- Cylindrical Waves, Equations of the Cylindrical Field, Wave Functions

Lecture

Это окончание невероятной информации про цилиндрические волны.

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and the eccentricity of the confocal ellipses
is e = 1/ch u. The transition to the case of circular coordinates is obtained in the limit as
c0→0 and u→∞. Then c0 ch u→ c0 sh u→ r, and v is, evidently, the angle
between the radius r and the x axis. For this reason we shall call f1(u)
the radial function, and f2(v)—the angular function. When u = 0
the eccentricity equals unity, and the ellipse degenerates into a straight-line segment of length 2c0,
joining the foci on the x axis.
The Mathieu equations (9) and (10) have been studied by many authors. Let us consider
first the angular function f2(v). Solutions of equation (10) exist, of course,
for any value of the separation constant b. But the electromagnetic field
is a single-valued function of the point, and therefore, if the properties of the medium are
homogeneous with respect to the variable v, then f2(v) must necessarily be
a periodic function of the angle v. But equation (10) admits periodic
solutions only for certain, characteristic values of the parameter b.
These characteristic values form a countable sequence
b1, b2, …, bm, • • . Their determination is rather complex, so that we shall confine ourselves
to a reference to tables *) and shall proceed directly to determining the functions
satisfying (9) and (10) when b coincides with the characteristic value
b^m.
Equation (10) admits, for the eigenvalues b, both even and
odd periodic solutions. Let us denote the countable sequence of
characteristic values yielding even solutions by bm(e);
the corresponding characteristic functions can be represented as series in
cosines:
Sem(c0λ, cos v)=Σ' Dn cos nv (m = 1, 2, 3...), (11)
n
where the summation with a prime is taken over even n if m is even,
and over odd n if m is odd. A recurrence formula relating the
coefficients Dn(c0λ) can be found by substituting (11) into (10). Thus
all the coefficients of the series are expressed in terms of one initial coefficient,
*) A very clear exposition of the theory of Mathieu functions is given by Whittaker and Watson,
loc. cit., ch. XIX; further details with detailed references to the literature
were published by Stratton in the monograph «Lamé, Mathieu and related functions in physics
and engineering», Kharkov — Kiev, GNTI of Ukraine, 1935. Tables of Mathieu functions and
characteristic values were published by Goldstein, Trans. Cambridge Phil. Soc. 23,
303—336, 1927. The functions Sem and Som, defined in the text, differ from the functions
cem and sem of these authors only by a proportionality coefficient. Whereas
Goldstein chooses his coefficients so that the normalizing factor is
equal to √π, we have found it more convenient to normalize the functions so that the even
function and the derivative of the odd function become unity at the pole v = 0.
See Stratton, Proc. Nat. Acad. Sci. U. S. 21, 51—56, 316—321, 1935 and Morse,
ibid., pp. 56—62. Detailed tables of the expansion coefficients Dn,
the characteristic values bm and the normalizing factors have been computed by Morse and will soon
appear in print.

which remains arbitrary. It is convenient to choose this initial
coefficient so that the function itself has unit value at v = 0, which
corresponds to η= cos v =1. For this we impose on Dn the condition
The odd periodic solutions of equation (10) correspond to the second
sequence of characteristic values, which we denote b^(o).
These functions can be represented as series in sines
Som(c0λ, cos v) = Σ' Fn sin nv, (13)
n
on whose coefficients (Fn(c0λ)) we impose the condition Σ n Fn == 1. Conse-
quently the unit value at v==0 will be possessed by the derivative of Som(c0λ, cos v).
[d/dv Som(c0λ, cos v)]v=0 =1. (14)
The characteristic functions Sem and Som form a complete system of
orthogonal functions. Let bi(e) and bj(e)—be two characteristic values,
and Sei and Sej—the corresponding functions. They satisfy the equations
d2Sei/dv2+(bi—c02λ2cos2v)Sei = 0, (15)
d2Sej/dv2+(bj—c02λ2cos2v)Sej=0. (16)
Multiply (15) by Sej, (16) by Sei and subtract one from the other:
d/dv(Sej dSei/dv—Sei dSej/dv)+(bi—bj)SeiSej= 0. (17)
Integrating (17) from 0 to 2π and taking into account the periodicity of the functions,
we obtain that
'J 0, i≠ j.
Sei(c0λ, cosv)Sej(c0λ, cosv)dv=' . (18)
0 Ni , i=j.
The normalizing factors Ni can be computed from the expansions of the functions,
in series.
Similarly one can derive that
'J Soi(c0λ, cosv)Soj(c0λ, cosv)dv=' ^ (19)
and
. . 'J Sei(c0λ, cos v) Soj(c0λ, cosv)dv=0, (20)-
0
this last relation holding both for i = j and for i≠j.
To each characteristic value bi(e) corresponds one and only^
one periodic solution Sei. But for this same number bi(e) there must ex-^
ist another independent solution of equation (10). Since the second
solution is non-periodic, it is inessential for physical problems^
at least so long as the medium is homogeneous' with respect to the angle v.

