Roots of two transcendental equations determining the frequency spectra of standing spherical electromagnetic waves
Authors: Pexton Robert L. , Steiger Arno D. 
Published: June 2010

Roots of the transcendental equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction j Subscript l Baseline left-parenthesis lamda right-parenthesis Over y Subscript l Baseline left-parenthesis lamda right-parenthesis EndFraction equals StartStartFraction j Subscript l Baseline left-parenthesis alpha lamda right-parenthesis StartFraction i Subscript l minus 1 Baseline left-parenthesis alpha lamda StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot right-parenthesis Over i Subscript l Baseline left-parenthesis alpha lamda StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot EndFraction minus StartFraction 1 Over StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot EndFraction j Subscript l minus 1 Baseline left-parenthesis alpha lamda right-parenthesis OverOver y Subscript l Baseline left-parenthesis alpha lamda right-parenthesis StartFraction i Subscript l minus 1 Baseline left-parenthesis alpha lamda StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot right-parenthesis Over i Subscript l Baseline left-parenthesis alpha lamda StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot EndFraction minus StartFraction 1 Over StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot EndFraction y Subscript l minus 1 Baseline left-parenthesis alpha lamda right-parenthesis EndEndFraction">

j
l

(
λ
)

y
l

(
λ
)

=

j
l

(
α
λ
)

i

l
−
1

(
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λ

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ε

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)

i
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1

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j

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)

i
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y

l
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1

(
α
λ
)

\frac {{{j_l}(\lambda )}}{{{y_l}(\lambda )}} = \frac {{{j_l}(\alpha \lambda )\frac {{{i_{l - 1}}(\alpha \lambda \sqrt {|\varepsilon |} )}}{{{i_l}(\alpha \lambda \sqrt {|\varepsilon |} }} - \frac {1}{{\sqrt {|\varepsilon |} }}{j_{l - 1}}(\alpha \lambda )}}{{{y_l}(\alpha \lambda )\frac {{{i_{l - 1}}(\alpha \lambda \sqrt {|\varepsilon |} )}}{{{i_l}(\alpha \lambda \sqrt {|\varepsilon |} }} - \frac {1}{{\sqrt {|\varepsilon |} }}{y_{l - 1}}(\alpha \lambda )}}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction eta j Subscript l minus 1 Baseline left-parenthesis eta right-parenthesis minus l j Subscript l Baseline left-parenthesis eta right-parenthesis Over eta y Subscript l minus 1 Baseline left-parenthesis eta right-parenthesis minus l y Subscript l Baseline left-parenthesis eta right-parenthesis EndFraction equals StartStartFraction StartFraction StartAbsoluteValue epsilon EndAbsoluteValue Over 1 plus StartAbsoluteValue epsilon EndAbsoluteValue EndFraction alpha eta j Subscript l minus 1 Baseline left-parenthesis alpha eta right-parenthesis minus l j Subscript l Baseline left-parenthesis alpha eta right-parenthesis plus StartFraction StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot Over 1 plus StartAbsoluteValue epsilon EndAbsoluteValue EndFraction alpha eta j Subscript l Baseline left-parenthesis alpha eta right-parenthesis StartFraction i Subscript l minus 1 Baseline left-parenthesis alpha eta StartRoot StartAbsoluteValue epsilon EndAbsoluteValue right-parenthesis EndRoot Over i Subscript l Baseline left-parenthesis alpha eta StartRoot StartAbsoluteValue epsilon EndAbsoluteValue right-parenthesis EndRoot EndFraction OverOver StartFraction StartAbsoluteValue epsilon EndAbsoluteValue Over 1 plus StartAbsoluteValue epsilon EndAbsoluteValue EndFraction alpha eta y Subscript l minus 1 Baseline left-parenthesis alpha eta right-parenthesis minus l y Subscript l Baseline left-parenthesis alpha eta right-parenthesis plus StartFraction StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot Over 1 plus StartAbsoluteValue epsilon EndAbsoluteValue EndFraction alpha eta y Subscript l Baseline left-parenthesis alpha eta right-parenthesis StartFraction i Subscript l minus 1 Baseline left-parenthesis alpha eta StartRoot StartAbsoluteValue epsilon EndAbsoluteValue right-parenthesis EndRoot Over i Subscript l Baseline left-parenthesis alpha eta StartRoot StartAbsoluteValue epsilon EndAbsoluteValue right-parenthesis EndRoot EndFraction EndEndFraction">

η

j

l
−
1

(
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)
−
l

j
l

(
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)

η

y

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1

(
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)
−
l

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)

=

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+

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j

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−
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+

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y

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y
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i

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i
l

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)

\frac {{\eta {j_{l - 1}}(\eta ) - l{j_l}(\eta )}}{{\eta {y_{l - 1}}(\eta ) - l{y_l}(\eta )}} = \frac {{\frac {{|\varepsilon |}}{{1 + |\varepsilon |}}\alpha \eta {j_{l - 1}}(\alpha \eta ) - l{j_l}(\alpha \eta ) + \frac {{\sqrt {|\varepsilon |} }}{{1 + |\varepsilon |}}\alpha \eta {j_l}(\alpha \eta )\frac {{{i_{l - 1}}(\alpha \eta \sqrt {|\varepsilon |)} }}{{{i_l}(\alpha \eta \sqrt {|\varepsilon |)} }}}}{{\frac {{|\varepsilon |}}{{1 + |\varepsilon |}}\alpha \eta {y_{l - 1}}(\alpha \eta ) - l{y_l}(\alpha \eta ) + \frac {{\sqrt {|\varepsilon |} }}{{1 + |\varepsilon |}}\alpha \eta {y_l}(\alpha \eta )\frac {{{i_{l - 1}}(\alpha \eta \sqrt {|\varepsilon |)} }}{{{i_l}(\alpha \eta \sqrt {|\varepsilon |)} }}}}

\]
for the spherical Bessel functions of the first and second kind,

j
l

(
x
)

{j_l}(x)

and

y
l

(
x
)

{y_l}(x)

, and for the modified spherical Bessel functions of the first kind,

i
l

(
x
)

{i_l}(x)

, have been computed. The ranges for the parameters

|

ε

|

\sqrt {|\varepsilon |}

and

α
\alpha

, the order l and the root index n are:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot StartAbsoluteValue epsilon EndAbsoluteValue EndRoot equals 1.0 comma 10.0 comma 100.0 comma 500.0 semicolon alpha equals 0.1 left-parenthesis 0.1 right-parenthesis 0.7 semicolon l equals 1 left-parenthesis 1 right-parenthesis 15 semicolon n equals 1 left-parenthesis 1 right-parenthesis 30 period">

|

ε

|

=
1.0
,
10.0
,
100.0
,
500.0
;

α
=
0.1
(
0.1
)
0.7
;

l
=
1
(
1
)
15
;

n
=
1
(
1
)
30.

\sqrt {|\varepsilon |} = 1.0,10.0,100.0,500.0;\quad \alpha = 0.1(0.1)0.7;\quad l = 1(1)15;\quad n = 1(1)30.

\]