# What exactly are complex sounds in terms of audio waves?

If you listen to a song through headphones you have a "right" sound wave and a "left" sound wave, correct?

But these waves have a lot of complex sounds combined in them and a digital sound wave is really just a series of sampled points anyway.

Questions:

Are sound waves really just single streams of audio that shift extremely quickly?

If you have a variety of instruments in a mix would the wave sampled at any one point just be a single sound?

• You are right that digital sound waves are made up of a series of sampled values, so it is not a continuous wave the same way an analog wave is. But once they reach your speakers/headphones the signals (one for left, one for right channel) have been converted back to analog, and will look more like the waves in JCPedroza's excellent answer. Nov 3 '14 at 19:36
• To be precise, the wave sampled at any one point would just be a single number. Sound consists of air pressure going up and down in certain patterns, tens or hundreds or thousands of times per second. Any piece of digital audio is simply a sequence of numbers, each representing the air pressure at one point in time. When you mix two audio signals together, you're just adding the numbers. Nov 4 '14 at 1:18

What you see and hear in the final waveform is the sum of all the instruments, the sum of all the individual sources.

All those sounds can be encoded in a single waveform. In the case of your first example (stereo sound) you have two channels (two waveforms, two signals) instead of one.

In other words, yes, it is a single sound (or two sounds in the case of stereo). It isn't different to what happens when you hear a sound live. Your ears pick up the sum of all the sounds in the room at that particular time and position. The same sum is taking place acoustically in the room and digitally or electronically in a waveform.

A 20Hz tone and a 200Hz tone can be summed into this single waveform: This interaction can be destructive too.

Any waveform summed with an identical waveform, but with inverted phase will cancel each other out. The same is happening in more musical scenarios, in everyday songs.

Kick snare and bass line can be all summed into one single waveform. • To complement Teental's great answer, if you want to break the complex waveforms back apart into their component sine waves then study the mathematics topic of Fourier Analysis. This is both very interesting theoretically and of great practical importance. Saying much more would be way off topic in a music group. The Wikipedia page is quite good but jumps quite quickly into fairly advanced maths. Fourier Analysis at Wikipedia Apr 13 '17 at 8:01

Any sound may be modeled as a combination of sinusoidal waveforms at different frequencies; in many cases, if a sound contains multiple frequencies which are all multiples or near-multiples of a common "fundamental" frequency, the union of those sounds will be perceived as one sound whose frequency is that of the fundamental, and whose "character" is determined by the frequency and amplitude ratios involved. Note that in some cases a listener will perceive the fundamental frequency even if no sound of that frequency is present. For example, a combination of 300Hz, 500Hz, and 700Hz would generally be perceived not as a collection of tones, but as a single 100Hz tone. Adding content at 200Hz, 600Hz, 900Hz, or other multiples of 100Hz wouldn't change the perceived pitch of the sound, but would likely make it sound "edgier".

Note that sound reproduction is seldom perfect; when vibrations in the air are converted to electrical signals or vice versa, or when electrical signals or vibrations are "moved around", there are four general kinds of things that can happen:

1. Simple linear scaling simply multiples the amplitude of all frequency content by a constant amount.

2. Other linear effects modify the signal in such a way that applying the linear effect to the combination of two signals would yield the same result as applying the effect to each signal individually and combining the result. Note that linear effects can be (and often are) affected by what the signal has been doing in the past. Consider, for example, a simple echo effect that adds to a signal 50% of the same signal, delayed by 0.001 second. Such an effect would boost by 50% the amplitude of any multiple of any frequency that was a multiple of 1000Hz (since the delayed signal would be in-phase with the non-delayed one) but cut by 50% the amplitude of any signal that was an odd-numbered multiple of 500Hz (since the delayed signal would have its high-points at the same time as the non-delayed one had its low points). Other frequencies would be boosted or reduced by differing amounts. Linear processes will change the amplitudes and phase relationships of frequencies which are present in the original signal, but will not change the frequencies themselves.

3. Harmonic distortion modifies a signal in such a way that the resulting instantaneous amplitude at any moment in time will depend in some arbitrary way on the instantaneous amplitude of the original signal at that moment in time. Unlike linear effects, the output of a harmonic distortion process does not depend in any way upon what the signal has done at any earlier moment in time. Given a signal that contains some combination of frequencies, the result of a harmonic distortion process will contain only frequencies that can be expressed as the sum of arbitrary (positive or negative) integer multiples of signals which are present in the original. If all frequencies which are present in the original are multiples of a common fundamental frequency, all frequencies which are present in the distorted signal will likewise be multiples of that frequency.

4. Effects that can neither be characterized as linear, nor as being purely time-independent harmonic distortion, can be thrown into an "everything else" category. Passing a signal through multiple linear effects is equivalent to passing them through one potentially-more-complicated one; likewise passing a signal through multiple kinds of harmonic distortion effects will yield some form of harmonic distortion. Combining effects, however, may yield results that cannot be simplified in such fashion.

Harmonic distortion effects can often be pleasing when all the frequencies in the original signal are multiples of a common fundamental; applying linear effects after such distortion may also yield a pleasant sound. Adding distortion after many kinds of linear effects, however, will frequently yield rather unpleasant sounds, since different frequencies will end up having distortion applied to them in different ways.

• I object a bit to saying "perceive the fundamental frequency even if no sound of that frequency is present". A combination of 300 Hz, 500 Hz and 700 Hz sinusoidals has 100 Hz as its fundamental frequency: the oscillation repeats every 10 milliseconds, the inverse of that time is by definition the frequency! It just so happens that 100 Hz is not contained in the signal's Fourier spectrum. But Fourier transform, though it's an extremely useful tool and the ear does something rather similar, is ultimately just an arbitrary basis switch. Nov 3 '14 at 23:45
• @leftaroundabout - a very interesting point. hadn't thought of things in that way before. Feb 20 '15 at 22:40