Coordinate Systems and Separation of Variables
Revisiting the wave equation…
1 ∂ 2
ψ
∇ ψ + 2
2 = 0
c ∂t
2
where previously in Cartesian coordinates, the Laplacian was given by
∂ 2
∂2
∂2
∇ = 2 + 2 +
2
∂z
∂y
∂x
2
We are now faced with a spherical polar coordinate system, with the
motivation that we might employ the separation of variables technique to solve
the wave equation where spherical symmetries are involved.
Spherical Polar Coordinates
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
(r , φ ,θ )
θ
φ
r = x 2
+ y 2 + z 2
r
y
[
θ = tan −1 x 2
+ y 2 z
φ = tan −1
[ y x ]
]
2
1
1
∂
1
∂
∂
∂
∂
⎞
⎛
⎞
⎛
∇2 = 2 ⎜ r 2 ⎟ + 2
⎟+ 2 2
⎜ sin θ
∂θ ⎠ r sin θ ∂φ 2
r r ⎝ ∂r ⎠ r sin θ ∂θ ⎝
Separation of Variables
Objective is to restate problems in alternative orthogonal coordinate systems
such that for the particular boundary conditions in force, the solutions can be
assumed to separate such that…
ψ (r,θ , φ , t) = R(r )Θ(θ )Φ (φ )T (
t )
Azimuth dependence:
Elevation dependence:
Radial dependence:
Time dependence:
d 2
Φ
2
+
m
Φ = 0
2
dφ
dΘ ⎞ ⎡
1 d ⎛
m2 ⎤
⎟ + ⎢n(
n + 1) − 2 ⎥ Θ = 0
⎜ sin θ
dθ ⎠ ⎣
sin θ dθ ⎝
sin θ ⎦
n(
n + 1)
1 d ⎛ 2 dR ⎞ 2
r
k
R
R=0
+
−
⎜
⎟
r 2 dr ⎝ dr ⎠
r2
1 d 2
T
2
+
k
T =0
2
2
c dt
Elevation Dependence: Legendre Functions
Legendre polynomials: Pn0 (x) = Pn (x)
Represent fields uniform in azimuth coordinate φ
Legendre polynomials with integer order m ≠ 0 represent other kinds of fields with harmonic variations in azimuth. The Associated Legendre polynomials P (x) = (−1) (1− x )
m
n
m
2 m2
dm
P (x)
m n
dx
Legendre functions available in MATLAB via call to legendre(N,X).
Selected Legendre Functions
Pn0 (x) = Pn (x)
n=1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
1
2
3
n=1, m=1
1
-0.2
0.5
-0.4
-0.6
0
1
2
3
-1
0
1
n=3
1
0.5
0.5
0
-0.5
-0.8
n=2
1
n=2, m=1
0
P12
1
P11
1
Pn
Pn
n=0
Pnm (x) for m = 1
2
n=3, m=1
3
0
1
2
-0.5
-1
-1
-2
3
0
1
1
1
0.5
0.5
3
-0.5
1
0
1
2
3
3
1
2
n=6, m=1
3
0
1
2
3
2
0
0
-1
2
n=5, m=1
2
-0.5
0
0
-2
P15
0
-1
2
0
-1
n=5
Pn
Pn
n=4
P14
0
0
P16
-1
0
3
1
P13
Pn
Pn
-0.5
2
n=4, m=1
2
1
0
1
-2
-2
0
1
2
3
0
0
1
2
3
Spherical Harmonics
Lump together azmuthal and elevation dependence to arrive at
spherical harmonics…
Ynm (θ , φ ) ≡
(2n + 1) (n − m)! m
Pn (cos θ ) exp imφ
4π (n + m)!
Radial Dependence - Spherical Bessel Functions
Solution to a differential equation that is closely related to the ordinary Bessel
equation...
d2 2 d
n(n + 1)
[ 2+
+ k2 −
] Rn (r) = 0
2
dr
r dr
r
Substituting Rn (r) =
1
u (r) gives
12 n
r
(n + 1 2) ]u (r) = 0
d 2
1 d
[ 2+
+ k2 −
n
dr
r dr
r2
2
Standing wave solutions
Traveling wave solutions
1
⎛π ⎞ 2
jn ( x) ≡ ⎜ ⎟ J n +1 2 ( x)
⎝ 2x ⎠
1
⎛π ⎞
yn ( x) ≡ ⎜ ⎟ Yn +1 2 ( x)
⎝ 2x ⎠
2
OR
⎛π ⎞
hn(1) (x) ≡ jn (x) + iyn (x) = ⎜ ⎟
⎝ 2x ⎠
1
⎛π ⎞
hn(2) (x) ≡ jn (x) − iyn (x) = ⎜ ⎟
⎝ 2x ⎠
1
2
2
[J
n +1 2
( x) + iYn +1 2 ( x)
[J
n +1 2
( x) − iYn +1 2 ( x)
]
]
Selected Spherical Bessel Functions
Spherical Bessel functions, jn(x) for n=0-3
Spherical Bessel functions, yn(x) for n=0-3
1.2
2
n=0
n=1
n=2
n=3
1
n=0
n=1
n=2
n=3
1
0.8
0
0.6
y n(x)
jn(x)
-1
0.4
-2
0.2
-3
0
-4
-0.2
-0.4
0
2
4
6
8
x
10
12
14
16
-5
0
2
4
6
8
x
10
12
14
16
Asymptotic forms for Bessel functions
In odd numbers of dimensions, we are able to express Bessel functions
exactly via trigonometric expansions.
Correspondence between planar and cylindrical expansions
P(k x , k y , z0 ) ↔ Pn (r , k z )
e
ik z ( z − z0 )
H n(1) (k r r )
↔ (1)
H n (k r a)
Important Connections
•
•
Infinite domains Æ Continuous transforms
Domain periodicity Æ Fourier series
•
•
Equivalence of coordinate representations given the right integrations
Basis functions result from solutions to differential equations
d 2 R 1 dR ⎛ 2 n 2 ⎞
+
+
⎜⎜
k r −
2 ⎟⎟
R
=
0
2
dr
r dr ⎝
r
⎠
•
Solutions to the Bessel differential equation
•
Are given by
•
These are Bessel functions of the first and second kind (the latter also referred to as
Neumann functions). As the domain is periodic (at least in this dimension), n must be an
integer.
J n(k r r ) and Yn (k r r )
Interior and Exterior problems
These functions are analogous to the plane wave exponentials, and in fact, have
the asymptotic forms
J n (x) ~
2
cos(x − nπ / 2 − π / 4)
πx
Yn (x) ~
2
sin( x − nπ / 2 − π / 4)
πx
Solution to interior problem:
p (r , φ , z, ω ) =
∞
∑e
n=−∞
inφ
∞
1
ik z z
C
(k
,
ω
)
e
J n (k r r ) dk z
n
z
∫
2π −∞
Need one measurement surface for each unknown coefficient function.
Review
•
•
Plane wave expansions etc
Plane wave solutions useful in Cartesian geometries.
Motivation
• To treat propagation and scattering problems involving spherical geometries
and symmetries.
Real-Parts of Selected Spherical Harmonics