot sin(x) * 2 from 0 to 10


However we usually deal with more complex sounds, more specific various frequencies and amplitudes. Varying amplitudes within same frequency etc. in the same time frame.

How are such sound patterns visualized? Say we have the previous tone + this one:

plot sin(5x) * 2 from 0 to 10


Would this be the end-result? Would a wave like this travel trough the air?

plot (sin(5x) + sin(x)) * 2 from 0 to 10


Even a mono track can have a rather detailed sound picture of say a philharmonic playing some composition. Would such a recording (or live observation for that matter – but that would give a different picture I guess as one would have multiple sound sources) be a complexified × ten folds version of the two sine waves?

My end goal is to better understand PCM audio and how digital audio works. A starting point is to better understand the physics behind audio. Then again a most of what I find deals with one frequency samples and the like. (I'm likely missing some terminology.)

Yes, you have the right idea. You will want to learn about Fourier analysis, which lets you take a complicated-looking waveform like your third figure and analyze it to say "this is two sine waves, frequencies 1 and 5, equal amplitudes, zero relative phase."

I like to think of a piano as an inverse Fourier transform machine: you push the keys to tell the piano "please generate frequencies C, E, and G, with the C having larger amplitude than the others" and the piano makes the air vibrate for you. Your auditory system then does the ordinary Fourier transformation: with ear training, you can take those vibrations and say "Oh, a major triad, with a strong root."

• Nice. Searching the web for "acoustic" / "sound wave fourier analysis" seems to give some hits more in the direction of what I'm after. – user129107 May 5 '14 at 5:52
• "I like to think of a piano as an inverse Fourier transform machine." This is one of the best analogies I have run across in a long time. – loneboat Aug 7 '14 at 21:26
• @loneboat Aww, shucks – rob Aug 7 '14 at 23:35
• @rob - I like the analogy and I would like to add something. Possibly one would remark "Oh, a major triad with a strong root." but the term Strong Root is generally used in connection with harmonic progression rather than wave amplitude (dynamical volume) of specific notes. For example you play a C Major triad on your Inverse Fourier Transform Machine (or Infotram - as some musicians call it). If you move from a C chord to an F chord (down a 5th) this would be considered a Strong Root Movement. Conversely from a C chord to a G chord (up a 5th) would be considered a Weak Root movement. – Wookie Jan 12 '20 at 13:07
• @Wookie I was thinking "forte" but wrote "strong" in an effort to avoid injecting musical jargon into a discussion with a possibly non-musical audience. A mistake, perhaps, given the other musical jargon in the answer, but a mistake I can live with. – rob Jan 12 '20 at 17:46