Boundary Value Problems in Spherical
Coordinates
Y. K. Goh
2009
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Outline
I Laplacian Operator in spherical coordinates
I Legendre Functions
I Spherical Bessel Functions
I Initial-value problem for heat flow in a sphere
I The three-dimensional wave equation
I Laplace Eq. in a sphere and exterior to a sphere
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplacian Operator in Spherical Coordinates
Laplacian Operator in SphericalCoordinates
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplacian Operator in Spherical Coordinates
Spherical Coordinates
I (r, θ, φ) : x = r cosφ sin θ, y = r sinφ sin θ, z = r cos θ
I ∇2u =1
r2
∂
∂r
(r2∂u
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂u
∂θ
)+
1
r2 sin2 θ
∂2u
∂φ2
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Legendre’s Equations and Legendre’s Functions
Legendre’s Equation arises naturally when solving some PDEin Spherical Coordinate systems. Usually it forms part of theSturm-Liouville problem which requires it to have boundedeigenfunctions over a fixed domain.
Definition (Legendre’s Equation)The Legendre’s Equations is a family of differential equationsdiffer by the parameter λ in the following form
(1− x2)y′′ − 2xy′ + λy = 0, (1)
ord
dx
[(1− x2)
dy
dx
]+ λy = 0. (2)
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Legendre’s Equations and Legendre’s Functions
The Legendre’s equation is a linear 2nd order ODE.
I x = ±1 are two singular points of the ODE.
I A solution near the ordinary point x = 0 is a power series
y =∞∑
n=−∞
anxn, an = 0,∀n < 0.
I The radius of convergence for the power series is thedistance from the centre of the series to the nearestsingular point, i.e. R = 1.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Legendre’s Equations and Legendre’s Functions
I Substitue the power series into the ODE, we will obtainthe recurrecnce relation
an+2 =n(n+ 1)− λ
(n+ 2)(n+ 1)an, n = 0, 1, 2, . . . .
I The recurrence relation gives two series solutions knownas the Legendre’s functions, where one is an oddfunction and the other one is an even function.
I By using convergence tests, we can show that the twoseries are convergence for |x| < 1.
I However, the series are generally not convergent atx = ±1, except if λ = `(`+ 1).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Legendre’s Polynomials
For the case of λ = `(`+ 1) :
I When n = `, an+2 =n(n+ 1)− `(`+ 1)
(n+ 2)(n+ 1)an = 0,
=⇒ a`+2 = a`+4 = a`+6 = · · · = 0.
I Thus, one of the series solution becomes a polynomial.
I After normalization, we obtain the LegendrePolynomial of degree `,
P`(x) =1
2`
N∑n=0
(−1)n(2`− 2n)!
n!(`− n)!(`− 2n)!x`−2n, (3)
where N = `/2 if ` is even or (`− 1)/2 is ` is odd.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Legendre function of the second kind
For the case of λ = `(`+ 1) :I The Legendre’s equation of order n, for ` = 0, 1, 2, . . .
(1− x2)y′′ − 2xy′ + `(`+ 1)y = 0,−1 < x < 1.
I A solution is the Legendre Polynomial of degree `, P`(x).I The other solution is a series solution known as the
Legendre function of the second kind, Q`(x).I Q`(x) converges on the −1 < x < 1 but unbounded in−1 ≤ x ≤ 1.
I Since P`(x) and Q`(x) are linearly independent, thus thegeneral solution to the ODE is
y = c1P`(x) + c2Q`(x).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Properties of Legendre Polynomials
I Symmetry: Pn(x) is even if n is even, and odd if n is odd.I End points: Pn(1) = 1, and P (−1) = (−1)n for all n.I Boundedness :|Pn(x)| ≤ 1 for all n and x in [−1, 1].I Zeros: Pn(x) has n distinct zeros in [−1, 1].
I Orthogonality:
∫ 1
−1
Pm(x)Pn(x) dx =2
2n+ 1δmn.
I Rodrigues’ Formula: Pn(x) =1
2nn!
dn
dxn(x2 − 1)n.
