Electromagnetic Theory

Cylinder Function

Q: When do you use Hankel functions instead of Bessel functions of the first kind?

The choice depends on the boundary conditions and the physics of the region. Bessel functions of the first kind J_n(x) are finite at the origin, so they are used inside a solid cylindrical region such as the air-filled core of a circular waveguide. Neumann functions Y_n(x) diverge to negative infinity as x approaches zero, so they are excluded from regions that contain the axis but are retained in annular regions such as a coaxial line where the axis is occupied by the inner conductor. Hankel functions, the complex combinations H_n^(1) = J_n + jY_n and H_n^(2) = J_n - jY_n, represent outward and inward traveling cylindrical waves. They are the natural basis for radiation and scattering problems because H_n^(2)(kr) behaves asymptotically like an outgoing cylindrical wave that satisfies the Sommerfeld radiation condition at large radius.

Q: How do Bessel function roots set the cutoff frequency of a circular waveguide?

The transverse field of a circular waveguide is expanded in cylinder functions, and the metallic wall at radius a imposes a boundary condition that selects discrete eigenvalues. For TM modes the axial electric field must vanish at the wall, giving J_n(k_c a) = 0, so k_c a equals the m-th root p_nm of the Bessel function. For TE modes the wall condition is on the derivative, giving J_n'(k_c a) = 0, so k_c a equals the m-th root p'_nm of the derivative. The cutoff frequency is then f_c = c times the root divided by (2 pi a). The dominant mode TE11 uses the smallest derivative root p'_11 = 1.841, so for a 10 mm radius air-filled guide f_c is about 8.79 GHz. The next mode TM01 uses p_01 = 2.405, giving f_c near 11.5 GHz.

Q: What is the asymptotic form of a cylinder function for large argument?

For large argument x, every cylinder function decays algebraically like one over the square root of x, slower than the one over r falloff of a spherical wave. Specifically J_n(x) is approximately sqrt(2/(pi x)) times cos(x - n pi/2 - pi/4), and Y_n(x) is the same amplitude times the sine of the same phase. This square-root amplitude rolloff reflects energy spreading over a cylindrical wavefront whose circumference grows linearly with radius, so the power per unit length stays constant while the field amplitude falls as 1/sqrt(r). The asymptotic forms are what let antenna and scattering codes replace exact Bessel evaluations with fast trigonometric approximations once kr exceeds roughly 10, keeping relative error below about 1 percent.

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Any solution of Bessel's differential equation falls under this umbrella name, which collectively covers the Bessel function of the first kind J_n(x), the Neumann (second kind) function Y_n(x), and the complex Hankel function combinations. These functions describe how a field varies with radius in cylindrical coordinates, which is exactly the geometry of a circular waveguide, a coaxial line, or a current filament radiating cylindrical waves. The zeros of J_n and its derivative set the eigenvalues p_nm = J_n root and p′_nm = J_n′ root that fix every TM and TE cutoff frequency, with the dominant TE₁₁ mode tied to p′₁₁ = 1.841 and the lowest TM mode TM₀₁ tied to p₀₁ = 2.405.

Category: Electromagnetic Theory

Governing ODE: Bessel equation, order n

Lowest TE root p′₁₁: 1.841

Why Cylindrical Geometry Demands Bessel Solutions

When the Helmholtz wave equation is separated in cylindrical coordinates (ρ, φ, z), the angular part yields harmonic functions e^jnφ with integer order n, the axial part yields traveling waves e^−jβz, and the radial part collapses to Bessel's equation. The radial solutions are the cylinder functions. This is the cylindrical analog of using sines and cosines in rectangular problems or Legendre functions in spherical problems. Because Bessel's equation is second order, it always has two linearly independent solutions: J_n(x), which is finite and oscillatory at the origin, and Y_n(x), which diverges logarithmically (n = 0) or as a negative power (n ≥ 1) as the argument approaches zero.

The physics of the region picks which combination survives. Inside a solid air-filled circular guide the field must remain finite on the axis, so only J_n is admissible and Y_n is discarded. In a coaxial structure the inner conductor removes the axis from the field region, so both J_n and Y_n are retained and combined to satisfy the conditions at both conductor radii. For radiation and scattering, the Hankel functions H_n⁽¹⁾ = J_n + jY_n and H_n⁽²⁾ = J_n − jY_n are preferred because they represent inward and outward traveling cylindrical waves, with H_n⁽²⁾ satisfying the radiation condition for an outgoing wave.

