Why can we assume the separation constant?

Question

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.

Answer 1

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.

Answer 2

Two functions that depend on different sets of variables and at the same time are equal to each other must be constant. If they weren't changing the values of the variables of one of the functions while keeping constant the values of the variables of the other would contradict the equality of the functions.