Wave equation, finding a constant that solves the equation?

Question

Reading an elementary book, I came across this problem:

Let $$f_{xx}+f_{yy}=f_{tt}$$

We have the solution:

$$f(x,y,t)=\sin(nx)\cos(nt)+\sin(my)\cos(mt)+\sin(nx+my)\cos(kt)$$

Where $m$ and $n$ are natural numbers.

For what value of $k$ does this satisfy the equation?

Answer 1

You can compute $f_{xx}+f_{yy}-f_{tt}$ for your specific choice of $f$. The result is the following combination of trigonometric functions: \begin{equation*} (k^2-n^2-m^2) \sin(nx+my) \cos(kt). \end{equation*} Since you want this to vanish for all $x$, $y$ and $t$, the only possibility is to choose \begin{equation*} k = \pm \sqrt{n^2+m^2}. \end{equation*}