Charles Yost wrote:
> I have been following the comments for several days now, but the Feb. 14
> comment by c. H. Thompson in reference to what are the fourth and fifth
> derivatives

Charles has written a wonderful and thought provoking article.
Here are some thoughts and analysis related to this problem.

=========================

In the electrostatic limit, as the wavevector k -> 0, it can
be shown that the conventional retarded wave solution of
the scalar potential of the oscillating electric dipole
becomes an instantaneous potential.

The conventional formulations for the retarded potentials
of the oscillating electric dipole are shown below.
In spherical coordinates, where: qa is the dipole moment,
e0 is vacuum permittivity, r is the radial distance, and
theta is the polar angle.

Scalar Potential:
Phi = {qa/4(Pi)(e0)r^2} cos(wt-kr) cos(theta)
-{kqa/4(Pi)(e0)r} sin(wt-kr) cos(theta)

Vector Potential (in Lorentz Gauge):
A = -{kqa/4(Pi)(e0)cr} sin(wt-kr) cos(theta) [r direction]
+{kqa/4(Pi)(e0)cr} sin(wt-kr) sin(theta) [theta direction]

If the scalar potential is to have an instantaneous factor,
it must have a term which looks like cos(wt), rather than
the retarded wave term cos(wt-kr).

Using trigonometric identities, cos(wt) can be expanded as:

cos(wt) = cos(kr)cos(wt-kr) - sin(kr)sin(wt-kr)

Writing the cos(kr) and sin(kr) terms as a power series:

cos(wt) = (1 - (k^2r^2)/2 + (k^4r^4)/4! - ... )cos(wt-kr)
-(kr -(k^3r^3)/3! +(k^5r^5)/5! - ... )sin(wt-kr)

Recognizing that the first term in each of the above power
series is identical to the two terms of the conventional
formulation for the retarded scalar potential --

Conventional Retarded Scalar Potential:
Phi = {cos(wt-kr) - (kr)sin(wt-kr)} {qa/4(Pi)(e0)r^2} cos(theta)

-- taking the power series expansion for cos(wt), and adding
back all terms except for the first two terms of each series,
such that everything cancels except the first terms:

Alternative Expression of the "Retarded" Scalar Potential:
Phi = cos(wt) + {(k^2r^2}/2 - (k^4r^4)/4! + ... }{cos(wt-kr)}
+ {- (k^3^r3)/3! + (k^5r^5)/5! +... }{sin(wt-kr)}

The above expression is a mathematical identity to the
two terms of the conventional retarded scalar potential.
In the limit as k -> 0, the potential becomes:

Phi = {qa/4(Pi)(e0)r^2} cos(wt) cos(theta)

For static fields, as k->0, the potential appears to be
instantaneous, not retarded.

Also interesting are all the terms in the above power
series expansion. The conventional solution for the
scalar potential of the oscillating electric dipole
contains source terms related only to the position of
charge and the velocity of charge -- cos(wt-kr) and
(kr)sin(wt-kr). The trigonometric identity for the
instantaneous potential, cos(wt), contains source terms
related to position, velocity, acceleration, jerk, and
all the other time derivatives of position --

Trigonometric Identity:
cos(wt) = (1 - (k^2r^2)/2 + (k^4r^4)/4! - ... )cos(wt-kr)
-(kr -(k^3r^3)/3! +(k^5r^5)/5! - ... )sin(wt-kr)


Another way to see that there is intantaneity in the scalar
potential of the oscillating electric dipole lies in the
solutions of the scalar wave equation:

del^2(Phi) = -k^2(Phi)

A solution of the wave equation for the scalar potential of
the oscillating electric dipole (first order Lengendre function
of cos(theta) with azimuthal symmetry, m=0), is presented below.
In spherical coordinates, and not showing in the formulation
the constant factor of qa/4(Pi)(e0), the time harmonic factor
exp(-iwt), or the polar angle variation cos(theta), and using i
for complex numbers, J for Bessel functions.

Solution of sclar wave equation:
Phi =
(For even powers of kr)
(1/r^2) {1 +(kr)^2/2 - (kr)^4/(4)(2) + ... + (kr)^n/n(n-2)!}
(And for odd powers of kr)
(1/r^2) {(kr)^3/3 - (kr)^5/(5)3! + ... (kr)^n/n(n-2)!}

Under the del^2 operation, each of the terms in the above two
series, generates the preceding term multiplied by a factor
of -k^2. The above operation terminates/begins at the first
term of each series, because the first term in each series
satisfies the Laplacian, del^2(Phi) = 0. Each of the above two
series satisfies the wave equation, (del^2 + k^2)(Phi) = 0.
Except for being an r-dilate, (a function of kr rather than r),
the above two series are identical to the spherical Bessel
function solutions of the wave equation, J(1) for even powers
of kr, and Y(1) for odd powers.