For the engineer, the practical payoff is that all the eigenvalue tables, mode charts, and cutoff frequency formulas of cylindrical structures reduce to the tabulated roots of these functions. Modern field solvers still evaluate cylinder functions either directly through recurrence relations or through the asymptotic large-argument forms once the radial argument kr grows beyond about 10.

Governing Equation and Key Roots

Bessel's differential equation (order n):
x²y″ + xy′ + (x² − n²)y = 0

General cylinder-function solution:
y(x) = A·J_n(x) + B·Y_n(x) ≡ C·H_n⁽¹⁾(x) + D·H_n⁽²⁾(x)

Hankel (traveling-wave) forms:
H_n⁽¹⁾(x) = J_n(x) + jY_n(x) H_n⁽²⁾(x) = J_n(x) − jY_n(x)

Circular-waveguide cutoff:
TM_nm: J_n(k_ca) = 0 → k_c = p_nm / a
TE_nm: J_n′(k_ca) = 0 → k_c = p′_nm / a f_c = c·k_c / (2π)

Large-argument asymptote:
J_n(x) ≈ √(2 / πx) × cos(x − nπ/2 − π/4)

Example: a = 10 mm air-filled guide, TE₁₁ with p′₁₁ = 1.841 → f_c ≈ 8.79 GHz; TM₀₁ with p₀₁ = 2.405 → f_c ≈ 11.5 GHz.

The Three Cylinder Functions Compared

Function	Symbol	Behavior at x → 0	Behavior at x → ∞	Typical Use
Bessel, first kind	J_n(x)	Finite (J₀=1, J_n≥1=0)	Decaying oscillation ∝ 1/√x	Solid circular guide, on-axis fields
Neumann, second kind	Y_n(x)	Diverges (−∞)	Decaying oscillation ∝ 1/√x	Coaxial / annular regions
Hankel, first kind	H_n⁽¹⁾(x)	Diverges (via Y_n)	Inward traveling wave e^+jx/√x	Incoming cylindrical waves
Hankel, second kind	H_n⁽²⁾(x)	Diverges (via Y_n)	Outgoing wave e^−jx/√x	Radiation, scattering, antennas
Modified Bessel	I_n, K_n	I_n finite, K_n diverges	I_n grows, K_n decays	Evanescent / below-cutoff fields

Common Questions

Frequently Asked Questions

When do you use Hankel functions instead of Bessel functions of the first kind?

J_n(x) is finite at the origin, so it is used inside a solid cylindrical region such as an air-filled circular guide. Y_n(x) diverges as x approaches zero, so it is dropped on the axis but kept in annular regions like a coaxial line. The Hankel forms H_n⁽¹⁾ = J_n + jY_n and H_n⁽²⁾ = J_n − jY_n represent inward and outward traveling cylindrical waves and are the natural basis for radiation and scattering, since H_n⁽²⁾(kr) satisfies the Sommerfeld radiation condition at large radius.

How do Bessel function roots set the cutoff frequency of a circular waveguide?

The wall at radius a selects discrete eigenvalues. TM modes need J_n(k_ca) = 0, so k_ca equals the root p_nm; TE modes need J_n′(k_ca) = 0, so k_ca equals p′_nm. Then f_c = c × root / (2πa). The dominant TE₁₁ uses p′₁₁ = 1.841, giving about 8.79 GHz for a 10 mm radius guide; TM₀₁ uses p₀₁ = 2.405, giving about 11.5 GHz.

What is the asymptotic form of a cylinder function for large argument?

For large x every cylinder function decays algebraically as 1/√x, a slower falloff than the 1/r of a spherical wave. J_n(x) ≈ √(2/πx) × cos(x − nπ/2 − π/4), and Y_n(x) uses the sine of the same phase. The 1/√x rolloff reflects energy spreading over a cylindrical wavefront whose circumference grows with radius, so power per unit length stays constant while amplitude falls as 1/√r. Solvers switch to these trigonometric forms once kr exceeds about 10, holding relative error under roughly 1 percent.

Related Terms

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