I Generating function:
Φ(x, t) = (1− 2xt+ t2)−1/2 =∞∑n=0
Pn(x)tn.
I Pn(x) =1
n!
dn
dtn[Φ(x, t)]t=0.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Graphs of Legendre Polynomials
Figure: Legendre Polynomials, Pn(x).
I P0(x) = 1
I P1(x) = x
I P2(x) = 12(3x2 − 1)
I P3(x) = 12(5x3 − 3x)
I P4(x) =18(35x4 − 30x2 + 3)
I P5(x) =18(63x5 − 70x3 + 15x)
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Graphs of Legendre functions of the second kind
Figure: Legendre functions, Qn(x).
I Q0(x) = 12
ln(
1+x1−x
)I Q1(x) = x
2ln(
1+x1−x
)− 1
I Q2(x) = 3x2−14
ln(
1+x1−x
)− 3x
2
I Q3(x) =5x3−3x
2ln(
1+x1−x
)− 5x2
2+ 2
3
I Q4(x) =35x4−30x2+3
16ln(
1+x1−x
)− 35x3
8+
55x24
.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Legendre Series Expansions
I Orthogonality of Legendre Polynomial
〈Pm|Pn〉 =
∫ 1
−1
Pm(x)Pn(x) dx =2
2n+ 1δmn.
I The Legendre series expansions for f(x),−1 ≤ x ≤ 1 is
f(x) =∞∑n=0
cnPn(x)
I Where the generalized Fourier coefficient cn is
cn =〈Pn|f〉‖Pn‖2
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Legendre Series Expansions
Examples
I Find the first three terms of the Legendre series expansionfor the function
f(x) =
{0, −1 ≤ x < 0,
1, 0 ≤ x ≤ 1.
I Find the Legendre series expansion for f(x) = x2 + x− 4,defined over the interval −1 ≤ x ≤ 1.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Associated Legendre Functions
DefinitionThe Associated Legendre’s Equation is defined as
(1−x2)y′′−2xy′+
[n(n+ 1)− m2
1− x2
]y = 0, −1 < x < 1,
(4)or
d
dx
[(1− x2)
dy
dx
]+
[n(n+ 1)− m2
1− x2
]y = 0, −1 < x < 1.
(5)
The solution to the associated Legendre’s equation is denotedas Pm
n (x) and Qmn (x) respectively.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Properties of Associated Legendre Functions
I Associated Legendre function of degree n and order m:Pmn (x), n = 0, 1, 2, . . . , |m| ≤ n .
I Legendre polynomial: P 0n(x) = Pn(x).
I Rodrigues’ Formula:
Pmn (x) = (−1)m(1− x2)m/2
dm
dxm[Pn(x)].
I Negative order: Pmn (x) = (−1)m
(n+m)!
(n−m)!P−mn (x).
I Orthogonality:∫ 1
−1
Pmk (x)Pm
l (x) dx =2
2k + 1
(k +m)!
(k −m)!δkl, |m| ≤ k, l.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Associated Legendre Functions
I P 00 (x) = 1
I P 11 (x) = −
√1− x2, P 0
1 (x) = x, P−11 (x) = −1
2P 1
1 (x)
I P 22 (x) = 3(1− x2), P 1
2 (x) = −3x√
1− x2,P 0
2 (x) = 12(3x2 − 1), P−1
2 (x) = −16P 1
2 (x),P−2
2 (x) = 124P 2
2 (x),
I P 33 (x) = −15(1− x2)3/2, P 2
3 (x) = 15x(1− x2),P 1
3 (x) = −32(5x2 − 1)(1− x2)1/2, P 0
3 (x) = 12(5x3 − 3x),
P−13 (x) = − 1
12P 1
3 (x), P−23 (x) = 1
120P 2
3 (x),P−3
3 (x) = − 1720P 3
3 (x).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Graphs of Associated Legendre Functionss
Figure: Associated LegendreFunctions, Pmn (x).
