# Non-Dimensionalizing Separation of Variables PDE

I have a problem where I'm considering a cylinder with a closed end at the bottom (x = 0) and open end at the top (x = L). The problem is looking at pressure disturbances throughout the air column. The governing PDE is: $$\frac{∂^2p}{∂t^2}− C^2∇^2p = 0$$

where p is pressure (p(x,r,t)) and the laplace operator is defined as $$∇^2(.) = \frac{∂^2}{∂x^2}(.) + \frac{1}{r}\frac{∂}{∂r}{r\frac{∂}{∂r} (.)}$$

The boundary conditions are: $$\frac{∂p}{∂x}(x = 0, r, t) = 0$$; $$p(x = L, r, t) = 0$$ $$\frac{∂p}{∂r} (x, r = a, t) = 0$$ $$p(x, r → 0, t)\, is\, finite$$

My strategy is Separation of Variables. This type of problem is very well-documented for p(x,t), but adding the radial coordinate confuses me a bit. I'm trying to non-dimensionalize the problem, but I'm having issue choosing scaling variables that are significant.

The ones that I chose now are: $$l = \frac{x}{l}$$ $$\alpha= \frac{r}{a}$$ $$s = \frac{Ct}{a}$$

I'm not sure if these are even correct for this type of problem, but even if, I'm not sure how to go about substituting into the PDE to make the PDE or boundary conditions non-dimensional. Any insight into the non-dimensional process in general and as it pertains to this problem would be greatly appreciated!