Individually, each of the above two series (even powers and
odd powers of kr) does not provide a complete solution for
the EM field. Individually, each does not satisfy Maxwell's
divergence equation divE = 0 -- no net flux out of a surface
enclosing an electric dipole. But, when put together they form
one solution. The Green's function: (u)del^2(v) - v(del^2)(u)
vanishes. This is well recognized by mathematicians, and the
first independent solution of the scalar wave equation is
generally written in the form of a spherical Hankel
function:

Phi = {qa/4(Pi)(e0)}cos(theta){cos{J(1)-iY(1)}exp(-iwt).

For which, the real part becomes the even powered series
multiplied by cos(wt), minus the odd powered series
multiplied by sin(wt).

The spherical bessel function solution of the scalar wave
equation, is explicitly instantaneous. It does not contain
any terms related to cos(wt-kr) or sin(wt-kr). But
conventionally, you see the scalar potential for the
oscillating electric dipole written in the retarded form:

Phi = (qa/4(Pi)(e0)r^2}cos(theta){cos(wt-kr)-(kr)sin(wt-kr)}.

The conventional retarded wave is indeed a solution of
the scalar wave equation. By mathematical sleight of hand
the explicitly instantaneous spherical bessel function
solution, can be made to look retarded. To see this clearly,
expand the conventional retarded solution as a power series.
The expansion results in many extra terms which do not
appear in the bessel function solution. But all of these
extra terms are added in, and subtracted out, in equal and
opposite amounts. The conventional retarded wave solution
is mathematically identical to the instantaneous spherical
bessel function solution.

The scalar wave equation is a second order linear differential
equation. It has two solution. The above spherical bessel
function solution is one of the two independent solutions for
the potential of the oscillating electric dipole. All math
and EM texts state that the second independent solution is
the complex conjugate of the Hankel function. Generally the
discussion stops at this point with little or no elaboration.

Complex Conjugate of First Independent Solution:
Phi = {qa/4(Pi)(e0)r^2}cos(theta){cos{J(1)+iY(1)}exp(+iwt).

The real part of the above expression is mathematically
identical to the real part of the first independent solution.
As such, it is clearly NOT the second independent solution
of the scalar wave equation for the oscillating electric dipole.

Another common belief is that the second independent solution
of the scalar wave equation for the oscillating electric
dipole is the advanced wave solution:

Phi = (qa/4(Pi)(e0)r^2}cos(theta){cos(wt+kr)+(kr)sin(wt+kr)}.

The same game has been played with the advanced wave
solution, as with the retarded wave solution. A lot of extra
terms exist in the power series expansion, which are added
and subtracted in equal amounts, leaving the spherical bessel
function solution. The advanced wave solution and the retarded
wave solution have substantially different physical
interpretations, yet they are mathematical identities. Both
are identical to the spherical bessel function solution.
The advanced wave solution is NOT the second independent
solution of the scalar wave equation.

Some texts provide suggestions that something may have
happened with this. Carrier, Crook and Pearson in the
text "Functions of Complex Variable", state the following.
(NOTE: The first order spherical Bessel functions are the
conventional (cylindrical) Bessel functions of order (3/2)
and (-3/2). In the comments below we have nu = 3/2 and -3/2.)

"Having found in this way a second solution for integral
values of nu and desiring to define a Bessel function of
the second kind in a way which would hold whether or not
nu was integral, Hankel suggested that the pair of basic
solutions of Bessel's equation be taken as J(nu) and the
expression:

{complex formulation omitted)

"This definition fails if nu = n+1/2, where n is integral,
so that this case is excluded. For all other nu, this
expression (or its limiting value if nu = n), is a solution
of Bessel's equation and is linearly independent of J(nu).
To avoid the nuisance of having the definition fail for nu
equal to half an odd integer, the standard definition
(suggested by Weber and Schlaffi as a modification of Hankel's
definition) of the Bessel function of the second kind is now
generally written:

{conjugate of first Hankel function substituted)"

It seems to me that "nuisance", is a peculiar choice of word
to find in a math text.