however, if the properties of the medium undergo discontinuities on the surfaces
u = const., then the boundary conditions may require application of functions
of the second kind.
Let us turn now to the radial functions. One can without much difficulty
«show that equation (9) is satisfied by a series of Bessel functions,
whose coefficients differ only by a factor from the coefficients of the series
(11) and (13). Thus, if the parameter b takes one of the
characteristic values bm(e), then the corresponding radial function will be
Rem(c0λ, ξ) = √(π/2) Σ' im-n Dn Jn(c0λξ), (21)
where ξ = ch u, im—n = exp i(m — n) π/2. Since all solutions of
Bessel's equation satisfy the recurrence relations (24) and (25), p. 317, then
•the even radial functions of the second kind are defined in terms of Bessel
functions of the second kind by the relation
Nem(c0λ, ξ) = √(π/2)Σ' im-nDnNn(c0λξ). (22)
The convergence of such series of functions is not easy to prove, but in the present case
it appears to be satisfactory. A great advantage of these
representations is that they lead us directly to asymptotic
expressions for very large values of c0λξ. According to (18) and (19), p. 317,
Rem(c0λ, ξ)∼ √(2/πc0λξ) cos(c0λξ— mπ/2— π/4), (23)
c0λξ → ∞.
Nem(c0λ, ξ) ∼ √(2/πc0λξ) sin(c0λξ — mπ/2—π/4). (24)
By analogy with the Hankel functions, we form linear combinations
Rem(1)(c0λ, ξ) = Rem+iNem = √(2/πc0λ) Σ' im-n Dn Hn(1)(c0λξ), (25)
Rem(2)(c0λ, ξ) = Rem — iNem = √(2/πc0λ) Σ' i-(m-n) Dn Hn(2)(c0λξ), (26)
n
whose asymptotic representation has the form
2m + 1
Rem(1)(c0λ, ξ)=√(2/πc0λξ) ei(c0λξ-π/4) (27)
√c0λ
—i(c0λξ—2m±1/4π)
c0λξ → ∞.
Rem(2)(c0λ, ξ)∼√(2/πc0λξ)e-i(…) ' . (28)
The function Rem satisfies the same equation as the angular
function Sem; both have no singularities except at the point at infinity.
Consequently, they must be proportional to one another:
Sem(c0λ, ξ) = √2 π Rem(c0λ, ξ), (29)
— Σ in-m..''/2 (for even m),
| 2 ''''
| >, n!'''-•''—. (for odd m).

On the left cos v is replaced by ξ. Consequently, at u = 0, ξ=1 we have
Rem(1)(c0λ, 1) = √2/π · 1/N_m [d/du Rem(c0λ, ch u)]u=0 = 0. .. (30)
' '
'' The corresponding relations Moore obtained also for the odd
radial functions. Let us define ' . •
Rom(c0λ, ξ) = √π/2 Σ' im-n Fn Nn(c0λξ), (31)
n
Nom(c0λ, ξ) = √π/2 Σ' im-n Fn Nn(c0λξ), (32)
n
Rom(1)(c0λ, ξ) = Rom+iNom, Rom(2)(c0λ, ξ) = Rom — iNom. (33)
The asymptotic expressions for the odd functions are identical with the
asymptotic expressions for the corresponding even functions. Finally,.
d/dv Som(c0λ, 1) == √2 π Rom(c0λ, 1), (34)
sin v
4 √n' c
—i i n——in-m (for even m),
'n'
i'''n''.' (for odd m).
at u = 0, ξ == 1 we have
Rom(c0λ, 1) = 0, [d/du Rom(c0λ, ch u)]=—i'. (35)
Now, having these functions at our disposal, we are in a position to
write the elementary wave function for the elliptic cylinder. The even
and odd wave functions, finite everywhere, and in particular along the axis^
joining the foci, are formed from the radial functions of the first kind
''em = Sem(c0λ, cos v) Rem(c0λ, ξ)e—iωt, (36)
''om = Som(c0λ, cos v) Rom(c0λ, ξ)e—iωt, (37)
If it is known that at large distances from the axis of the cylinders the field
propagates radially outward, then the elementary wave functions are formed
from radial functions of the third kind
ψem = Sem(c0λ, cos v) Rem(3)(c0λ, ξ) e , (38)
ψom = Som(c0λ, cos v) Rom(3)(c0λ, ξ)e . . (39)
The components of the electric and magnetic vectors are found by the
rules given in section 6.3.
6.13. Integral representations. According to (19), p. 321, in any
system of cylindrical coordinates
f(u1, u2) = 'J g(β) eiλ(u1 cos β+u2 sin β) dβ. (40)
If we replace the rectangular coordinates by elliptic ones by means of the trans-
formation x= c0 ch u cos v, y=c0 sh u sin v, then (40) can be rewritten
in the form
f(u,v)= 'J g(β)eiρβ dβ, (41)