Figure: Associated LegendreFunctions, Qmn (x).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Associated Legendre Series Expansions
I Orthogonality of associated Legendre functions
〈Pmk |Pm
l 〉 =
∫ 1
−1
Pmk (x)Pm
l (x) dx =2
2k + 1
(k +m)!
(k −m)!δkl.
I The associated Legendre series expansions of order m forf(x),−1 ≤ x ≤ 1 is
f(x) =∞∑n=m
cnPmn (x)
I Where the generalized Fourier coefficient cn is
cn =〈Pm
n |f〉‖Pm
n ‖2, n = m,m+ 1,m+ 2, . . . .
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Associated Legendre Series Expansions
Example
I The associated Legendre series expansion of order m = 2for
f(x) =
{0, −1 ≤ x < 0,
1, 0 ≤ x ≤ 1,
is given by f(x) =∞∑n=m
cnPmn (x). The first few
coefficients are
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Spherical Harmonics Functions
DefinitionThe spherical harmonic function is defined as
Y mn (θ, φ) =
√2n+ 1
4π
(n−m)!
(n+m)!Pmn (cos θ)eimφ. (6)
I Orthogonality:
〈Y mn |Y m′
n′ 〉 =
∫ π
θ=0
∫ 2π
φ=0
Y m∗n Y m′
n′ sin θ dθ dφ = δnn′δmm′
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Spherical Harmonics Series Expansions
A function f(θ, φ) defined in 0 ≤ θ < π and 0 ≤ φ < 2π canbe expressed as a spherical harmonics expansion
f(θ, φ) =∞∑n=0
n∑m=−n
cmn Ymn (θ, φ),
where the coefficient cmn is given by
cmn =〈Y m
n |f〉‖Y m
n ‖2= 〈Y m
n |f〉,
since Y mn has be normalized to ‖Y m
n ‖2 = 〈Y mn |Y m
n 〉 = 1.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Spherical Harmonics Functions
Figure: Spherical Bessel’s function, jn(x).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Spherical Bessel Functions
I Spherical Bessel’s equation
x2y′′ + 2xy′ + [x2 − n(n+ 1)]y = 0 (7)
I Two linearly independent solutions are the sphericalBessel’s functions of the first and second kinds.
I jn(x) =
√π
2xJn+1/2(x) = (−x)n
(1
x
d
dx
)nsinx
x.
I yn(x) =
√π
2xYn+1/2(x) = −(−x)n
(1
x
d
dx
)ncosx
x.
I Similar to the Bessel’s functions, jn(x) is bounded nearx = 0, but not bounded at x = 0 for yn(x).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Spherical Bessel’s Functions of the first kind
Figure: Spherical Bessel’s function,jn(x).
I j0(x) =sinx
x
I j1(x) =sinx
x2− cosx
xI j2(x) =(
3
x2− 1
)sinx
x− 3 cosx
x2
I j3(x) =
(15
x3− 6
x
)sinx
x−(
15
x2− 1
)cosx
x
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Spherical Bessel’s Functions of the second kind
Figure: Spherical Bessel’s function,yn(x).
I y0(x) = −cosx
x
I y1(x) = −cosx
x2− sinx
xI y2(x) =(
1− 3
x2
)cosx
x− 3 sinx
x2
I y3(x) =(6
x− 15
x3
)cosx
x−(
15
x2− 1
)sinx
x
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. inside a sphere (radial symmetry)
A Dirichlet’s problem inside a sphere with radial symmetry.
I u is radial symmetry, i.e. u(r, θ, φ) = u(r, θ)
I PDE: Laplace Eq. ∇2u(r, θ) = 0, 0 < r < L, 0 ≤ θ < π.
I1r2
∂
∂r
(r2∂u
∂r
)+
1r2 sin θ
∂
∂θ
(sin θ
∂u
∂θ
)= 0.
I Boundary ConditionsI u(L, θ) = f(θ), 0 ≤ θ < π.
