Problem - Consider the 2-dimensional wave equation in polar coordinates u_tt = c²∇²u modeling the displacement vibrations u(r, θ, t) of a circular membrane of radius R.

Assume that the following is fixed.

Use separation of variables to find all eigenfunctions, i.e., all solutions of the form u(r, θ, t) = Q(r)Θ(θ)T(t). You only need to find the functions Q(r), Θ(θ) and T(t). You do not need to put together to a series. You may assume that the constant Θ''(θ)/ Θ(θ) is negative and also that the constant T''(t)/T(t) is also negative. This way you do not need to consider several cases.