# Why can we assume the separation constant?

I am starting a project using the wave equation and I don't understand why when doing separation of variables, we can assume the following, where the equation has already been separated.

$$\frac{1}{c^2h} \frac{d^2h}{dt^2}=\frac{1}{\phi}\frac{d^2\phi}{dx^2}=-\lambda$$

This site gives a description 1/4 of the way down but it doesn't click 100% for me. Why can we assume these functions are constants? If anyone can provide a more detailed explanation or simple/intuitive examples I would appreciate it.

Assume that $\frac{1}{c^2h(t)} \frac{d^2 h}{d t^2}(t) = \frac{1}{\varphi(x)}\frac{d^2 \varphi}{d x^2}(x)=-\lambda(x,t)$, this is the most general assumption you can make at first. But you can differentiate $\lambda(x,t)$ with respect to $x$, using the fact that $\frac{1}{c^2h(t)} \frac{d^2 h}{d t^2}(t) = -\lambda(x,t)$ you conclude that $\lambda_x(x,t)=0$, similarly you can say that $\lambda_t(x,t)=0$, therefore, by fundamental theorem of calculus, you conclude that $\lambda$ is a constant.