See also - SPHERICAL WAVES: Vector and Scalar Wave Equations

Lecture

Это окончание невероятной информации про сферические волны.

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coordinates equation (2) is satisfied by the
function
M = rot(RΨ) = [L, R] = … rot N, (7)
whose components are equal to
The components of the third solution can easily be found from the relation
N=rot M:
M_R— … , M_θ—kR …, M_φ —kR sinθ … (9)
Since Ψ must satisfy equation D), p. 352, the radial
component reduces to the simpler form 1)
N_R = … (10)
In order to obtain expanded expressions for the vector wave
functions M and N, we need only carry out the
differentiation of (1) in accordance with formulas (8) and (9). We separate off the time
1) The first complete investigation of the electromagnetic problem of the sphere was
carried out by Mie, Ann. Physik 25, 377, 1908, and by Debye, Ann. Physik, 30,
57, 1909. Both authors used a pair of potential functions, which lead
directly to our vectors M and N. The connection between these solutions and the
radial Hertz vector was pointed out by Sommerfeld in the book Frank–Mises,
Integral and Differential Equations of Mathematical Physics,

factor, writing M = m e^{-iωt}, N = n e^{-iωt}, and obtain
m_{mn} = … Ψ_n(kR) … P_n^m(cosθ) sin, cos mφ i_2 +
… Ψ_n(kR) … P_n^m(cosθ) cos, sin mφ i_3. (12)
7.12. Integral representations. The wave functions l, m and n
can be represented as integrals of plane vector waves,
similar to the integrals (18)—(20), p. 347. According to relation (60),
p. 361, for functions of the first kind, which are finite at the origin,
we have
l^{(1)} = … ∫∫ e^{ikR cosγ} P_n^m(cosα) k sinα dα dβ. (13)
0 0 0
First of all, by differentiating under the integral sign, we shall find the expressions
∂f/∂R, (1/R)∂f/∂θ, (1/R sinθ)∂f/∂φ,
for the components of the gradient. From
Fig. 70, p. 358, it is seen that if the vector k(α,β) is directed along the line OQ,
and R(θ,φ) along the line OP, then
k i_1 =k cosγ = k [sinα sinθ cos(φ — β)+cosα cosθ],
k i_2 = k [sinα cosθ cos(φ—β) — cosα sinθ],
k i_3 = — k sinα sin(φ — β),
(14)
where the unit coordinate vectors of the spherical coordinate system
are taken, of course, at the observation point P. Then, without further calculation,
it is clear that
l^{(1)}_{mn} = … ∫∫ k(α,β)j_n^{-1}(…)P_n^m(cosα) cos, sin mβ sinα dα dβ. (15)
0 0 0
We obtain the corresponding representations for m^{(1)} and n^{(1)} by noting that
[R, k] = kR{ sinα sin(φ — β)i_2 + [sinα cosθ cos(φ —β) — cosα sinθ] i_3 },
[[k, R], k] = k^2R { sin^2γi_1 — cosγ [sinα cosθ cos(φ — β)—cosα sinθ] i_3+ (16)
+ cosγ sinα sin(φ—β) i_2},
and differentiating according to (8) and (9), in the form:
m^{(1)}_{mn} = … ∫∫ [k,R] j_n^{-1}(…) P_n^m(cosα) cos, sin mβ sinα dα dβ, (17)
0 0
2π π
…
n^{(1)}_{mn}=… ∫∫ [[k, R], k] … ×
0 0
cos
× P_n^m(cosα) sin mβ sinα dα dβ. (18)

To obtain the integral representations of the functions of the third
and fourth kind, one must replace the integration with respect to α from 0 to π
by integration along the corresponding contour in the complex domain, as was
described on pp. 361—362.
With the help of these integral representations one can, without difficulty,
compute the rectangular components of the vectors l, m and n, although the resulting
expressions are rather long and cumbersome. The vector expressions under the integral
are resolved into their rectangular components, after which
we obtain them in the form of expressions containing α, β, θ, φ explicitly. Thus,
for example, k_1 = k sinα cosβ. Then, by means of the recurrence relations for
spherical harmonics, these factors are eliminated, and the integrals reduce
to a form in which they can be computed with the help of (13).
7.13. Orthogonality. It is evident that the scalar product of any
odd vector with any even vector, or with any vector of a different
index m, is orthogonal; it is necessary, therefore, to consider
only those products which do not vanish upon integration with respect to φ
from 0 to 2π. Thus
0 0… 0
……
+{… …+ …P_n P_n'…n_n(kR)n_{n'}(kR)}, (19)
where δ = 0 for m>0 and δ = 1 for m ==0. To transform
(19), we shall make use of the formula
0
[ 0, for n ≠ n',
= { 2 (n+m)!
…n(n+1) for n=n'. (20)
2n+1 (n—m)!
*
Consequently, when integrating (19) with respect to θ for n≠n' we obtain zero,
and for n = n'
2π π
∫∫ l_{mn} l_{mn} sinθdθdφ= …
0 0
…
This expression can be transformed to its final form by using
the recurrence relations (33) and (34), p. 357; then the normalizing
factor we obtain in the form
∫∫ m_{mn}, n_{mn} sinθdθdφ=(1 +δ) (… n(n+1) [j_{n-1}(kR)]^2 +
+(n + 1)[j_{n+1}(kR)]^2). (22)

