A guitar is a plucked instrument and it is played by plucking a string at an off-centre point fixed at two ends. In general, Fourier analysis tells that all harmonics (the resonant frequencies of the string) will be excited and the string will vibrate in a superposition of different harmonics. It is true that for $n^{th}$ harmonic the amplitude goes like $1/n$ suggesting that the fundamental contributes the most. In this situation, I am confused. How does a guitar string produce a pure tone/pure frequency sound instead of a noise?
[Physics] If all harmonics are generated by plucking, how does a guitar string produce a pure frequency sound
acousticsharmonicsstringvibrationswaves
Related Solutions
When you release the plucked string, its shape is momentarily triangular: tied down at the ends and pointed at the location of your finger. But the solutions to the wave equation are not triangle functions, but sinusoidal functions, whose displacements from rest obey $$y_n(x) \propto \sin \frac{2\pi x}{\lambda_0 / n},$$ where $\lambda_0$ is twice the length of the string. These waves, whose frequencies are integer multiples of the fundamental frequency, are the harmonics.
There is a theorem that you can add up all of these well-behaved $y_n(x)$ and generate any shape $y(x)$ for the real string that you want. The subject is called Fourier analysis. And that's just what happens when you release your guitar string. From the string's perspective you've just excited a whole bunch of different modes with different $n$, and they all begin to oscillate at their own frequencies.
It's worth pointing out that you have some control over which harmonics you excite by choosing where you pluck the string. Here's how the harmonics up to $n=16$ (four octaves above the fundamental) contribute to the shape of a guitar string plucked near the middle, near the sound hole, and near the nut:
The "exact" triangle shape is in blue; the fundamental excitation is in green; the fundamental plus first harmonic in red, then cyan, magneta, yellow, etc. as more harmonics are included. Plucking a guitar string near the nut (bottom figure) excites lots and lots of the higher harmonics. This is a thing you can hear on a guitar: strumming close to the nut produces a harsh, pinched sort of an "eeee" sound. By contrast, if you pluck the guitar string very near the center of the string, you put very little energy into the 1st, 3rd, 5th, harmonics, which have a node at the middle of the string. This gives a sort of rounder, "oooo" sound to the strings. Give it a try!
This effect is known as inharmonicity, and it is important for precision piano tuning.
Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force.
The solutions are of the form $\sin(kx - \omega t)$, where $\omega = kv$. Applying fixed boundary conditions, the allowed values of the wavenumber $k$ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. This predicts evenly spaced harmonics.
However, piano strings are made of thick wire. If you bend a thick wire, there's an extra restoring force in addition to the wire's tension, because the inside of the bend is compressed while the outside is stretched. One can show that this modifies the wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} - A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^2}.$$ Upon taking a Fourier transform, we have the nonlinear dispersion relation $$\omega = kv \sqrt{1 + (A/v^2)k^2}$$ which "stretches" evenly spaced values of $k$ into nonuniformly spaced values of $\omega$. Higher harmonics are further apart. We can write this equation in terms of the harmonic frequencies $f_n$ as $$f_n \propto n \sqrt{1+Bn^2}$$ which should yield a good fit to your data. Note that the frequencies have no dependence on the amplitude, as you noted, and this is because our modified wave equation is still linear in $y$.
This effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. This results in stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect. That is, a piano whose fundamental frequencies really were tuned to simple ratios would sound out of tune!
Best Answer
Usually a guitar does not produce a pure tone/frequency. If so, its sound would be very close to a diapason. The difference between noise and a musical tone is not that a tone is made by a unique frequency, but there is a continuum between a pure tone (one frequency) and noise (all frequencies, not only multiple of a fundamental, without any regular pattern among their weights), where many non-pure tones are still recognized as dominated by a fundamental frequency. The additional frequencies add what we call the tone color or timbre of the sound.
In general, the exact weight of each harmonics can be somewhat varid according to how and where the chord is plucked. You might find interesting this study on the subject.