where, as before, λ=√k2—h2 = k sin α, and where
p(u,v,β) = x cos β+y sin β = c0(ch u cos v cos β+sh u sin v sin β) (42)
The function f(u, v) must satisfy the equation
d2f/du2+d2f/dv2+c02λ2(ch2u-cos2v)f=0, (43)
but since f=f1(u)f2(v), it must, evidently, also satisfy
the equation
d2f/du2+(c02λ2ch2u-b)f=0. (44)
which is obtained by multiplying (9), p. 331, by f2(v). Differentiating (41)
with respect to u, we obtain
consequently, for (41) to satisfy (44), it is necessary that
The quantity ρ is a function of β and u, and it is easy to verify that
d2ρ/du2 = d2ρ/dβ2 (48)
Consequently, (46) is equivalent to the equation
'J g(β) [d2g/dβ2+(b —c02λ2cos2β) gρ] dβ == 0. (49)
Finally, according to (30), p. 322,
Consequently, (41) is a solution of (44), if g(β) satisfies
the equation
d2g/dβ2+(b~c02λ2cos2β)g==0, (51)
and the path of integration is chosen so that
•g'-' g'''| =0. (52)
Note that the equation for the amplitude g(β) is identical with the equation
for the angular function f2(v)
and differs only inessentially from the
equation for f1(u), from which we started. This property is
common to Fourier representations of all solutions of equations belonging
to the group defined by equation (6), p. 330.
The results obtained hold for any values of the separation constant hm2.
But if we restrict ourselves to the even and odd sequences
of characteristic values bm and b^m, then it is convenient to take as g(β) the
periodic solutions of equation (51), denoted by Sem(c0λ, cos β),
Som(c0λ, cosβ).
Since p(u, v, β) is also periodic in β with period 2π, then,^
evidently, (52) will be satisfied if for the path of integration one takes any

«segment of the real axis of length 2π. Then the integral representation
f(u, v), valid for b belonging to the sequence bm(e), has the form
f2(e)(u, v) = 'J Sem(c0λ, cosβ)eic0λchu cosβ dβ. (53)
0
From this one can immediately find the integral representation for f1(u),
•setting v = 0.
f1(u) = 'J Sem(c0λ, cosβ)eic0λchu cosβ dβ. (54)
0
It remains to find the connection of the particular solution defined by (54), with the
previously known functions. Introducing the expansion of Sem in a series of cosines and setting
for brevity c0λchu = ρ, we obtain
2π
f1(u) = Σ'Dn 'J eip cos β cos nβ dβ = 2πΣ' in Dn Jn(ρ). (55)
0 '^
Note further that 'J-n=(—1)nJn, and recall that the summation
is taken over even n when m is even and only over odd n when m is odd.
Therefore (—1)n = (—1)nm, and, by virtue of definition (21), p. 333,
Rem(c0λ, ξ) = 1/√2π 'J Sem(c0λ, cosβ) eic0λξ cosβ dβ, ( 56)
0
where ξ==ch u. Moreover, Sem(c0λ, cos β) is proportional to Rem(c0λ, cos β),
satisfies the integral equation
Rem(c0λ,ξ) = i-m 'J Rem(c0λcosβ)eic0λξcosβdβ. (57)
0
One can obtain another representation of Rem, by expanding the integ-
..rand in (57) in a series. We have
Rem(c0λ, ξ) = √π/2 Σ' i-(m-n) 'J Jn(c0λ cos β) eic0λξcosβ dβ. (58)
« 0
.But ' •
π
Jn(c0λ cos β) == 1/π 'J cos nσ ei c0λ cosβcosσ dσ, (59)
0
' whence
2π 2π
'J Jn(c0λcosβ)eic0λξcosβdβ = i-n 'J cos nσ 'J e ic0λ[ξ+cosσ]cosβdβdσ. (60)
0 0
Recalling once more that (—1)n = (—1)nm, we obtain
_ 2π
Rem(c0λ, ξ) = i-m 'J-n √π/2 'J J0[c0λ(ξ+cosβ)]Sem(c0λ, cosβ)dβ.