I Regularity Conditions: |u| <∞ in 0 < r < L.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. inside a sphere (radial symmetry)
I Separation of variables: u(r, θ) = R(r)Θ(θ)
I ODEs:{r2R′′ + 2rR′ − λR = 0, 0 < r < L; (Euler Eq.)
Θ′′ + cot θΘ′ + λΘ = 0, 0 ≤ θ < π. (Legendre Eq.)
I B.C.: u(L, θ) = f(θ) acts like initial condition.I The ODE for θ can be converted to Legendre by making
a change of variable s = cos θ.I =⇒ (1− s2)Θ′′ − 2sΘ′ + λΘ = 0, −1 < s < 1,I Bounded solution is only possible for Legendre’s
equation when λ = n(n+ 1), n = 0, 1, 2, . . . , i.e.Θ ∝ Pn(s) = Pn(cos θ).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. inside a sphere (radial symmetry)
I Re-write the set of ODEs and corresponding boundedsolutions inside a sphere for n = 0, 1, 2, . . . are{r2R′′ + 2rR′ − n(n+ 1)R = 0, =⇒ R(r) = Cnr
n
Θ′′ + cot θΘ′ + n(n+ 1)Θ = 0, =⇒ Θ(θ) = Pn(cos θ)
I General solution:
u(r, θ) =∞∑n=0
CnrnPn(cos θ), 0 < r < L, 0 <≤ θ < π.
I “Initial” condition gives: CnLn =〈Pn|f〉‖Pn‖2
=
2n+ 1
2
∫ π
0
f(θ)Pn(cos θ) sin θ dθ, n = 0, 1, 2, . . . .
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. inside a sphere (radial symmetry)
Example
I Solve the Dirichlet problem for the Laplace Equationinside a sphere with f(θ) = L cos2 θ.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. exterior to a sphere (radial symmetry)
A Dirichlet’s problem outside a sphere with radial symmetry.
I u is radial symmetry, i.e. u(r, θ, φ) = u(r, θ)I PDE: Laplace Eq. ∇2u(r, θ) = 0, r > L, 0 ≤ θ < π.
I1r2
∂
∂r
(r2∂u
∂r
)+
1r2 sin θ
∂
∂θ
(sin θ
∂u
∂θ
)= 0.
I Boundary ConditionsI u(L, θ) = f(θ), 0 ≤ θ < π.
I Regularity Conditions: |u| <∞ in r > L.
I Separation of variables: u(r, θ) = R(r)Θ(θ) =⇒{r2R′′ + 2rR′ − n(n+ 1)R = 0, =⇒ R(r) = Dnr
−n−1
Θ′′ + cot θΘ′ + n(n+ 1)Θ = 0, =⇒ Θ(θ) = Pn(θ).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. exterior to a sphere (radial symmetry)
I General solution:
u(r, θ) =∞∑n=0
Dnr−n−1Pn(cos θ).
I “Initial” condition gives:
DnL−n−1 =
〈Pn|f〉‖Pn‖2
.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. exterior to a sphere (radial symmetry)
Example
I Find the solution to the Dirichlet’s problem for Laplaceequation exterior to a sphere with the initial conditionu(L, θ) = sin θ.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. interior to a sphere
A Dirichlet’s problem inside a sphere.
I PDE: Laplace Eq. ∇2u(r, θ) = 0, 0 < r < L, 0 ≤ θ < π.
I 1r2
∂∂r
(r2 ∂u∂r
)+ 1
r2 sin θ∂∂θ
(sin θ ∂u∂θ
)+ 1
r2 sin2 θ∂2u∂φ2 = 0.
I Boundary ConditionsI Periodic B.C.: u(r, θ, φ) = u(r, θ, φ+ 2π).I u(L, θ, φ) = f(θ, φ), 0 ≤ θ < π, 0 ≤ φ < 2π.
I Regularity Conditions: |u| <∞ in 0 < r < L.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. interior to a sphere
I Separation of variables: u(r, θ, φ) = R(r)Θ(θ)Φ(φ) =⇒Φ′′ + λΦ = 0, 0 ≤ φ < π
r2R′′ + 2rR′ − µR = 0, 0 < r < L
Θ′′ + cot θΘ′ + (µ− λ csc2 θ)Θ = 0, 0 ≤ θ < π.
