acoustics
What are those characteristics by which every sound can identified uniquely?
For example, pitch is one of the characteristics of sound, but let’s say a note C# can also be played on a guitar and piano with same pitch but the resulting sound that we hear is different so what are those characteristics which defines every sound.
After much investigation, simulation and a deep literature search, I've figured out the true answer.
You perceive a chirp because you are being hit with the echos of the sharp noise that generated the sound. The times between the arrival of those echos is decreasing inversely with time, so it sounds as if it were a tone with a fundamental frequency increasing linearly in time, hence the chirp.
To get a feel for the phenomenon, consider a simulation:
Above you see a slowed down version of the simulated pressure wave inside a 2D racquetball court. I threw up the generated sound on soundcloud.
If you watch the simulation, pick a particular point and watch the reflected sounds go by, you'll notice the different instances of the multiple echos arrive faster and faster as time goes on.
You can clearly hear the chirps in the generated sound, and if you listen closely you can hear secondary chirps as well. These are also visible in the spectrogram:
This phenomenon was studied and published recently by Kenji Kiyohara, Ken'ichi Furuya, and Yutaka Kaneda: "Sweeping echoes perceived in a regularly shaped reverberation room ," J. Acoust. Soc. Am. Vol.111, No.2, 925-930 (2002). more info
In particular, they explain not only the main sweep, but the appearance of the secondary sweeps using some number theory. Worth reading in full. This suggests that for the best sweep one should both stand and listen in the center of the room, though they should be generic at any location.
Following the paper, we can give a simple geometric argument. If you imagine standing in the middle of a standard racquetball court, which is twice as long as it is tall or wide, and clap, your clap will start propagating and reflecting off the walls. A simple way to study the arrival times is with the method of images, so you imagine other claps generated by reflecting your clap across the walls, and then reflections of those claps and so on. This will generate a whole set of "image" claps, located at positions $$ ( m, l, 2k) L $$ where $m,l,k$ are integers and $L$ is 20 feet for a racquetball court, the time for any particular clap to reach you is $t = d/c$ and so we have $$ t = \sqrt{m^2 + l^2 + 4k^2} \frac{L}{c} $$ for our arrival times. If we look at how these distribute in time:
It becomes clear why we perceive a chirp. The various sets of missing bars, which themselves are spaced like a chirp, give rise to our perceived subchirps.
For the simulation, I numerically solved the wave equation: $$ \frac{\partial^2 p}{dt^2} = c^2 \nabla^2 p $$ and used impedance boundary conditions on the walls $$ \nabla p \cdot \hat n = -c \eta \frac{\partial p}{\partial t} $$ I used a collocation method spatially, with a Chebyshev basis of order 64 in the short axis and 128 on the long axis. and used RK4 for the time integration.
I modeled the room as 20 feet by 40 feet and started it of with a gaussian pressure pulse in one corner of the room. I listened near the back wall towards the top corner.
I put up an ipython notebook of my code, with the embedded audio and video. I recommend playing with it yourself. On my desktop it takes about minute to do a full simulation of the sound.
I've updated the code to generate sound at multiple locations, and generate their sounds. I can't seem to embed audio on stackexchange, but if you click through to the IPython notebook view, you can listen to all of the generated sounds. But what I can do here is show the spectrograms:
These are laid out in roughly their locations inside of the room. Here the noise was generated in the lower left, but the chirps should be generic for any listening and generation location.
When you pluck a string or hit a drum or sound a not on a flute, the instrument and the air in and around it vibrate and this vibration propagates as sound waves in the air to your hear drum.
When you hear an instrument being played, what you recognise as the note is the base frequency. 'C' corresponds to $261.6$ Hz and is the same for a piano or a guitar. But a 'C' played from a guitar, played from a piano or simply a $261.6$ Hz sounwave played from a computer speaker sound totally different. This is because of the overtones.
Let's look at the case of a string for a concrete example.
If you pluck the 'C' string on the guitar, you will hear the characteristic sound of it makes. This is because the string is vibrating at $261.6$ Hz, but also at a bunch of higher frequencies. These higher frequencies are called "overtones", and they are determined by the shape and build of the body of the guitar as well as the way you set the string in motion.
This is guitars with different shapes sound different. You can also try plucking or strumming the guitar string in various different places, and you will hear different tones of 'C'.
Overtones of the vibrating instrument are what makes each instrument (or voice, for that matter) sound different. The material, shape, and way the instrument is played all contribute to determine which overtones will be present.
The reason instruments sound more similar while holding a long note is that the overtones dissipate energy faster. Higher frequency vibrations generally lose energy quicker. So once the string is plucked or hit, the overtones start losing energy (thus lowering volume) faster than the base note, and after a while you only hear the base note.
TL;DR: Instrument sound different because of the overtones they produce. These are higher frequencies than the note being played, and are determined by factors such as shape, material and way of playing the instrument and give the characteristic flavour to each instrument.
Best Answer
For a long time, timbre was believed to be based on the relative amplitudes of the harmonics. This is a hypothesis originally put forward by Helmholtz in the 19th century based on experiments using extremely primitive lab equipment. E.g., he used Helmholtz resonators to "hear out" the harmonics of various sounds. In reality, the relative amplitudes of the harmonics is only one of several factors that contribute to timbre, and it's far from sufficient on its own, as you can tell when you listen to a cheap synthesizer. Flicking the switch from "flute" to "violin" doesn't actually make the synthesizer sound like a flute or a violin enough that you could tell what it was intended to be.
A lot of different factors contribute to timbre:
relative amplitudes of the harmonics
the manner in which the harmonics start up during the attack of the note, with some coming up sooner than others (important for trumpet tones)
slight deviations from mathematical perfection in the pattern $f$, $2f$, $3f$, ... of the harmonics (important for piano tones)
the sustain and decay of the note (guitar versus violin)
vibrato
Some sounds, such as gongs and most percussion, aren't periodic waveforms, in which case you don't even get harmonics that are near-integer multiples of a fundamental.
Because there are so many different factors that combine to determine timbre, it's remarkably difficult to synthesize realistic timbres from scratch. Modern digital instruments meant to sound like acoustic instruments often use brute-force recording and playback. For example, digital pianos these days just play back tones recorded digitally from an acoustic piano.