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I know that a string when plucked shows harmonics but I do not really understand how. Like, I understand resonating air column and how standing waves are formed there, but I can't seem to get the step by step process of what happens in a string plucked at some point on it. Does the string produce a pulse which propagates and reflects? That doesn't seem to be the case to me. Does the plucked part keep oscillating between crests and troughs? But as far as I know, it will cause the amplitude to increase and the amplitude is constant for a specific harmonic over time (please correct me if I'm wrong). What exactly causes that standing wave pattern while plucking a point $\frac 1x ^{th}$ of the length of the string (where $x$ is even)?

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Here is an easier way to think about this.

A stretched string, once excited into oscillation, will follow a sine curve as it vibrates up and down. But when you pluck a string, you snag it on your fingernail, pull it sideways into not a sine wave shape but instead a triangle shape (two straight lines of string extending to the ends with an offset in the center), and then release it.

If you perform a fourier decomposition of a triangle or sawtooth wave like this, you will discover that the triangle wave contains a family of harmonic overtones which are absent in the case of the fundamental sine wave, and these harmonics are responsible for the particular tonal quality of a plucked string.

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The best way to think about string harmonics is through the "superposition principle."

The allowed harmonics (a better term is "normal mode") for a string of length $L$ fixed at both ends are of the form $$\psi_n(x)=\sin(k_nx);\quad k_n=\frac{n\pi}{L};\quad n\in \mathbb{N}\tag{A}.$$The time evolution of each of these harmonics is found using the wave equation. We have $$\Psi_n(x,t)=\psi_n(x)\times\cos(\omega_nt+\phi_n);\quad\omega_n=\frac{n\pi v}{L}\tag{B},$$ where $v$ is the speed in the wave equation.

The key argument of the superposition principle is this: if we deform the string at $t=0$ and then allow it to oscillate on its own, there exists a set of time-independent constants $c_n$ such that the state of the string at time $t$ is given by $$\Psi(x,t)=\sum_nc_n\Psi_n(x,t)\tag{C}.$$

To understand this equation, we first consider what happens if $c_1$ is nonzero, and $c_n=0$ for all other $n$. This is a familiar pure harmonic with amplitude $c_1$. Thus, the entire equation means that at every time $t$, the state of the entire string is just a the "sum" of the harmonics, each of which is oscillating independently at its own frequency; the amplitude of each harmonic is $c_n$.

We may now apply this model to the case at hand: a plucked string. We first assume that the "pluck" involves deforming the string such that at $t=0$, it is completely stationary and it takes the form $\Psi_0(x)$. This could be triangle-like if we consider the image of a stringed instrument, but it could be anything sufficiently small. Since the wave is initially stationary, we must have $\phi_n=0$ (this can be proved by considering the derivative of $\Psi_n$ with respect to $t$). Our goal is to find $c_n$ at $t=0$, then everything that the string does after that follows from the superposition principle. Considering the projection of each harmonic e have $$c_n=\int_0^L\mathrm{d}x\,\psi_n(x)\Psi_0(x)\tag{D}.$$

To summarize, if we pluck a string by moving it $\Psi_0(x)$ at $t=0$ and then letting go of it, the state of the string at any future time $t$ is given by equation (C), with variables therein defined by equations (A), (B), and (C), with $\phi_n=0$ in (B).

Numerous special cases, including the case where the string is plucked at $x=L/m$ for even $m$, can be examined by evaluating the integral in (C).


This may initially look like it's just a lot of math math, but actually deeply reflects the mechanisms of the wave propagation. The fact that equation (C) holds follows from the linearity of the wave equation for strings, which is, in turn, a result of the linear relation between the restoring force on a particle of the string and the displacement of the particle. Visual effects, like the apparent initial propagation of the pluck's perturbation, are just results of this description of time evolution. We haven't actually proved it here, but any instantaneous configuration $\Psi(x,t_0)$ that the wave could possibly take, i.e. any displacement that respects the boundary conditions, can be expressed as a weighted sum of functions given by equation (A). And since each of these $\psi_n$ must be made to evolve in agreement with the wave equation, the net evolution of the wave must follow the superposition thereof.

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