Integral representations of the radial functions of the third and fourth
kind can be obtained by choosing other contours of integration. To
simplify, let ch u = ξ, v = 0, cos β = t. Then the various radial
functions are represented by integrals of the form
fn(1)(ξ) = 'J Sen(c0λ, t) eic0λξt (1 — t2)—1/2 dt, (62)
c
where the contour C is such that the vanishing of the so-called
«bilinear concomitant» (52) is ensured, which here has the form
(1—t2)'[V'Sen(',t)—ASen(',t)>—t=«. (63)
When t becomes very large, the asymptotic representation of Sen will be
Sen(c0λ, t) ∼ √2/πc0λt cos(c0λt— nπ/2— π/4) ; (64)
consequently, the vanishing of (63) at infinity is ensured by the
factor exp[ic0λ(t—1)]. Therefore, if the real part of c0λ(t—1)
is greater than zero, then the beginning or end of the contour can be taken at t = i∞,
and for the other limit one can take +1 or—1. As a result we obtain
Rem(3)(c0λ, ξ) = i-m √2/π 'J Sem(c0λ, t) eic0λξt (1 — t2)—1/2 dt, (65)
i∞
Rem(4)(c0λ, ξ) = i-m √2/π 'J Sem(c0λ,t)eic0λξt(1—t2)—1/2dt, (66)
—1
provided that the real part of [c0λi(ξ—1)]>0. Returning to the
complex β-plane, we find that (65) is equivalent to the equality
--i∞
Rem(3)(c0λ, ξ) = i-m √2/π 'J Sem(c0λ, cosβ) eic0λξ cosβ dβ; (67)
0
a corresponding integral holds also for Rom(3)(c0λ, ξ).
With the help of (67) one can derive a series of other integrals of type (61).,So,
instead of (58) one can write
— '—i∞
Rem(3)(c0λ,ξ) = √π/2Σ'i-(m-n) 'J Jn(c0λcosβ)eic0λξcosβ dβ =
--i∞
= √π/2Σ'(—1)ni-m 'J i-ncos nσ dσeic0λ(ξ+cosσ)cosβ. (68)

if the real part of ρ > 0, consequently,
Rem(3)(c0λ, ξ) = i-m (- 1)n √π/2 'J J0[c0λ(ξ + cos σ)] Sem(c0λ, cos σ) dσ, (70)
0
if the real part of [c0λ(ξ—1)]>0. Combination of (61) and (70) gives
2π
Rem(3)(c0λ, ξ) = i-m(—1)n √π/2 'J J0[c0λ(ξ + cos σ)]Sem(c0λ, cosσ) dσ. (71)
0
To find the integral representations of the odd functions, the
starting point is the relation
2π 2π
'J eic0λξ cos β g' dβ sin β dβ = — 'J eic0λξ cos β dβ cos β sin β dβ (72)
0 0
which is obtained by integration by parts. The proof of the relations
, 2π
Rom(c0λ, ξ) = 1/√2 i-m √π/2 'J Som(c0λ, cos β) eic0λξcos β sin β dβ =
= 1/√2 i-m √π/2 'J Rom(c0λ, cos β) eic0λξ cos β sin β dβ =
0
2«
= c0λ Σ' (-1)n i-m √π/2 'J J1[c0λ(ξ+cos σ)] ×
0
×Som(c0λ,cosσ)·pσ dσ, (73)
• 2«
Rom(c0λ, ξ) = c0λi-m(—1)n √π/2/(ξ2-1) 'J J1[c0λ(ξ+ cos σ)] ×
0
×Som(c0λ, cosσ)·_''Jσ dσ (74)
is left to the reader.
6.14. Expansion of plane and circular waves. The periodic functions
Sem(c0λ, cos v) and Som(c0λ, cos v) form a complete orthogonal system of
functions. , Consequently, if f(u, v)—is a function periodic in v
with period 2π, piecewise continuous in the argument v together with its first
derivative df/dv, then it can be expanded in a series
∞ ∞
f(u, v) = Σ f2m(e)(u) Sem(c0λ, cos v) + Σ f2m(o)(u) Som(c0λ, cos v), (75)
whose coefficients are determined by the equalities
f2m(e)(u)=1/πN 'J f(u, v)Sem(c0λ, cos v)dv,
'o
(76)
This expansion can be applied to represent a plane wave
with propagation constant λ, travelling in the direction defined by-

