Here are six different ways of sounding the same note.  (The note is C1, or the C one octave up from middle C on the piano; frequency around 520 Hz.)  Listen by clicking on it:

U	V	W
X	Y	Z

Most of these sound very different from one another.  (Maybe you can name the instruments for some.  Identification is at the bottom of the page, or may come sooner.)  The reasons for their differences are pictured in their waveforms.  The waveform of the guitar sound

is a picture of the up and down motion of the string while it is sounding.  The five hundredths of a second selected are blown up here:

For an accordion, the waveform pictures the up/down vibration of the reed; for others, the waveform always shows the motion of the moving part that produces the sound (maybe just vibrating air, if you’re tooting a flute).

There are two properties of the waveform that account for most differences you hear – the spectrum of the waveform, and its envelope.

Spectrum

The simplest sound has a waveform that looks like this (this is W’s)

all the time it’s sounding.  That smooth, unscraggy shape in the blowup means that W (but not the scraggy guitar) is producing a pure tone, i.e., it’s emitting only the note C1, frequency 520-ish Hz, and no others.  A Fourier analysis of the waveform pictures the spectrum of the sound (variously called its harmonic content or timbre).  Here’s W’s spectrum:

A spike in the spectrum means a pure note of that frequency is present in the sound.  The big spike, actually at 530 Hz, is the C1 note.  There are other spikes, each showing the presence of another pure tone in W’s sound.  The one at 1058 Hz, reaching about -30 dB on the vertical scale, is C2 on the piano, in musical terms one octave higher than C1.  But the vertical scale tells us that C1 is well over 100 times louder in W’s sound than C2 is, so I’m safe in calling W’s sound a pure tone.

The sound Y is also pretty nearly a pure tone, as a blowup of its waveform shows:

Listen to W and Y again to hear their similarity.

Now listen to V again.  Its waveform resembles W’s on a large scale:

but not in detail:

Here’s the spectrum of this raggedy Andrew:

Pipe all those other pure tones!  This is what gives most instruments their unique sound: which higher tones are present in the spectrum, and how loud they are.  (In musical terms, the second spike is C2, an octave above C1; the 3^rd spike is G2, an octave-and-a-fifth above C1; the 4^th, two octaves up, C3; the 8^th spike is C4, the highest note the piano can play.)

Envelope

Here are two waveforms from a viola, first when it is bowed and second when the same string is plucked:

The second envelope is so different, with its sudden onset (the attack) and relatively rapid pooping-out (decay), that everyone recognizes a plucked string is behind the sound, and then the spectrum helps decide what plucked instrument it is.  But it really is the envelope and not the spectrum of the waveform that is the strongest identifier for sound with this second kind of envelope (like the guitar back at the start).  Convince yourself with this example:

A	Ode to Joy snippet played on a piano	Listen
B	Same music as A but played backwards on a piano, i.e., starting with the last note and then reading to the left on the sheet music	Listen
C	Recording produced in B reversed in time electronically (waveform of C is mirror image of B’s)	Listen

Identification:

U	V	W
X	Y	Z

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