Bessel function of the second kind
/ BESS-el /
A second-order differential equation always has two independent solutions, and the Bessel equation is no exception. The Bessel function of the first kind J_n is the one that stays finite at the centre; its partner, the Bessel function of the second kind Y_n, is the other independent solution — the one that blows up at the centre but is essential everywhere the centre is excluded.
Y_n(x), also called the Neumann function, is the second solution of Bessel's equation that is linearly independent of J_n. Its defining feature is a singularity at the origin: as x tends to 0, Y_0(x) behaves like (2 over pi) ln x, diverging to minus infinity, and Y_n for n at least 1 blows up like 1 over x^n. For large x it again becomes a decaying, phase-shifted sinusoid, Y_n(x) is approximately square root of (2 over pi x) times sin(x minus n pi/2 minus pi/4) — the sine partner to the cosine-like J_n, so together they span all oscillatory cylindrical solutions just as sine and cosine span the flat case.
Y_n is needed whenever the region does not include the axis x = 0 — the field between two coaxial cylinders, the vibration of an annular (ring-shaped) membrane, an acoustic duct with a central obstacle, or a waveguide with a hollow core. Combining J_n and Y_n into the Hankel functions J_n plus or minus i Y_n gives inward- and outward-travelling cylindrical waves, the natural language for scattering and radiation. The Bessel equation that produces both kinds belongs to the series-solutions topic.
For a membrane shaped like a flat ring (an annulus), the displacement is a combination A J_n(k r) plus B Y_n(k r); the Y_n term, illegal for a full disk because it is infinite at r = 0, is now allowed and needed because r never reaches 0.
Whether the Y_n solution is kept or discarded is decided entirely by whether the domain includes the singular axis r = 0.
Y_n is unbounded at the origin, so it must be discarded for any region containing the axis (like a full disk) — keeping it there would predict an infinite physical quantity at the centre.