Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Figure 7.4.2 shows $J_{0}$ , $J_{1}$ and $J_{1}^{'}$ . All of them oscillate, with an amplitude that decreases as $1/ x$ . $J_{0}$ starts at $1$ , all the others at $0$ . The zeros are what matter for the waveguide, because the wall conditions will put the wall at a zero of $J_{m}$ (TM) or of $J_{m}^{'}$ (TE). We write

J_{m} (x_{mn}) = 0, J_{m}^{'} (x_{mn}^{'}) = 0, n = 1, 2, 3, \dots (7.4.11)

for the $n$ -th zero of $J_{m}$ and of its derivative. Table 7.4.1 lists the first ones. Other books write $p_{mn}$ and $p_{mn}^{'}$ , or $χ_{mn}$ .

The Bessel functions J_0 (orange), J_1 (blue) and the derivative J_1' (green, dashed). Filled dots: the zeros x_{0n} and x_{1n} of J_0 and J_1, which fix the cutoffs of the TM_{0n} and TM_{1n} modes. Open circles: the zeros x'_{1n} of J_1', the maxima and minima of J_1, which fix the TE_{1n} modes. The first of them, x'_{11}=1.841, belongs to the dominant TE_{11} mode. The zeros of J_1 are also the zeros of J_0'=-J_1. — **Figure 7.4.2.** The Bessel functions $J_{0}$ (orange), $J_{1}$ (blue) and the derivative $J_{1}^{'}$ (green, dashed). Filled dots: the zeros $x_{0 n}$ and $x_{1 n}$ of $J_{0}$ and $J_{1}$ , which fix the cutoffs of the TM $_{0 n}$ and TM $_{1 n}$ modes. Open circles: the zeros $x_{1 n}^{'}$ of $J_{1}^{'}$ , the maxima and minima of $J_{1}$ , which fix the TE $_{1 n}$ modes. The first of them, $x_{11}^{'} = 1.841$ , belongs to the dominant TE $_{11}$ mode. The zeros of $J_{1}$ are also the zeros of $J_{0}^{'} = - J_{1}$ .

Table 7.4.1. The first zeros of the Bessel functions $J_{m}$ (first table) and of their derivatives $J_{m}^{'}$ (second table). Since $J_{0}^{'} = - J_{1}$ , the zeros $x_{0 n}^{'}$ coincide with $x_{1 n}$ (the root $x = 0$ of $J_{0}^{'}$ is not counted).

$x_{mn}$	$n = 1$	$n = 2$	$n = 3$
$m = 0$	$2.405$	$5.520$	$8.654$
$m = 1$	$3.832$	$7.016$	$10.173$
$m = 2$	$5.136$	$8.417$	$11.620$
$m = 3$	$6.380$	$9.761$	$13.015$

$x_{mn}^{'}$	$n = 1$	$n = 2$	$n = 3$
$m = 0$	$3.832$	$7.016$	$10.173$
$m = 1$	$1.841$	$5.331$	$8.536$
$m = 2$	$3.054$	$6.706$	$9.970$
$m = 3$	$4.201$	$8.015$	$11.346$

For large $x$ the asymptotic form of the box above, $J_{m} (x) \approx 2/ (π x) cos (x - π /4 - mπ /2)$ , shows that the zeros become nearly $π$ apart: the Bessel functions behave more and more like the sines of the rectangular guide, with an amplitude that decays because the circumference grows with $ρ$ and the same power is spread over a longer ring.

Quick check

Use the asymptotic form to estimate the third zero of $J_{0}$ and compare with the table.

$cos (x - π /4) = 0$ for $x - π /4 = π /2 + (n - 1) π$ , that is $x = 3 π /4 + (n - 1) π$ . For $n = 3$ : $x = 2.75 π = 8.639$ , against the exact $8.654$ , an error of $0.2 %$ . Even for $n = 1$ the estimate $3 π /4 = 2.356$ is only about $2 %$ below $2.405$ .

7.4.3 Bessel functions and their zeros