We are going to solve the classical wave equation $$ c^2\partial^2_x y=\partial^2_t y. $$ with boundary conditions $y(-\pi/2,t)=y(\pi/2,t)=0$ and initial condition $y(x,0)=f(x)=1-\frac{2|x|}{\pi}$.
The procedure is pretty standard, but I find the result quite unsatisfying. We have eigensolutions of the form $$ A\cos nx \cos\frac{nt}{c}, n\text{ odd},\\ A\sin nx \cos\frac{nt}{c}, n\text{ even}. $$ When I expand $f(x)$ into its Fourier series, I find that there is a constant term $1/2$, so we have something like $$ f(x)=1/2+\sum (a_k \cos (2kx)+ b_k \sin(2kx)). $$ The term $1/2$ corresponds to $n=0$ above, so at any time $t$, the term $1/2$ will remain there: $$ y(x,t)=1/2+\sum (a_k \cos (2kx)\cos\frac{nk}{c}+ b_k \sin(2kx)\cos\frac{nk}{c}). $$ I find this quite strange: how could we have a $1/2$ hanging out there? Although the string starts at a position above the $x$ axis, it should move to a position below it very soon, so how could it still have an $1/2$ above the y axis?
Did I do something incorrectly?