To find spherical solutions to the wave equation it is natural to use
spherical coordinates.
x
=
r
sin
θ
cos
φ
x
=
r
sin
θ
cos
φ
y
=
r
sin
θ
sin
φ
y
=
r
sin
θ
sin
φ
z
=
r
cos
θ
z
=
r
cos
θ
There is a very nice discussion of Spherical Coordinates at:
http://mathworld.wolfram.com/SphericalCoordinates.html
There is also a nice discussion of Cylindrical coordinates at the same site
http://mathworld.wolfram.com/CylindricalCoordinates.html
Beware the confusion about
θ
θ
and
φ
φ
.
We are calling the polar angle
θ
.
θ
.
All other mathematical disciplines get it wrong and call it
φ
.
φ
.
The Laplacian can be written in spherical coordinates, but where does that
come from?looking at just the
x
x
term
∂
∂
x
=
∂
r
∂
x
∂
∂
r
+
∂
θ
∂
x
∂
∂
θ
+
∂
φ
∂
x
∂
∂
φ
∂
∂
x
=
∂
r
∂
x
∂
∂
r
+
∂
θ
∂
x
∂
∂
θ
+
∂
φ
∂
x
∂
∂
φ
Then you take the second derivative to get
∂
2
∂
x
2
∂
2
∂
x
2
which as you can imagine is a tremendously boring and tedious thing to
do.Since this isn't a vector calculus course lets just accept the
solution.In the case of spherical waves it is not so difficult
since the
θ
θ
and
φ
φ
derivative terms all go to
0
0
.
∇
2
ψ
(
r
)
=
1
r
2
∂
∂
r
(
r
2
∂
ψ
∂
r
)
=
∂
2
ψ
∂
r
2
+
2
r
∂
ψ
∂
r
=
1
r
∂
∂
r
(
ψ
+
r
∂
ψ
∂
r
)
=
1
r
∂
2
(
r
ψ
)
∂
r
2
∇
2
ψ
(
r
)
=
1
r
2
∂
∂
r
(
r
2
∂
ψ
∂
r
)
=
∂
2
ψ
∂
r
2
+
2
r
∂
ψ
∂
r
=
1
r
∂
∂
r
(
ψ
+
r
∂
ψ
∂
r
)
=
1
r
∂
2
(
r
ψ
)
∂
r
2
Thus for spherical waves, we can write the wave equation:
1
r
∂
2
(
r
ψ
)
∂
r
2
=
1
v
2
∂
2
ψ
∂
t
2
1
r
∂
2
(
r
ψ
)
∂
r
2
=
1
v
2
∂
2
ψ
∂
t
2
Now we can multiply both sides by
r
r
and since
r
r
does not depend upon
t
t
write
∂
2
(
r
ψ
)
∂
r
2
=
1
v
2
∂
2
∂
t
2
(
r
ψ
)
∂
2
(
r
ψ
)
∂
r
2
=
1
v
2
∂
2
∂
t
2
(
r
ψ
)
This is just the one dimensional wave equation with a harmonic solution
r
ψ
(
r
,
t
)
=
A
e
i
k
(
r
∓
v
t
)
r
ψ
(
r
,
t
)
=
A
e
i
k
(
r
∓
v
t
)
or
ψ
(
r
,
t
)
=
A
r
e
i
k
(
r
∓
v
t
)
ψ
(
r
,
t
)
=
A
r
e
i
k
(
r
∓
v
t
)
