7.4.2 The Helmholtz equation in cylindrical coordinates

Two families of modes can propagate in the pipe:

TM $_{mn}$ modes, with $H_{z} = 0$ and $E_{z} \neq = 0$ ,
TE $_{mn}$ modes, with $E_{z} = 0$ and $H_{z} \neq = 0$ .

In both cases the longitudinal component that is not zero obeys the Helmholtz equation $\nabla^{2} E_{z} + k^{2} E_{z} = 0$ (or the same for $H_{z}$ ), derived from the wave equation for time-harmonic fields in Section 5.6. Write $E_{z} = Ψ (ρ, ϕ) e^{- j τ z}$ and use the Laplacian in cylindrical coordinates (Section 2.3),

\nabla^{2} = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial}{\partial ρ}) + \frac{1}{ρ ^{2}} \frac{\partial ^{2}}{\partial ϕ ^{2}} + \frac{\partial ^{2}}{\partial z ^{2}} .

The $z$ derivative of $e^{- j τ z}$ twice gives the factor $- τ^{2}$ , and with $k^{2} - τ^{2} = h^{2}$ from Eq. (7.4.1) the common factor $e^{- j τ z}$ drops out:

\frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial Ψ}{\partial ρ}) + \frac{1}{ρ ^{2}} \frac{\partial ^{2} Ψ}{\partial ϕ ^{2}} + h^{2} Ψ = 0, (7.4.5)

where $Ψ$ stands for $E_{z}$ (TM) or $H_{z}$ (TE). The first two terms are the transverse Laplacian $\nabla_{T}^{2}$ in polar coordinates, so Eq. (7.4.5) is the equation $\nabla_{T}^{2} Ψ + h^{2} Ψ = 0$ of Section 7.1, written for a round cross-section.

No TEM wave in a hollow pipe

A TEM wave has $E_{z} = H_{z} = 0$ , and the transverse-field formulas then require $h = 0$ . The transverse electric field becomes the gradient of a potential that obeys Laplace's equation in the cross-section, $\nabla_{T}^{2} Φ = 0$ (Section 7.1). Inside a single closed conductor the whole wall is at one potential, and the only solution of Laplace's equation with a constant boundary value is a constant: the field is zero. A TEM wave needs two separate conductors, which is exactly what the coaxial line of Section 7.5 adds.

Separation of variables

We look for solutions in the product form $Ψ = R (ρ) Φ (ϕ)$ , as we did with $X (x) Y (y)$ in the rectangular guide. Substituting into Eq. (7.4.5) and multiplying by $ρ^{2} / (R Φ)$ ,

function of ρ only \frac{ρ}{R} \frac{d}{d ρ} (ρ \frac{d R}{d ρ}) + h^{2} ρ^{2} = function of ϕ only - \frac{1}{Φ} \frac{d ^{2} Φ}{d ϕ ^{2}} . (7.4.6)

The left side does not change when $ϕ$ changes, the right side does not change when $ρ$ changes, and they are equal everywhere: both must be the same constant, which we call $m^{2}$ .

The angular part. $Φ^{''} + m^{2} Φ = 0$ has the solutions

Φ (ϕ) = cos m ϕ or sin m ϕ . (7.4.7)

The angles $ϕ$ and $ϕ + 2 π$ describe the same point of the cross-section, so the field must be periodic with period $2 π$ . This forces $m$ to be an integer, $m = 0, 1, 2, \dots$ Unlike the rectangular guide, where the walls quantise both wavenumbers, here the closure of the circle itself quantises the angular one.

The radial part. With the constant $m^{2}$ the left side of Eq. (7.4.6), divided by $ρ^{2}$ and multiplied by $R$ , gives Bessel's equation of order $m$ :

\frac{d ^{2} R}{d ρ ^{2}} + \frac{1}{ρ} \frac{d R}{d ρ} + (h^{2} - \frac{m ^{2}}{ρ ^{2}}) R = 0. (7.4.8)

With the new variable $x = h ρ$ it becomes $x^{2} R^{''} + x R^{'} + (x^{2} - m^{2}) R = 0$ , the standard form found in every table of special functions. It is a second-order equation with two independent solutions,

R (ρ) = A J_{m} (h ρ) + B Y_{m} (h ρ), (7.4.9)

$J_{m}$ , the Bessel function of the first kind, finite everywhere and in particular at $ρ = 0$ ,
$Y_{m}$ , the Bessel function of the second kind or Neumann function (also written $N_{m}$ ), which tends to $- \infty$ as $ρ \to 0$ .

The axis $ρ = 0$ lies inside the hollow pipe, and the fields there are finite. Therefore $B = 0$ . (In the coaxial line of Section 7.5 the axis lies inside the inner conductor, outside the field region, and $Y_{m}$ stays.)

Longitudinal field of a hollow circular guide

In a hollow circular guide the longitudinal field of every mode has the form

Ψ (ρ, ϕ) = C J_{m} (h ρ) cos m ϕ (or sin m ϕ), m = 0, 1, 2, \dots, (7.4.10)

with $Ψ = E_{z}$ for TM modes and $Ψ = H_{z}$ for TE modes, times $e^{- j τ z}$ . The wall condition will fix the allowed values of $h$ .

In plain words

Around the pipe the field repeats itself $m$ times, like a sine wave wrapped on a circle. Across the pipe it follows a Bessel function, a wave that starts finite at the centre and oscillates with slowly shrinking amplitude. The integers $m$ (around) and $n$ (across, still to come) label the modes, as $m$ and $n$ did in the rectangular guide.

Common mistake · Treating m as a free real number

Separation of variables alone allows any constant $m^{2}$ . It is the single-valuedness of the field, $Ψ (ρ, ϕ + 2 π) = Ψ (ρ, ϕ)$ , that makes $m$ an integer. A non-integer $m$ would give a field that jumps along some radius, which no source-free region can sustain. (In a guide with a wedge-shaped cross-section, which does not close on itself, non-integer orders do occur.)