I Periodic B.C. :u(r, θ, φ) = u(r, θ, φ+ 2π), 0 ≤ φ < 2π =⇒
I SL-problem Φ′′ + λΦ = 0, Φ(φ) = Φ(φ+ 2π)I Eigenvalues, λ = λm = m2,m = 0, 1, 2, . . . .I Eigenfunctions, Φm(φ) = Am cosmφ+Bm sinmφ; or
more often for the case of Spherical coordinate problems,the eigenfunctions are writen as Φm(φ) = eimφ.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. interior to a sphere
I ODE for Θ :Θ′′ + cot θΘ′ + (µ−m2 csc2 θ)Θ = 0, 0 ≤ θ < π.
I Could be converted to associated Legendre’s equation bychanging the variable s = cos θ.
I (1− s2)Θ′′ − 2sΘ′ +[µ− m2
1−s2
]Θ = 0
I The associated Legendre’s equation has boundedsolution only when µ = n(n+ 1).
I The bounded solution is the associated Legendrefunction of the first kind, Θmn(θ) = Pmn (cos θ).
I Combining eigenfunctions for φ and θ:I Θmn(θ)Φm(φ) ∝ Y m
n (θ, φ).I The spherical harmonics function is defined as
Y mn (θ, φ) =
√2n+14π
(n−m)!(n+m)!P
mn (cos θ)eimφ.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Laplace Eq. interior to a sphere
I ODE for r: r2R′′ + 2rR′ − n(n+ 1)R = 0 is an Eulerequation:
I The bounded solution is R(r) ∝ rn, 0 < r < L.
I General solution:
u(r, θ, φ) =∞∑n=0
n∑m=−n
CnmrnY m
n (θ, φ).
I “Initial” condition: u(L, θ, φ) = f(θ, φ), gives thegeneralized Fourier coefficients
cnmLn =〈Y m
n |f〉‖Y m
n ‖2=
∫ 2π
0
∫ π
0
Y m∗n (θ, φ)f(θ, φ) sin θ dθ dφ
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Initial-value problem for heat flow in a sphere
Consider the Dirichlet’s problem for heat equation in a sphere
I PDE:∂u∂t
= c2(
1r2
∂∂r
(r2 ∂u
∂r
)+ 1
r2 sin θ∂∂θ
(sin θ ∂u
∂θ
)+ 1
r2 sin2 θ∂2u∂φ2
),
0 < r < L, 0 ≤ θ < π, 0 ≤ φ < 2π, t > 0.
I Boundary conditions:I periodic boundary: u(r, θ, φ, t) = u(r, θ, φ+ 2π, t).I u(L, θ, φ, t) = 0.
I Regularity condition: |u| <∞ inside the sphere.
I Initial condition: u(r, θ, φ, 0) = f(r, θ, φ).
I Separation of variables : u = T (t)R(r)Θ(θ)Φ(φ).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Initial-value problem for heat flow in a sphere
I ODEsT ′ + c2λ2T = 0 t > 0
Φ′′ +m2Φ = 0 0 ≤ φ < 2π
Θ′′ + cot θΘ′ + (n(n+ 1)−m2 csc2 θ)Θ = 0, 0 ≤ θ < π
r2R′′ + 2rR′ + (λ2r2 − n(n+ 1))R = 0 0 < r < L.
