Timbre depends strongly on envelope: on how the sound varies over time. (Timbre also depends on spectrum.) This background page to the multimedia chapter Quantifying Sound gives an introduction to envelope and timbre
A demonstration of the importance of envelope to timbre
This simple experiment takes a recording of a harpsichord and reverses it in time. The envelope is shown in the soundtrack, which is attached below the music. When we play forwards, we hear and see that each note begins quickly, reaches its maximum amplitude after short time, and then fades steadily away – a characteristic envelope for the harpsichord. This music, from Bach's Musikalisches Opfer (Musical Offering), is written so that the notes are the same in the reverse direction. So, playing it in reverse, the tune is the same but each note is reversed in time. When we play in reverse, we hear the reversed envelopes: each note gradually increases to a maximum, then decays suddenly.
The player is Paul Dyer Artistic Director/Harpsichordist, Australian Brandenburg Orchestra
You probably agree that the timbre is completely different: in the forwards direction, it sounds like a harpsichord – an instrument in which strings are plucked by a plectrum, and are then free to vibrate, losing enery as they do so. In the reverse direction, it doesn't sound (to me) like a string instrument at all: if anything, I'd say it resembles an old pedal organ*. Note, however, that if we take a sound spectrum over any short time interval, the distribution of power among the different frequencies is exactly the same for the forwards and backwards sound. This is an important observation, because some sources give the reader the impression that, once one knows the proportions of the different frequency components+, one knows the timbre. This demonstration shows how false that statement is.
* How could this string instrument sound to me like an organ? Well, the resonances of the harpsicord string are very nearly harmonic, so its sound spectrum is very nearly harmonic. The organ's is almost exactly harmonic. In an old pedal organ, sometimes notes get louder as they are played. So that is the nearest example I could find for this timbre.
+ Note, however, that the spectrum of the entire sound sample (including both amplitude and phase) does contain all the time information and so does contain the envelope, just as the time recording V(t) contains all frequency information.
Attack, decay, sustain and release
The envelope is complicated and varies greatly among different sounds. In musical notes, one can sometimes identify different stages: a rapid increase in loudness, called the attack, a subsequent fall in loudness, called the decay, a period during which the loudness varies little, called the sustain, and finally a period during which the loudness falls to zero, called the release. This example shows those in a note produced by a synthesiser. The length of the attack and release are quite slow, so that we can hear them more clearly.
Not all musical notes have all of these. The harpsicord has effectively no sustain: just a rapid attack and a slow decay.
A very important part of time are the starting transients: the short-lived sounds at the start of a note. These often contain substantial components that are not harmonics of the note. Broad band noise may be present, and the fundamental may be relatively slow to start. Without the starting (and sometimes finishing) transients, it is very difficult to recognise different musical instruments.
I'll add more to this later. In the meantime, see also