Complete set of solutions to the wave equation

[Thomas Rigby]

TL;DR Summary

How to form the most general solution to the wave equation when using separation of variables?

I am solving the wave equation in z,t with separation of variables. As I understand it, Z(z) = acos(kz) + bsin(kz) is a complete solution for the z part. Likewise T(t) = ccos(ω t) + dsin(ωt) forms a complete solution for the t part. So what exactly is ZT = [acos(kz) + bsin(kz)][ccos(ωt) + dsin(ωt)]?
It does not appear to be the most general solution; I can only get a subset of the possible solutions of the form

Qcos(kz)cos(ωt) + Rcos(kz)sin(ωt) + Ssin(kz)cos(ωt)+Tsin(kz)sin(ωt).

I would think this latter would be the most general solution.

 

[phyzguy]

I believe the most general solution to the 1D wave equation is F(x, t) = f(x-ct) + g(x+ct) , where f and g are any arbitrary (differentiable) functions.

 

[Thomas Rigby]

That is known as d'Alembert's solution. I asked about separation of variables.

 

[hutchphd]

You neglected to mention that the omega on k had to equal the speed.
But any linear combination with various k is also is a solution and that will reproduce d'Alambert's form (via Fourier synthesis) and so both descriptions are complete and consistent..