I B.C. :Φ(0) = Φ(2π), =⇒ Φm(φ) = Ame
imφ,m = 0,±1,±2, . . .
|Θ| <∞ =⇒ Θmn(θ) = Pmn (cos θ), n ≥ |m|
R(L) = 0. =⇒ Rk(r) = jn (λnkr) , k = 1, 2, . . .
here, λnk = αnk/L and αnk is the kth root of nth orderspherical Bessel function, jn(x).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Initial-value problem for heat flow in a sphere
I The remaining ODE for t has the solutionTkn(t) ∝ e−(cλn
k )2t
I General solution:
u(r, θ, φ, t) =∞∑k=1
∞∑n=0
n∑m=−n
Akmne−(cλn
k )2tjn(λnkr)Pmn (cos θ)eimφ
or
u(r, θ, φ, t) =∞∑k=1
∞∑n=0
n∑m=−n
akmne−(cλn
k )2tjn(λnkr)Ymn (θ, φ)
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
Initial-value problem for heat flow in a sphere
I The generalized Fourier coefficient
akmn =
∫ L0
∫ 2π
0
∫ π0f(r, θ, φ)jn(λnkr)Y
mn (θ, φ) r2 sin θ dθ dφ dr∫ L
0
∫ 2π
0
∫ π0|jn(λnkr)Y
mn (θ, φ)|2 r2 sin θ dθ dφ dr
,
k = 1, 2, . . . , n = 0, 1, 2, . . . ,m = 0,±1,±2, . . . ,±n.
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
The three-dimensional wave equation
Consider the Dirichlet’s problem for wave equation in a sphere
I PDE: ∂2u∂t2
=
c2(
1r2
∂∂r
(r2 ∂u
∂r
)+ 1
r2 sin θ∂∂θ
(sin θ ∂u
∂θ
)+ 1
r2 sin2 θ∂2u∂φ2
),
0 < r < L, 0 ≤ θ < π, 0 ≤ φ < 2π, t > 0.I Boundary conditions:
I periodic boundary: u(r, θ, φ, t) = u(r, θ, φ+ 2π, t).I u(L, θ, φ, t) = 0.
I Regularity condition: |u| <∞ inside the sphere.I Initial condition:
I u(r, θ, φ, 0) = f(r, θ, φ).I ∂u
∂t (r, θ, φ, 0) = g(r, θ, φ).I Separation of variables : u = T (t)R(r)Θ(θ)Φ(φ).
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
The three-dimensional wave equation
I ODEsT ′′ + c2λ2T = 0 t > 0
Φ′′ +m2Φ = 0 0 ≤ φ < 2π
Θ′′ + cot θΘ′ + (n(n+ 1)−m2 csc2 θ)Θ = 0, 0 ≤ θ < π
r2R′′ + 2rR′ + (λ2r2 − n(n+ 1))R = 0 0 < r < L.
I B.C. :Φ(0) = Φ(2π), =⇒ Φm(φ) = Ame
imφ,m = 0,±1,±2, . . .
|Θ| <∞ =⇒ Θmn(θ) = Pmn (cos θ), n ≥ |m|
R(L) = 0. =⇒ Rk(r) = jn (λnkr) , k = 1, 2, . . .
I ODE for t has the solution T (t) = A cos cλnkt+B sin cλnkt
Y. K. Goh
Boundary Value Problems in Spherical Coordinates
The three-dimensional wave equation
I General solution: u(r, θ, φ, t) =∞∑k=1
∞∑n=0
n∑m=−n
(akmn cos cλnkt+ bkmn sin cλnkt)jn(λnkr)Ymn (θ, φ).
I Fourier coefficients:
akmn =
∫ L0
∫ 2π
0
∫ π0f(r, θ, φ)jn(λnkr)Y
mn (θ, φ) r2 sin θ dθ dφ dr∫ L
0
∫ 2π
0
∫ π0|jn(λnkr)Y
mn (θ, φ)|2 r2 sin θ dθ dφ dr
,
bkmn =
∫ L0
∫ 2π
0
∫ π0g(r, θ, φ)jn(λnkr)Y
mn (θ, φ) r2 sin θ dθ dφ dr
cλnk∫ L
0
∫ 2π
0
∫ π0|jn(λnkr)Y
mn (θ, φ)|2 r2 sin θ dθ dφ dr
,
k = 1, 2, . . . , n = 0, 1, 2, . . . ,m = 0,±1,±2, . . . ,±n.Y. K. Goh
Boundary Value Problems in Spherical Coordinates