Questions: Cylindrical Harmonics and Bessel Functions

Question 1

When solving Laplace's equation inside a solid cylinder (including the axis r = 0), why is the Neumann function Y_n(kr) discarded as a basis function?

• A. Y_n oscillates too rapidly near the center, violating the boundary condition at r = 0
• B. Y_n diverges as r → 0, making the field infinite at the cylinder axis, which has no physical justification
• C. Y_n fails to satisfy Bessel's equation at nonzero radii
• D. Y_n is not square-integrable and cannot be normalized

Answer: (B)

Y_n(kr) → −∞ as r → 0, so including it in the solution would imply an infinite field on the cylinder axis — physically unacceptable unless there is an actual singularity (like a line charge) located on the axis. This is directly analogous to dropping the 1/r^{l+1} term in spherical solutions when the domain includes the origin. If the domain is an annular region not including r = 0 (e.g., between two coaxial cylinders), both J_n and Y_n must be kept, because neither boundary contains the axis.

Question 2

A circular waveguide has inner radius a. The TM₀₂ mode requires J₀(ka) = 0, where j_{0,2} ≈ 5.52 is the second zero of J₀. If the radius is doubled to 2a, what happens to the cutoff wavenumber k for this mode?

• A. k doubles — the larger cylinder supports higher wavenumbers
• B. k is halved — since k = j_{0,2}/a and a has doubled, k = j_{0,2}/(2a)
• C. k stays the same — the Bessel zero j_{0,2} is fixed, so k is independent of a
• D. k changes by a factor of √2 — waveguide cutoff scales with the square root of area

Answer: (B)

The boundary condition at the wall requires J_n(ka) = 0, so the allowed values of k are k_{n,m} = j_{n,m}/a, where j_{n,m} is the m-th zero of J_n. If a doubles to 2a, then k = j_{0,2}/(2a), which is half the original value. Physically, a larger cylinder supports the same mode shape but at a lower frequency (longer wavelength). This is exactly analogous to a Cartesian cavity of length L having k = nπ/L — doubling the length halves the wavenumber.

Question 3

The zeros of Bessel functions J_n are not evenly spaced, unlike the evenly-spaced zeros of sin(kx).

• T. True
• F. False

Answer: True

The zeros of sin(kx) are evenly spaced at multiples of π/k. Bessel functions also oscillate and have infinitely many zeros, but their zeros are not evenly spaced — they asymptotically approach equal spacing (≈ π apart for large arguments), but the early zeros are irregularly distributed and must be tabulated. This is one reason cylindrical problems are more complex than Cartesian ones: you cannot write a simple formula for the n-th zero of J_n the way you can for sin.

Question 4

In cylindrical coordinates, the azimuthal part of the separated solution to Laplace's equation (the φ-dependence) is a Bessel function.

• T. True
• F. False

Answer: False

Bessel functions arise from the *radial* separated equation, not the azimuthal one. When Laplace's equation is separated in cylindrical coordinates (R, Φ, Z), the azimuthal Φ equation is simply Φ'' + n²Φ = 0, whose solutions are familiar trigonometric functions sin(nφ) and cos(nφ) (or e^{inφ}), with n an integer required by single-valuedness around the full angle. The radial equation is what produces Bessel's equation and its Bessel function solutions. The z-equation gives exponentials or trigonometric functions depending on boundary conditions.

Question 5

Explain why Bessel functions appear in cylindrical boundary value problems instead of sines and cosines, and what determines which order J_n is needed for a given problem.

Model answer: In Cartesian coordinates, the separated Laplace equation in x is simply f'' + k²f = 0, yielding sines and cosines. In cylindrical coordinates, the radial equation acquires extra terms from the coordinate geometry: r²R'' + rR' + (k²r² − n²)R = 0. This is Bessel's equation, and its solutions J_n(kr) and Y_n(kr) are the appropriate 'sines and cosines' for radial oscillations in a cylindrical geometry. The order n is determined by the azimuthal mode number: if the solution has e^{inφ} azimuthal dependence (n lobes around the cylinder), the corresponding radial function is J_n. Cylindrically symmetric problems (no φ-dependence) use J₀; one-lobe variations use J₁; and so on.

The geometric origin of Bessel functions is the radial Laplacian in cylindrical coordinates, which differs from the flat Cartesian Laplacian by curvature terms. These curvature terms change the eigenvalue equation from constant-coefficient (yielding exponentials/trig) to variable-coefficient (yielding Bessel functions). The analogy to sines/cosines is exact: J_n oscillates, has zeros that determine allowed wavenumbers, and forms a complete orthogonal basis for radial functions on an interval.