When we pluck a string, it vibrates in all possible modes of vibrations. The lowest frequency possible is the fundamental frequency and it is the most significant part of sound.
But why do the amplitude of higher harmonics decrease? Which formula is responsible?
Also, how is the energy of wave distributed among different modes?
A Google search didn't give any explained answer.
Best Answer
Why not calculate it?
Consider a string of length $L$, with its ends fixed at $x=\pm\frac{L}{2}$. Let's assume for convenience that at time $t=0$ the string is "plucked" at $x = 0$, so that the string displacement relative to its equilibrium position is given by $$f(x)=A\left|1-\frac{2x}{L}\right|.$$
The standing wave solutions to the wave equation obeying the boundary conditions are $$\psi_n(x)=\cos\left(\frac{(n-\frac{1}{2})2\pi x}{L}\right) $$ with $n\ge1$, $n=1$ corresponding to the fundamental, $n=2$ to the third harmonic, $n=3$ to the fifth harmonic and so on. Note that I haven't included the odd solutions (even harmonics) here, because these modes won't be excited since $f(x)$ is even.
It is a straightforward exercise to show that $\psi_n$ are orthogonal: $$\int\limits_{-L/2}^{L/2}\psi_m(x)\psi_n(x)dx=\frac{L}{2}\delta_{mn}$$ where $\delta_{mn}$ is the Kronecker delta. If $$f(x)=\sum\limits_{m=1}^\infty a_m\psi_m(x),$$ multiplying by $\psi_n$, integrating and using the orthogonality relation yields $$a_n = \frac{2}{L}\int\limits_{-L/2}^{L/2}f(x)\psi_n(x)dx=\frac{4A}{L}\int\limits_{0}^{L/2}\left(1-\frac{2x}{L}\right)\cos\left(\frac{(n-\frac{1}{2})2\pi x}{L}\right)dx.$$ Evaluating the integral gives $$a_n=\frac{2A}{\pi^2\left(n-\frac{1}{2}\right)^2}.\tag{1}$$ So the amplitude of the harmonics decreases roughly as $1/n^2$.
You find that if you pluck the string closer to the ends, the amplitude of the harmonics goes down slower, i.e. there are more "overtones". Specifically, if the string is plucked a distance $\ell$ from one of the ends, the amplitudes are $$ b_n = \frac{2AL^2}{\pi^2\ell(L-\ell)n^2}\sin\left(\frac{n\pi\ell}{L}\right)\tag{2}$$ where the sine factor accounts for the slower decay of $b_n$ when $\ell$ is small. $(2)$ is more general than $(1)$ as it is also valid when the string is not plucked in the middle, and is also consistent with how a guitar string is normally picked.
Note: the meaning of $n$ in $b_n$ is different from before: here, $n=1$ is the fundamental, $n=2$ is the second harmonic, $n=3$ is the third harmonic and so on. The difference is because when the string is plucked in the middle, the even harmonics are not excited.
As for the energy distribution, the energy in the $n$'th harmonic is $$ E_n = \frac{1}{4}M\omega_n^2b_n^2 = \frac{1}{4}M\omega_1^2n^2b_n^2$$ where $M$ is the total mass of the string and $\omega_n=n\omega_1$ is the angular frequency of the $n$'th harmonic.