Jackson ("Classical Electrodynamics", 1965), provides the
following comments relating to the second independent solution
of the wave equation (for the oscillating magnetic dipole field).

"The coefficients A(lm) are not completely arbitrary.
The divergence condition divB=0 must be satisfied. Since
the radial functions are linearly independent, the
condition divB=0 must hold for the two sets of terms
in (16.35) separately."
.
.
"The assumption (16.39) that the field is transverse to
the radius vector, together with (16.40), is sufficient
to determine a unique set of vector angular functions of
order l, one for each m value."
.
.
"Any linear combination of these fields, summed over l
and m, satisfies the set of equations (16.33). They have
the characteristic that the magnetic induction is
perpendicular to the radius vector. They therefore do
not represent a general solution to equations (16.33).
They are, in fact, the spherical equivalent of the
transverse magnetic (TM), or electric (TE), cylindrical
fields of chapter 8."
.
.
"If we had started with the set of equations (sic: the
electrical field wave equations), we would have obtained
an alternative set of multipole fields in which E is
transverse to the radius vector."
.
.
"Just as for the cylindrical wave-guide case, the two sets
of multipole fields can be shown to form a complete set
of vector solutions to Maxwell's equations. The terminology
electric and magnetic multipole fields will be used, rather
than TM and TE, since the sources of each type of field will
be seen to be the electric-charge density and the magnetic
moment density, respectively."

In the above, Jackson recognizes the need to find two independent
solutions of the wave equation. Jackson's suggested second
independent solution is not the same second independent solution
suggested by the mathematicians working in spherical bessel
function.

Jackson presumes only transverse wave solutions. The TM and TE
wave solutions are presented as the two independent solutions.
Without doubt, the TM wave is an electromagnetic dipole field
solution, and the TE wave is also solution. But each represents
a single solution for one of two different types of sources --
the oscillating electric dipole, or the oscillating magnetic
dipole. The second independent solution for the oscillating
electric dipole source is clearly NOT the TE wave.

The second independent solution for the scalar potential
of the oscillating electric dipole appears to be missing in
action. I believe the second solution is the longitudinal
wave solution.

It is possible to write a power series which is a second
independent solution of the scalar wave equation for the
oscillating electric dipole. This power series solution
is similar in form to the second independent solution of
the bessel equation in cylindrical coordinates. It is a
mystery to me, why we are able to see the second independent
solution in the cylindrical bessel equation, yet believe
that it is not needed in a spherical coordinate system.

The second independent solution of the wave equation for
the oscillating electrical dipole is messy and semi-divergent.
It contains a power series of positive exponents along with
terms related to ln(kr), and it also contains an infinite
series of terms with negative exponents. The series of terms
with negative exponents resembles 1/f noise, also known as
pink noise. It is peculiar that the words "pink noise" are
on the ECHELON watch list.

We are missing one of the solutions of the wave equation for
the oscillating electric dipole. We are also missing both
of the evanescent wave solutions. In effect playing
with a quarter of a deck.

Real Waves:
del^2(Phi) = -k^2(Phi) One solution missing.

Evanescent Waves:
del^2(Phi) = +k^2(Phi) Both solutions missing.

These are linear differential equations. When we find
all of the four independent solutions for Phi, which
simultaneiously satisfy both of these wave equations.
We should see a potential which satisfies the Laplacian.

del^2(Phi) = 0

A suggested total solution for the potential of the
oscillating electric dipole, might look something like this:

Phi = {qa/4(Pi)(e0)r^2}{cos(wt)-(ik^3r^/3!)sin(wt)}

An instantaneous real electrical field due to position,
which varies as 1/r^3, and an instantaneous, purely reactive
(complex) electric field which is not distance dependent.
The second part of the field is due to a source term
related to jerk -- the time derivative of acceleration.
The suggested reactive electric field has a cubic frequency-
power spectrum and a formulation which is essentially similar
to that suggested for the zero point field.

What do we believe is the source of the zero point field?
A field with a cubic frequency spectrum will require a
source term related to jerk. No such source term currently
exists in classical or quantum electrodynamics. We have
source terms related to position, velocity, and acceleration
-- i.e. terms related to +cos(wt), -sin(wt), -cos(wt). Why
stop here? One can see that a term related to +sin(wt),
a jerk term, may be needed in the group of operations.

Regards,
Robert Stirniman