Variable separation method for solving wave equation

Question

In variable separable method we assume the solution to be the product of such functions each of which is function of only one variable. What is the basis for that assumption? What allows us to assume that the solution is in such a product form?

Answer 1

Fourier developed separation of variables. He looked for solutions that had the form $$ X(x_1,x_2,\cdots,x_n)= X_1(x_1)X_2(x_2)\cdots X_n(x_n) $$ where these are the variables of the equation (some variables could be time, and others would be spatial.) Then he would divide the resulting equation by $X$, and try to isolate on the left side of the equation all terms that depended on only one variable. He would then conclude that because the left side did not depend on any variables on the right and vice-versa, there would be a constant $\lambda$ such that $$ \mbox{LEFT HAND SIDE} = \lambda = \mbox{RIGHT HAND SIDE} $$ That's how he ended up with two equations; the one on the left would be an ODE in a single variable $x_k$, and the one on the right would be a PDE that would no longer depend on $x_k$. So the left side was an ODE in one variable. He would continue this process until the PDE was reduced to a system of ODES, which was possible for only a limited class of equations, but that class included many equations that were very important to Physics. After that, Fourier used "orthogonality" relations to be able to fully sole the ODEs using what we now call Fourier series and Fourier integral representations of functions.

There was no basis for the assumption. Fourier tried something, and he was able to make it work for some very important equations. It took a lot original ideas to make it work, and his Fourier series ideas were so controversial at the time that his original manuscript on The Analytical Theory of Heat where these techniques were developed was banned from publication for many years, until he would gain such prominence that he would force its publication in basically in its original form. Here's a free translation of Fourier's original work at Google; it's a legitimate free copy. https://www.google.com/books/edition/The_Analytical_Theory_of_Heat/No8IAAAAMAAJ?hl=en&gbpv=1&dq=Fourier,+analytical+theory+of+heat+conduction&printsec=frontcover