Electromagnetic Theory

Cylindrical Wave

Q: How does cylindrical wave amplitude decay compare with spherical and plane waves?

A cylindrical wave radiates power across an expanding cylindrical surface whose area grows linearly with radius (circumference 2 pi rho times unit length), so power density falls as 1/rho and field amplitude falls as 1/sqrt(rho). A spherical wave spreads over a surface area of 4 pi r squared, so its power density falls as 1/r squared and amplitude as 1/r. A plane wave has no spreading loss because its constant-phase surfaces are flat and infinite. In decibel terms, doubling the distance reduces a cylindrical wave by 3 dB, a spherical wave by 6 dB, and a plane wave by 0 dB. This 3 dB per octave behavior is why long line sources such as slotted waveguide arrays and leaky-wave antennas exhibit cylindrical spreading in their transverse plane before transitioning to spherical spreading in the far field.

Q: Why are Hankel functions used to represent cylindrical waves instead of Bessel functions?

The radial part of the cylindrical wave equation is Bessel's equation, whose general solution combines Bessel functions of the first kind J_n and second kind Y_n. The Hankel functions are defined as complex combinations H_n^(1) = J_n + jY_n and H_n^(2) = J_n - jY_n. With the e^(j omega t) time convention, H_n^(2)(k rho) has the asymptotic form e^(-jk rho)/sqrt(rho), representing an outward-traveling wave that satisfies the Sommerfeld radiation condition, while H_n^(1) represents an inward-traveling wave. Bessel functions J_n alone produce standing waves and are used for fields that must remain finite on the axis, such as inside a circular cavity or dielectric rod. So a purely radiating field uses H_n^(2), a purely incoming field uses H_n^(1), and a confined resonant field uses J_n.

Q: At what distance does a cylindrical wave approximation break down for a finite line source?

A real radiator of finite length L behaves like a cylindrical wave only in the region where the observation distance rho is large compared with the wavelength but small compared with the Fresnel distance set by the source length. The transition to spherical spreading occurs near rho approximately equal to 2L squared divided by lambda, the same far-field criterion used for apertures. For a 1 meter slotted-array antenna at 10 GHz (lambda = 30 mm), this transition is near 67 meters; closer than that the field in the array plane spreads cylindrically at 3 dB per distance doubling, and beyond it the pattern collapses to spherical 6 dB per doubling behavior. The cylindrical model also requires rho to exceed roughly one wavelength so the 1/sqrt(rho) asymptotic form of the Hankel function is accurate to better than a few percent.

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Radiating outward from an idealized infinite line source, this field has constant-phase surfaces that form a family of coaxial cylinders, distinguishing it from the flat fronts of a plane wave and the expanding shells of a spherical wave. Because the radiated power crosses a cylindrical surface whose area grows only linearly with radius ρ, the power density decays as 1/ρ and the field amplitude as 1/√ρ, giving a 3 dB drop per distance doubling. The radial dependence solves Bessel's equation, so the exact field is written with Hankel functions: H_n⁽²⁾(kρ) for outgoing radiation and J_n(kρ) for the standing fields inside a circular resonator. Cylindrical waves describe long slotted-waveguide arrays, leaky-wave antennas, and any 2D electromagnetic problem with translational symmetry along one axis.

Category: Electromagnetic Theory

Amplitude decay: 1/√ρ (3 dB per 2× range)

Radial solution: Hankel / Bessel functions

Line-Source Radiation and Coaxial Wavefronts

A cylindrical wave is the natural radiation pattern produced by a source that is long in one dimension and concentrated in the other two, the idealized infinite line current. Separating the homogeneous wave equation in cylindrical coordinates (ρ, φ, z) yields three ordinary differential equations: a harmonic equation in φ that forces integer azimuthal order n, a propagation equation in z, and Bessel's equation in the radial coordinate ρ. The radial solutions are Bessel functions J_n and Y_n, or equivalently their Hankel combinations. Choosing H_n⁽²⁾(kρ) selects a purely outward-traveling field that obeys the Sommerfeld radiation condition, which is the defining feature of a true cylindrical wave as opposed to a standing-wave resonance.

