Exercise 19 and 20(page 622) in differential equations and boundary value problems EDWARDS & PENNEY says the following:
Problems $19$ and $20$ deal with the vibrations of a string under the influence of the downward force $F(x)=-pg$ of gravity. According to Eq. ($1$), its displacement function satisfies the partial differential equation
$$\frac{\partial^{2} y}{\partial t^{2}}=a^{2}\frac{\partial^{2} y}{\partial x^{2}}-g \quad(40)$$
with endpoint conditions $y(0,t)=y(L,t)=0$.
$19$. Suppose first that the string hangs in a stationary position, so that $y=y(x)$ and $y_{tt}=0$. Hence its differential equation of motion takes the simple form $a^{2}y''=g$. Derive the stationary solution $$y(x)=\Phi (x)=\frac{gx}{2a^{2}}(x-L)$$
$20$. Now suppose that the string is released from rest in equilibrium. Consequently the initial conditions are $y(x,0)=0$ and $y_{t}(x,0)=0$. Define
$$v(x,t)=y(x,t)-\Phi (x)$$
where $\Phi (x)$ is the stationary solution of problem $19$. Deduce from Eq. ($40$) that $v(x,t)$ satisfies the boundary value problem
\begin{align} v_{tt} &= a^{2}v_{xx} \\ v(0,t) &= v(L,t) = 0 \\ v(x,0) &= -\Phi (x)\\ v_{t}(x,0) &= 0 \end{align}
Conclude from Eqs. ($22$) and ($23$) that
$$y(x,t) - \Phi (x) = \sum _{n=1}^{\infty} A_{n} \cos \left(\frac{n\pi at}{L} \right) \sin \left(\frac{n\pi x}{L} \right)$$
where the coefficients $A_{n} $ are the Fourier sine coefficients of $f(x)=-\Phi (x)$. Finally, explain why it follows that the string oscillates between the positions $y = 0$ and $y = 2 \Phi (x)$.
I don't know how to show that the string oscillates between $y=0$ and $y=2\Phi (x)$.
I know that
$$y(x,t)=\sum _{n=1}^{\infty} A_{n} \cos \left(\frac{n\pi at}{L} \right) \sin \left(\frac{n\pi x}{L} \right)+\Phi (x)$$
and
$$\Phi (x)= \sum -A_{n} \sin \left(\frac{n\pi x}{L} \right)$$
is Fourier sine series of $\Phi$, then
\begin{align} y(x,t) &= \sum _{n=1}^{\infty } - A_{n} \sin \left(\frac{n\pi x}{L} \right) \left[1 - \cos \left(\frac{n\pi at} {L} \right) \right] \\ &= 2 \sum _{n=1}^{\infty } - A_{n} \sin \left(\frac{n\pi x}{L} \right) \sin^{2} \left(\frac{n\pi at}{2L} \right) \end{align}
I would appreciate if anyone can help