Consider the wave equation $\frac{\partial^2u}{\partial x^2} = a^2\frac{\partial^2 u}{\partial y^2}$. Show that $u(x,y) = \sin(y-ax)$ is a solution of the wave equation. More generally, show that $u(x,y) = f(x-ay) + g(x+ay)$ satisfies the wave equation.
Hi, so in this problem for the second part, I got $f_{xx}(x-ay)+g_{xx}(x-ay)$ for the left hand side and $a^2( f_{yy}(x-ay)+g_{yy}(x-ay))$ for the right hand side and I don't get why the derivatives with respect to $x$ and $y$ have to be equal or am I confusing and the $x$ and $y$ don't matter when taking the derivative of the whole thing?
Notice that $f$ is a function of one variable, so when you compute $\frac{\partial u}{\partial x}$ you should get $$\frac{\partial u}{\partial x} = f'(x-ay)+g'(x+ay)$$ instead of $$f_{x}(x-ay) + g_{x}(x+ay).$$