The order n sets the azimuthal field shape. An n = 0 mode is omnidirectional in φ and corresponds to a simple line current, the 2D analog of an isotropic radiator and the basis of the two-dimensional Green's function. Higher orders carry an e^jnφ dependence that produces dipole-like (n = 1), quadrupole (n = 2), and progressively more structured angular patterns. Any 2D field can be expanded as a weighted sum of these cylindrical modes, the same idea formalized in the cylindrical wave expansion used to post-process cylindrical near-field antenna measurements.

The hallmark of cylindrical spreading is its 1/√ρ amplitude law. Energy launched by the line source flows through a cylinder of circumference 2πρ per unit length, so conservation of power forces |E| ∝ 1/√ρ. This is gentler than the 1/r decay of a spherical wave because the wave spreads in only two dimensions rather than three. In practice a finite radiator behaves cylindrically only in an intermediate zone, beyond about one wavelength but inside the far-field distance 2L²/λ set by its length L, after which the pattern reverts to spherical spreading.

Governing Equations of the Cylindrical Wave

Separated cylindrical field (outgoing):
E_z(ρ,φ,z) = E₀ H_n⁽²⁾(k_ρρ) e^jnφ e^−jk_zz, with k_ρ² + k_z² = k²

Radial (Bessel) equation:
ρ² R″ + ρ R′ + (k_ρ²ρ² − n²) R = 0

Far-field asymptote (k_ρρ >> 1):
H_n⁽²⁾(k_ρρ) ≈ √(2 / (πk_ρρ)) × e^{−j(k_ρρ − nπ/2 − π/4)}
so amplitude ∝ 1/√ρ (3 dB per range doubling)

Where k = 2π/λ is the free-space wavenumber, k_ρ = √(k² − k_z²) the radial wavenumber, ρ the radial distance, n the integer azimuthal order, k_z the axial propagation constant, and H_n⁽²⁾ the Hankel function of the second kind. A pure 2D cylindrical wave has k_z = 0 so k_ρ = k, and the zeroth-order line source uses H₀⁽²⁾(kρ).

Wave Geometry Comparison

Wave type	Wavefront shape	Source	Power density	Amplitude decay	Loss per 2× range
Cylindrical	Coaxial cylinders	Infinite line current	∝ 1/ρ	∝ 1/√ρ	3 dB
Spherical	Concentric spheres	Point source	∝ 1/r²	∝ 1/r	6 dB
Plane	Flat planes	Infinite aperture / far field	Constant	Constant	0 dB
Conical / beam	Collimated bundle	Focused aperture	Near-constant in beam	Slow (diffraction)	< 1 dB

Common Questions

Frequently Asked Questions

How does cylindrical wave amplitude decay compare with spherical and plane waves?

A cylindrical wave spreads its power across a cylinder whose area grows linearly with radius, so power density falls as 1/ρ and amplitude as 1/√ρ, a 3 dB drop per distance doubling. A spherical wave spreads over 4πr², giving 1/r amplitude and 6 dB per doubling, while a plane wave has flat fronts and zero spreading loss. Long line sources such as slotted-waveguide and leaky-wave antennas show cylindrical 3 dB-per-octave spreading in their transverse plane before reverting to spherical behavior in the far field.

Why are Hankel functions used to represent cylindrical waves instead of Bessel functions?

The radial equation is Bessel's equation, with general solution made of J_n and Y_n. The Hankel combinations H_n⁽²⁾ = J_n − jY_n and H_n⁽¹⁾ = J_n + jY_n separate traveling waves: with e^jωt convention H_n⁽²⁾ is outgoing and satisfies the radiation condition, H_n⁽¹⁾ is incoming. The Bessel function J_n alone stays finite on the axis and represents the standing fields inside a circular cavity or dielectric rod.

At what distance does a cylindrical wave approximation break down for a finite line source?

A radiator of length L behaves cylindrically only where ρ exceeds about one wavelength yet stays inside the far-field distance 2L²/λ. For a 1 m slotted array at 10 GHz (λ = 30 mm) that transition is near 67 m; closer than that the in-plane field spreads cylindrically at 3 dB per doubling, and beyond it the pattern collapses to spherical 6 dB-per-doubling behavior. The 1/√ρ Hankel asymptote also needs kρ >> 1 to be accurate to within a few percent.

Related Terms

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