Consider the $1+1$ wave equation $$ u_{tt} -c^2 u_{xx} = 0 $$
The following page from my lecture notes explains d'Alembert's solution to this
As you can see, they have used the substitution $$ \xi = x + ct \hspace{5 mm} \eta = x - ct $$
I understand how they have deduced that $u_x = u_\xi + u_\eta$. However, I'm unsure of how they obtained their result for $u_{xx}$.
I would have worked this out using the chain rule, as follows $$ u_{xx} = \frac{\partial}{\partial \xi} \left( u_\xi + u_\eta \right) \cdot \frac{\partial \xi}{\partial x} + \frac{\partial}{\partial \eta} \left( u_\xi + u_\eta \right) \cdot \frac{\partial \eta}{\partial x} = u_{\xi \xi} + u_{\eta \xi} + u_{\xi \eta} + u_{\eta \eta} $$ This is only equal to their derived result for $u_{xx}$ if we assume that $u_{\eta \xi} = u_{\xi \eta}$. This leads me on to my first question: (why) can we say that $u_{\eta \xi} = u_{\xi \eta}$?
Secnd question: further down in these notes, it says that the boundary condition $u_t(x,t) = \psi (x)$ tells us that $$ f'(x) - g'(x) = \frac{1}{c} \psi(x) $$
How was this result obtained? I cannot see how they have deduced this?