The same formulas lead directly to the integrals:
2π π
∫∫ n_{mn}, n_{mn} sinθdθdφ=(1+δ)… { n(n+1)[j_n(kR)]^2 (23)
0 0 0 0
2π π
∫∫ n_{mn} n_{mn} sinθdθdφ=
0 0 0 0
+n(n+1)[j_{n+1}(kR)]^2), (24)
while all other products, in which the indices n differ,
give zero.
Considering products of vectors of different types, we obtain,
first of all,
2π
0
whence
π 2π
∫∫ m_{mn} n_{mn'} sinθdθdφ = 0 (26)
0 0…
regardless of the values of n and n'. Similarly *
π 2π
∫∫ m_{mn} n_{mn'} sinθdθdφ = 0. (27)
0 0 0 0 …
As in the cylindrical coordinate system, the orthogonality turns out
incomplete because of the product l_{mn}n_{mn'}, whose integral over the whole sphere
at n =
π 2π
∫∫ l_{mn}
0 0
n' is not
n_{mn}
0
equal
sinθdθdφ =
0
to
zero;
in
this
case
= … (2n+1) … (n-m)! … n(n+ 1)k{[j_{n-1}(kR)]^2 — [j_{n+1}(kR)]^2}. (28)
In order to obtain complete orthogonality, we may, as in
section 7.2, treat k' as a variable parameter and integrate with respect to k' and R;
however, in most cases such an operation proves unnecessary.
7.14. Expansion of a plane vector wave. In considering
the diffraction of a plane wave with given polarization on a spherical
obstacle, we must first expand the incident plane vector wave
in the spherical wave functions l_{mn}, m_{mn}, n_{mn}
0 0 0
Let us consider the vector function
f(R) = a e^{ikz} = a e^{ikR cosθ}, (29)
where a is an amplitude vector, oriented arbitrarily with respect
to the rectangular frame of reference. Let us resolve the vector a along the three unit

vectors directed respectively along the x, y and z axes. Then
a_θ = sinθ cosφ i_1 + cosθ cosφ i_2—sinφ i_3; 1
a_φ = sinθsinφi_1+cosθsinφi_2+cosφi_3; > (30),
a_R = cosθ i_1—sinθ i_2, )
where i_1, i_2, i_3 are, as before, the unit vectors shown for the spherical
coordinate system in Fig. 8, p. 56. The divergence of the vector functions
a_θ exp(ikz) and a_φ exp(ikz) is zero, and, consequently, they can be
expanded only in the characteristic functions m and n. At R==0 the field is
finite, and we shall therefore need functions of the first kind. It is evident,
further, that the φ-dependence of the form (30) restricts us to the value m=1;
if we now consider the even and odd properties of (11) and (12),
we establish that the expansion must have the form
00
a_θe^{ikz} = Σ (c_n m_{1n}^{(1)} + d_n n_{1n}^{(1)}). (31)
n = 0
To determine the coefficients of the expansion we apply the orthogonality
relations of the preceding section
π 2π
∫∫ a_θm_{1n}^{(1)} e^{ikR cosθ} sinθ dθ dφ=2πi^n n(n +1) [j_n (kR)]^2, (32)
0 0
whence, according to (23),
c_n= …/n(n+1). (33)
Similarly
∫∫ a_θn_{1n}^{(1)} e^{ikR cosθ} sinθdθdφ== …
0 0
=—2πi^{n+1} … {(n+1)[j_{n-1}(kR)]^2 + n[j_{n+1}(kR)]^2}, (34)
which gives, by virtue of (24),
Consequently,
∞
a_θe^{ikz} = Σ i^n …{m_{1n}^{(1)}—i n_{1n}^{(1)}}. (36)
n = 0 …
In the same way, for a plane wave polarized in the direction of the y axis,
we obtain
∞
Since the divergence of the longitudinal wave function a_R exp(ikz) is not equal to
zero, its expansion must also contain the functions l. In fact
it turns out that only these functions are needed, and one can readily