Given a plucked string which is taken on the interval $[0,\pi]$, and it satisfies the wave equation with $c=1$. The initial position of the string is: $\ f(x) = \frac{xh}{p}$ ($0\leq x\leq p$), and $\ f(x) = \frac{h(\pi-x)}{\pi - p}$ for $p\leq x\leq pi$. Choose an initial velocity $g(x) = 0$.
Question: Using the formula to compute Fourier sine coefficients, I was able to obtain $f(x) = \sum_{m=1}^{\infty} A_m\ sin(mx)$ where $A_m=\frac{2h}{m^2}\frac{sin(mp)}{p(\pi-p)}.$ But I cannot find the positions $p$ for which the $2$nd, $4$th,... harmonics are missing and the $3$rd, $6$th,... harmonics are missing, respectively. Can anyone please help me with this issue?
It looks like you want $\sin(mp)=0$ when $m$ is a multiple of 2 and 3 respectively. One way of achieving this is to set $p=\pi/2$ and $p=\pi/3$.