the unit vector n, forming with the fixed frame of reference
the polar angles α and β, as shown in Fig. 66, p. 319. We have
Ψ = eik sin α (x cos β+y sin β) · eikz cos α-iωt (77)
Expressing x and y in terms of u and v and setting λ = k sin α, we obtain
f(u, v, β)=eip, p=c0(chu·cosv·cosβ+shu·sinv·sinβ). (78)
Since expression (78) is completely symmetric with respect to ψ and β,
the expansion (75) can be written in the form
∞
eip = Σ am(u) Sem(c0λ, cosβ) Sem(c0λ, cos v) +
m = 0
∞
+ Σ bm(u) Som(c0λ, cosβ) Som(c0λ, cos v), (79)
m = 0
where the coefficients am(u) and bm(u) depend only on u. By virtue of (76) we have
2π
am(u) Nm Sem(c0λ, cosβ) = 'J Sem(c0λ, cos v) eip dv. (80)
0
Since the coefficients am(u) do not depend on the value of β, we may set
β = 0, Sem(c0λ, 1)= 1, p = c0 ch u cos v. Then, according to (56),
am(u) = ''''' Rem(c0λ, ch u). (81)
Similarly we find
bm(u) Nm Som(c0λ, cosβ) = 'J Som(c0λ, cos v) eip dv. (82)
0
Let us now differentiate this last equation with respect to β, and then set
β = 0. Then, by virtue of (14), the coefficient bm(u) turns out to be equal
to bm(u) = 1/der Som(c0λ, ch u). ( 83)
Thus, the complete expansion for a plane wave of arbitrary
direction has the form
∞
eip = Σ 2/π·1/Nm Rem(c0λ, ch u) Sem(c0λ, cos v) Sem(c0λ, cos β)+
+ 1/Nm Rom(c0λ, ch u) Som(c0λ, cos v) Som(c0λ, cos β)]. (84)
This last result allows us to immediately write the integral
representation for those elementary elliptic wave functions which
remain finite along the axis. For this we need only multiply (84) by
Sem(c0λ, cos β) and integrate over the period. Then from the orthogonality
of the functions it follows that
2π
1—4»
Rem(c0λ, ch u) Sem(c0λ, cos v)= '—' 'J Sem(c0λ, cos β)eip dβ; (85)
0
and similarly for the odd functions
Rom(c0λ, ch u) Som(c0λ, cos v) = — 'J Som(c0λ, cos β) eip dβ.

With the help of (67) one can also show that the elementary functions of
elliptic waves, travelling outward from the axis, are represented by the contour integral
Rem(3)(c0λ, ch u) Sem(c0λ, cos v) == i-m √ 'J Sem(c0λ, cos β) e±ip dβ. (86)
0
The upper sign in the exponential function is taken for —π/2 < v < •π/2,
the negative sign — for π/2<'v'<3π/2. A similar equation can be derived
for the odd functions.
From section 6.8 it is easy to see that the even circular wave function
cos nθ Jn(λr), remaining finite on the axis, can be represented by the integral
2πin cos nθJn(λr) = 'J cos nβeip dβ, (88 )
where p = λxcosβ+λy sin β = λrcos(β — θ). Multiplying by Dn and summing over even
«or odd n, we obtain the equality
Rem(c0λ, ch u) Sem(c0λ, cos v) = 1/Nm Σ' in Dn cos nθJn(λr),
(89)
n
which expresses the elliptic function as an expansion in circular
wave functions. A similar expression can be found for the odd
functions.
The addition theorem for circular and elliptic waves, referred
to two parallel axes, was derived by Morse ^).