= Solution
Assume $0<a<b<l$ for the interior pulse; endpoint variants use the same coefficient formula. Orthogonality of the <Fourier sine basis> gives
$$
b_n=\frac2l\int_a^b\sin\frac{n\pi x}{l}\,dx=\frac{2}{n\pi}\left(\cos\frac{n\pi a}{l}-\cos\frac{n\pi b}{l}\right),\qquad
f(x)\sim\sum_{n\ge1}b_n\sin\frac{n\pi x}{l}.
$$
The <Fourier sine series> equals $f$ at its interior continuity points and takes value $1/2$ at the jumps $a,b$; the sine series is zero at both endpoints. Thus it does not retain an arbitrarily assigned value at a jump.
For the string, write $y(x,t)=\sum_{n\ge1}q_n(t)\sin(n\pi x/l)$. The <wave equation on a string> gives
$$
q_n''+\left(\frac{n\pi c}{l}\right)^2q_n=0,\qquad q_n(0)=0,\qquad q_n'(0)=\frac2l\sin\frac{n\pi}{4}.
$$
The last equality is the <Fourier sine series> of the <Dirac delta> at $l/4$. Solving the modal <ordinary differential equations> gives
$$
\boxed{y(x,t)=\frac{2}{\pi c}\sum_{n=1}^{\infty}\frac{\sin(n\pi/4)\sin(n\pi x/l)\sin(n\pi ct/l)}n.}
$$
This <impulsively struck fixed-end string> is understood as a <distributional weak solution>; the initial velocity is a distribution, not a classical continuous function.
To evaluate the requested time, take the odd $2l$-periodic extension $v$ of $\delta(x-l/4)$ and use the <D'Alembert formula> $y=(2c)^{-1}\int_{x-ct}^{x+ct}v(s)\,ds$. At $ct=l/2$, the interval contains the positive impulse at $l/4$. For $x<l/4$, it also contains the negative reflected impulse at $-l/4$, so the contributions cancel. For $l/4<x<3l/4$ only the positive impulse contributes. For $x>3l/4$ neither contributes. Thus the midpoint-valued <Fourier series> gives
$$
\boxed{y\left(x,\frac{l}{2c}\right)=\begin{cases}0,&0\le x<l/4\text{ or }3l/4<x\le l,\\1/(2c),&l/4<x<3l/4,\\1/(4c),&x=l/4\text{ or }x=3l/4.\end{cases}}
$$
The discontinuity values are a convention for the Fourier representative and do not affect the distribution. The sketch shows the reflected negative front cancelling the left portion of the initially generated plateau.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ib/paper-3-struck-string.png]
{title=Fixed-end string at time l divided by 2c, showing the plateau and midpoint values at the two jump fronts}
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