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"Bessel Functions for Spherical Cavity of Radius x=kR"
j[n_, x_] := Sqrt[Pi / 2 / x]BesselJ[n + 1 / 2, x]
Plot[Evaluate[Table[j[k, x], {k, 0, 2}]], {x, 0.001, 8}]
2 4 6 8
-0.2
0.2
0.4
0.6
0.8
1.0
FullSimplify [PowerExpand[Table[j[k, x], {k, 0, 2}]]] //
ColumnForm
Sin[x]
x
-x Cos[x]+Sin[x]
x2
-3 x Cos[x]+-3+x2 Sin[x]
x3
DSin[x]
x, x
Cos[x]
x
-Sin[x]
x2
D - x Cos[x] + Sin[x]x2
, x
Sin[x]
x
-2 (-x Cos[x] + Sin[x])
x3
D-3 x Cos[x] + - 3 + x2Sin[x]
x3, x
-3 Cos[x] + -3 + x2 Cos[x] - x Sin[x]
x3
+33 x Cos[x] + -3 + x2 Sin[x]
x4
"The first root simplyfies to:"
"Tan[x]=x"
"Which wolfram alpha gives a list of values from looking at the
plot I chose:"
"x=4.49"
Printed by Wolfram Mathematica Student Edition
7/24/2019 Frequency the Lowest Modes for Spherical Cavity
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"Second Root"
FindRootSin[x]
x-
2(- x Cos[x] + Sin[x])
x3, {x, 2}
Second Root
{x
2.08158}
"Third Root"
FindRoot
-3 Cos[x] + - 3 + x2Cos[x] - x Sin[x]
x3+
33 x Cos[x] + - 3 + x2Sin[x]x4
, {x, 3.5}
Third Root
{x 3.34209}
"Therefore the three lowest modes are:"
"Having:" w = kc; x = kR
w1 = 4.49 c / R
w2 = 2.08 c / R
w3 = 3.34 c / R
2 Frequency the Lowest modes for Spherical Cavity.nb
Printed by Wolfram Mathematica Student Edition
