Mathematically the shape of the string is the superposition of multiple standing waves. In general it has the form:
$$ y(x,t) = \sum_{i=1}^{\infty} \sin \left(i\frac{ \pi x}{\ell} \right)
\left(
A_i \sin \left( i\frac{ \pi c t}{\ell} \right)+ B_i \cos \left( i\frac{ \pi c t}{\ell} \right)
\right) $$
where $\ell$ is the length from end to end, and $c$ is the speed of wave propagation along the string.
Now given an initial pluck of the string, of triangular shape you find the coefficients $A_i$ and $B_i$ using Fourier transform. The result is:
$$ \begin{align} A_i &= 0 \\
B_i &= Y \frac{2 \ell^2 \sin \left( \frac{i \pi x_p}{\ell} \right)}{\pi^2 i^2 x_p (\ell-x_p)} \end{align} $$
where $x_p$ is the position of the pluck point, $Y$ is the pluck amplitude, and $i=1 \ldots \infty$
To arrive at this use the pluck shape $y_0(x) = Y\,{\rm if}(x \leq x_p, \tfrac{x}{x_p}, 1-\tfrac{x-x_p}{\ell-x_p})$ and the known initial conditions
$$ \begin{aligned}
\lim \limits_{t\rightarrow 0} y(x,t) & = y_0(x) = {\rm triangle} \\
\lim \limits_{t\rightarrow 0} \frac{\partial}{\partial t} y(x,t) & = v_0(x) =0 \end{aligned}$$
Now pre-multiply with $\sin\left(i \frac{\pi x}{\ell} \right)$ and integrate over the length of the string
$$ \begin{aligned}
\int \sin\left(i \frac{\pi x}{\ell} \right) y_0(x) {\rm d}x &= \int \sin\left(i \frac{\pi x}{\ell} \right) \lim_{t\rightarrow 0} y(x,t) {\rm d}x = B_i \frac{\ell}{2} \\ \int \sin\left(i \frac{\pi x}{\ell} \right) v_0(x) {\rm d}x &= \int \sin\left(i \frac{\pi x}{\ell} \right) \lim_{t\rightarrow 0} \frac{\partial}{\partial t} y(x,t) {\rm d}x = A_i \frac{i\,\pi\, c}{2}
\end{aligned} $$
or
$$\begin{aligned}
A_i &= \frac{2}{i\,\pi\,c} \int \sin\left(i \frac{\pi x}{\ell} \right) v_0(x) {\rm d}x = 0 \\
B_i & = \frac{2}{\ell} \int \sin\left(i \frac{\pi x}{\ell} \right) y_0(x) {\rm d}x = \frac{2 Y \ell^2}{\pi^2 i^2 x_P (\ell-x_p)} \sin\left(i \frac{\pi x_p}{\ell} \right)\end{aligned} $$
